**Singular Stochastic Control in Option Hedging with Transaction Costs**

Tze Leung Lai and Tiong Wee Lim

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/46154

## **1. Introduction**

102 Stochastic Modeling and Control

**Acknowledgement** 

Technical Report ITL – 98 – 6.

Vol. 34, No. 2, pp. 101 – 107.

Vol.3, No. 5, pp. 407-411.

Vol.14, Bund. F, pp. 1-12.

**5. References** 

– 43

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I like to acknowledge my Supervisor, Professor J.O. Afolayan, for his guidance, teaching, patience, encouragement, suggestions and continued support, particularly for arousing my

Ayyub, B.M. and Patev, C.R. (1998)."Reliability and Stability Assessment of Concrete Gravity Structures". (RCSLIDE): Theoretical Manual, US Army Corps of Engineers

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Afolayan, J. O. (2003) "Improved Design Format for Doubly Symmetric Thin-Walled Structural Steel Beam-Columns", Botswana Journal of Technology, Vol. 12, No.1, pp. 36

Afolayan, J. O. and Opeyemi, D.A. (2008) "Reliability Analysis of Static Pile Capacity of Concrete in Cohesive and Cohesionless Soils", Research Journal of Applied Sciences,

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Coduto, D.P. (2001). Foundation Design: Principles and Practices. 2nd ed., Prentice Hall Inc.,

Farid Uddin, A.K.M. (2000) "Risk and Reliability Based Structural and Foundation Design of a Water Reservoir (capacity: 10 million gallon) on the top of Bukit Permatha Hill in Malaysia", 8th ASCE Specialty Conference on Probabilistic Mechanics and Structural

Melchers, R.E. (2002). Structural reliability analysis and prediction, John Wiley & Sons,

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interest in Risk Analysis (Reliability Analysis of Structures).

ISBN: 0-471-98771-9, Chichester, West Sussex, UK

An option written on an underlying asset (e.g., stock) confers on its holder the right to receive a certain payoff before or on a certain (expiration) date *T*. The payoff *f*(·) is a function of the price of the underlying asset at the time of exercise (i.e., claiming the payoff), or more generally, a functional of the asset price path up to the time of exercise. We focus here on European options, for which exercise is allowable only at *T*, which are different from American options, for which early exercise at any time before *T* is also allowed. For example, the holder of a European call (resp. put) option has the right to buy (resp. sell) the underlying asset at *T* at a certain (strike) price *K*. Denoting by *ST* the asset price at *T*, the payoff of the option is *<sup>f</sup>*(*ST*), with *<sup>f</sup>*(*S*)=(*<sup>S</sup>* <sup>−</sup> *<sup>K</sup>*)<sup>+</sup> and (*<sup>K</sup>* <sup>−</sup> *<sup>S</sup>*)<sup>+</sup> for a call and put, respectively.

Black & Scholes [1] made seminal contributions to the theory of option pricing and hedging by modeling the asset price as a geometric Brownian motion and assuming that (i) the market has a risk-free asset with constant rate of return *r*, (ii) no transaction costs are imposed on the sale or purchase of assets, (iii) there are no limits on short selling, and (iv) trading occurs continuously. Specifically, the asset price *St* at time *t* satisfies the stochastic differential equation

$$dS\_l = aS\_{l'}dt + \sigma S\_{l'}dW\_{l'} \quad S\_0 > 0,\tag{1}$$

where *α* ∈ **R** and *σ* > 0 are the mean and standard deviation (or volatility) of the asset's return, and {*Wt*, *t* ≥ 0} is a standard Brownian motion (with *W*<sup>0</sup> = 0) on some filtered probability space (Ω, F, {F*t*, *t* ≥ 0}, **P**). The absence of transaction fees permits the construction of a continuously rebalanced portfolio consisting of ∓Δ unit of the asset for every ±1 unit of the European option such that its rate of return equals the risk-free rate *r*, where Δ = *∂c*/*∂S* (resp. *∂p*/*∂S*) for a call (resp. put) whose price is *c* (resp. *p*). By requiring this portfolio to be *self-financing* (in the sense that all subsequent rebalancing is financed entirely by the initial capital and, if necessary, by short selling the risk-free asset) and to perfectly replicate the outcome of the European option at expiration, Black & Scholes [1] have shown that the

©2012 Lim and Lai, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Lim and Lai, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 104 Stochastic Modeling and Control Singular Stochastic Control in Option Hedging

"fair" value of the option in the absence of arbitrage opportunities is the initial amount of capital *<sup>E</sup>*ˆ{*e*−*rT <sup>f</sup>*(*ST*)}, where *<sup>E</sup>*<sup>ˆ</sup> denotes expectation under the equivalent martingale measure (with respect to which *St* has drift *α* = *r*). Instead of considering geometric Brownian motion *St* <sup>=</sup> *<sup>S</sup>*<sup>0</sup> exp{(*<sup>r</sup>* <sup>−</sup> *<sup>σ</sup>*2/2)*<sup>t</sup>* <sup>+</sup> *<sup>σ</sup>Wt*}, it is convenient to work directly with Brownian motion *Wt*. The fact that *σWt* and *Wσ*2*<sup>t</sup>* have the same distribution suggests the change of variables

$$s = \sigma^2 (t - T), \quad z = \log(\mathcal{S}/K) - (\rho - 1/2)\text{s} \tag{2}$$

with Transaction Costs 3

Singular Stochastic Control in Option Hedging with Transaction Costs 105

if necessary, for the more general singular stochastic control problem of option hedging even though it is not reducible to an equivalent optimal stopping problem because of the presence

In Section 2, we review the equivalence theory between singular stochastic control and optimal stopping. We also outline the development of the computational schemes of Lai & Lim [18] to solve stochastic control problems that are equivalent to optimal stopping. In Section 3, we introduce the utility-based option hedging problem, outline how the algorithm in Section 2 can be modified to solve stochastic control problems for which equivalence does not exist, and provide numerical examples to illustrate the use of the coupled backward induction algorithm to compute the optimal buy and sell boundaries of a short European

Bather & Chernoff [10] pioneered the study of singular stochastic control in their analysis of the problem of controlling the motion of a spaceship relative to its target on a finite horizon with an infinite amount of fuel. A key idea in their analysis is the reduction of the stochastic control problem to an optimal stopping problem via a change of variables. This spaceship control problem is an example of a *bounded variation follower problem* and the equivalence between singular stochastic control and optimal stopping has since been established for a general class of bounded variation follower problems by Karatzas & Shreve [7, 8], Karatzas & Wang [9] and Boetius [2]. In this section, we review this equivalence for a particular formulation of the bounded variation follower problem in which the control *<sup>ξ</sup>*<sup>+</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>−</sup> is not applied additively to the Brownian motion {*Zu*} and outline the backward induction

**2.1. A bounded variation follower problem and its equivalent optimal stopping**

Suppose that the state process **S** = {*St*, *t* ≥ 0} representing the underlying stochastic environment (in the absence of control) is given by (1). In our subsequent application to option hedging, **S** represents the price of the asset underlying the option, whereas in other applications such as to the problem of reversible investment by Guo & Tomecek [21], **S** represents an economic indicator reflecting demand for a certain commodity. A singular control process is given by a pair (*ξ*+, *ξ*−) of adapted, nondecreasing, LCRL processes such that *dξ*<sup>+</sup> and *dξ*<sup>−</sup> are supported on disjoint subsets. We are interested in problems with a finite time horizon, i.e., we consider the time interval [0, *T*] for some terminal time *T* ∈ (0, ∞).

*<sup>s</sup>* and *ξ*<sup>−</sup>

and decrease, respectively, in control level resulting from the controller's decisions over the

A pair (*ξ*+, *ξ*−) is an *admissible* singular control if, in addition to the above requirements, *xt* ∈ I for all *t* ∈ [0, *T*], where I is an open, possibly unbounded interval of **R** and I is its

*xt* = *x*<sup>0</sup> + *ξ*<sup>+</sup>

*<sup>t</sup>*<sup>+</sup> − *ξ*<sup>−</sup>

*<sup>t</sup>* − *ξ*<sup>−</sup>

<sup>0</sup> = 0. The total control value is therefore given by the finite

*<sup>s</sup>* represent the cumulative increase

*<sup>t</sup>* . (3)

*<sup>t</sup>*<sup>+</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>+</sup>

of additional value functions.

call option. We conclude in Section 4.

**2. Singular stochastic control and optimal stopping**

algorithm for solving the equivalent Dynkin game.

Given any times 0 <sup>≤</sup> *<sup>s</sup>* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>T</sup>*, *<sup>ξ</sup>*<sup>+</sup>

<sup>0</sup> = *ξ*<sup>−</sup>

time interval [*s*, *t*], with *ξ*<sup>+</sup>

variation process

closure.

**problem**

with *<sup>ρ</sup>* <sup>=</sup> *<sup>r</sup>*/*σ*2. The Black-Scholes option pricing formulas *<sup>E</sup>*ˆ{*e*−*r*(*T*−*t*) *<sup>f</sup>*(*ST*)<sup>|</sup> *St* <sup>=</sup> *<sup>S</sup>*} are given explicitly by *c*(*s*, *z*) = *Keρs*[*ez*−*s*/2Φ(*z*/ √−*<sup>s</sup>* <sup>−</sup> √−*s*) <sup>−</sup> <sup>Φ</sup>(*z*/ √−*s*)] for the call and *<sup>p</sup>*(*s*, *<sup>z</sup>*) = *Keρs*[Φ(−*z*/ √−*s*) <sup>−</sup> *<sup>e</sup>z*−*s*/2Φ(−{*z*/ √−*<sup>s</sup>* <sup>−</sup> √−*<sup>s</sup>*})] for the put, where <sup>Φ</sup> is the standard normal distribution function. Correspondingly, the option deltas are Δ(*s*, *z*) = ±Φ(±{*z*/ √−*<sup>s</sup>* <sup>−</sup> √−*<sup>s</sup>*}) with <sup>+</sup> for the call and <sup>−</sup> for the put.

In the presence of transaction costs, perfect hedging of a European option is not possible (since it results in an infinite turnover of the underlying asset and is, therefore, ruinously expensive) and trading in an option involves an essential element of risk. This hedging risk can be characterized as the difference between the realized cash flow from a hedging strategy which uses the initial option premium to trade in the underlying asset and bond, and the desired option payoff at maturity. By embedding option hedging within the framework of portfolio selection introduced by Magill & Constantinides [12] and Davis & Norman [13], Hodges & Neuberger [15] used a risk-averse utility function to assess this shortfall (or "replication error") and formulated the option hedging problem as one of maximizing the investor's expected utility of terminal wealth. This involves the value functions of two singular stochastic control problems, for trading in the market with and without a (short or long) position in the option, and the optimal hedge is given by the difference in the trading strategies corresponding to these two problems. The nature of the optimal hedge is that an investor with an option position should rebalance his portfolio only when the number of units of the asset falls "too far" out of line. For the negative exponential utility function, Davis et al. [14], Clewlow & Hodges [11], and Zakamouline [20] have developed numerical methods to compute the optimal hedge by making use of discrete-time dynamic programming on an approximating binomial tree for the asset price; see Kushner & Dupuis [3] for the general theory of Markov chain approximations for continuous-time processes and their use in the numerical solution of optimal stopping and control problems. More recently, Lai & Lim [18] introduced a new numerical method for solving the singular stochastic control problems associated with utility maximization, yielding a much simpler algorithm to compute the buy and sell boundaries and value functions in the utility-based approach.

The new method is motivated by the equivalence between singular stochastic control and optimal stopping, which was first observed in the pioneering work of Bather & Chernoff [10] on the problem of controlling the motion of a spaceship relative to its target on a finite horizon with an infinite amount of fuel and has since been established for the general class of bounded variation follower problems by Karatzas & Shreve [7, 8], Karatzas & Wang [9] and Boetius [2]. By transforming the original singular stochastic control problem to an optimal stopping problem associated with a Dynkin game, the solution can be computed by applying standard backward induction to an approximating Bernoulli walk. Lai & Lim [18] showed how this backward induction algorithm can be modified, by making use of finite difference methods if necessary, for the more general singular stochastic control problem of option hedging even though it is not reducible to an equivalent optimal stopping problem because of the presence of additional value functions.

In Section 2, we review the equivalence theory between singular stochastic control and optimal stopping. We also outline the development of the computational schemes of Lai & Lim [18] to solve stochastic control problems that are equivalent to optimal stopping. In Section 3, we introduce the utility-based option hedging problem, outline how the algorithm in Section 2 can be modified to solve stochastic control problems for which equivalence does not exist, and provide numerical examples to illustrate the use of the coupled backward induction algorithm to compute the optimal buy and sell boundaries of a short European call option. We conclude in Section 4.

## **2. Singular stochastic control and optimal stopping**

2 Will-be-set-by-IN-TECH

"fair" value of the option in the absence of arbitrage opportunities is the initial amount of capital *<sup>E</sup>*ˆ{*e*−*rT <sup>f</sup>*(*ST*)}, where *<sup>E</sup>*<sup>ˆ</sup> denotes expectation under the equivalent martingale measure (with respect to which *St* has drift *α* = *r*). Instead of considering geometric Brownian motion *St* <sup>=</sup> *<sup>S</sup>*<sup>0</sup> exp{(*<sup>r</sup>* <sup>−</sup> *<sup>σ</sup>*2/2)*<sup>t</sup>* <sup>+</sup> *<sup>σ</sup>Wt*}, it is convenient to work directly with Brownian motion *Wt*. The fact that *σWt* and *Wσ*2*<sup>t</sup>* have the same distribution suggests the change of variables

with *<sup>ρ</sup>* <sup>=</sup> *<sup>r</sup>*/*σ*2. The Black-Scholes option pricing formulas *<sup>E</sup>*ˆ{*e*−*r*(*T*−*t*) *<sup>f</sup>*(*ST*)<sup>|</sup> *St* <sup>=</sup> *<sup>S</sup>*} are

standard normal distribution function. Correspondingly, the option deltas are Δ(*s*, *z*) =

In the presence of transaction costs, perfect hedging of a European option is not possible (since it results in an infinite turnover of the underlying asset and is, therefore, ruinously expensive) and trading in an option involves an essential element of risk. This hedging risk can be characterized as the difference between the realized cash flow from a hedging strategy which uses the initial option premium to trade in the underlying asset and bond, and the desired option payoff at maturity. By embedding option hedging within the framework of portfolio selection introduced by Magill & Constantinides [12] and Davis & Norman [13], Hodges & Neuberger [15] used a risk-averse utility function to assess this shortfall (or "replication error") and formulated the option hedging problem as one of maximizing the investor's expected utility of terminal wealth. This involves the value functions of two singular stochastic control problems, for trading in the market with and without a (short or long) position in the option, and the optimal hedge is given by the difference in the trading strategies corresponding to these two problems. The nature of the optimal hedge is that an investor with an option position should rebalance his portfolio only when the number of units of the asset falls "too far" out of line. For the negative exponential utility function, Davis et al. [14], Clewlow & Hodges [11], and Zakamouline [20] have developed numerical methods to compute the optimal hedge by making use of discrete-time dynamic programming on an approximating binomial tree for the asset price; see Kushner & Dupuis [3] for the general theory of Markov chain approximations for continuous-time processes and their use in the numerical solution of optimal stopping and control problems. More recently, Lai & Lim [18] introduced a new numerical method for solving the singular stochastic control problems associated with utility maximization, yielding a much simpler algorithm to compute the buy

given explicitly by *c*(*s*, *z*) = *Keρs*[*ez*−*s*/2Φ(*z*/

√−*s*) <sup>−</sup> *<sup>e</sup>z*−*s*/2Φ(−{*z*/

and sell boundaries and value functions in the utility-based approach.

The new method is motivated by the equivalence between singular stochastic control and optimal stopping, which was first observed in the pioneering work of Bather & Chernoff [10] on the problem of controlling the motion of a spaceship relative to its target on a finite horizon with an infinite amount of fuel and has since been established for the general class of bounded variation follower problems by Karatzas & Shreve [7, 8], Karatzas & Wang [9] and Boetius [2]. By transforming the original singular stochastic control problem to an optimal stopping problem associated with a Dynkin game, the solution can be computed by applying standard backward induction to an approximating Bernoulli walk. Lai & Lim [18] showed how this backward induction algorithm can be modified, by making use of finite difference methods

√−*<sup>s</sup>* <sup>−</sup> √−*<sup>s</sup>*}) with <sup>+</sup> for the call and <sup>−</sup> for the put.

*<sup>p</sup>*(*s*, *<sup>z</sup>*) = *Keρs*[Φ(−*z*/

±Φ(±{*z*/

*<sup>s</sup>* <sup>=</sup> *<sup>σ</sup>*2(*<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*), *<sup>z</sup>* <sup>=</sup> log(*S*/*K*) <sup>−</sup> (*<sup>ρ</sup>* <sup>−</sup> 1/2)*<sup>s</sup>* (2)

√−*<sup>s</sup>* <sup>−</sup> √−*s*) <sup>−</sup> <sup>Φ</sup>(*z*/

√−*s*)] for the call and

√−*<sup>s</sup>* <sup>−</sup> √−*<sup>s</sup>*})] for the put, where <sup>Φ</sup> is the

104 Stochastic Modeling and Control Singular Stochastic Control in Option Hedging

Bather & Chernoff [10] pioneered the study of singular stochastic control in their analysis of the problem of controlling the motion of a spaceship relative to its target on a finite horizon with an infinite amount of fuel. A key idea in their analysis is the reduction of the stochastic control problem to an optimal stopping problem via a change of variables. This spaceship control problem is an example of a *bounded variation follower problem* and the equivalence between singular stochastic control and optimal stopping has since been established for a general class of bounded variation follower problems by Karatzas & Shreve [7, 8], Karatzas & Wang [9] and Boetius [2]. In this section, we review this equivalence for a particular formulation of the bounded variation follower problem in which the control *<sup>ξ</sup>*<sup>+</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>−</sup> is not applied additively to the Brownian motion {*Zu*} and outline the backward induction algorithm for solving the equivalent Dynkin game.

## **2.1. A bounded variation follower problem and its equivalent optimal stopping problem**

Suppose that the state process **S** = {*St*, *t* ≥ 0} representing the underlying stochastic environment (in the absence of control) is given by (1). In our subsequent application to option hedging, **S** represents the price of the asset underlying the option, whereas in other applications such as to the problem of reversible investment by Guo & Tomecek [21], **S** represents an economic indicator reflecting demand for a certain commodity. A singular control process is given by a pair (*ξ*+, *ξ*−) of adapted, nondecreasing, LCRL processes such that *dξ*<sup>+</sup> and *dξ*<sup>−</sup> are supported on disjoint subsets. We are interested in problems with a finite time horizon, i.e., we consider the time interval [0, *T*] for some terminal time *T* ∈ (0, ∞). Given any times 0 <sup>≤</sup> *<sup>s</sup>* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>T</sup>*, *<sup>ξ</sup>*<sup>+</sup> *<sup>t</sup>*<sup>+</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>+</sup> *<sup>s</sup>* and *ξ*<sup>−</sup> *<sup>t</sup>*<sup>+</sup> − *ξ*<sup>−</sup> *<sup>s</sup>* represent the cumulative increase and decrease, respectively, in control level resulting from the controller's decisions over the time interval [*s*, *t*], with *ξ*<sup>+</sup> <sup>0</sup> = *ξ*<sup>−</sup> <sup>0</sup> = 0. The total control value is therefore given by the finite variation process

$$\mathfrak{x}\_{t} = \mathfrak{x}\_{0} + \mathfrak{f}\_{t}^{+} - \mathfrak{f}\_{t}^{-} \,. \tag{3}$$

A pair (*ξ*+, *ξ*−) is an *admissible* singular control if, in addition to the above requirements, *xt* ∈ I for all *t* ∈ [0, *T*], where I is an open, possibly unbounded interval of **R** and I is its closure.

Let *F*(*t*, *S*, *x*), *κ*±(*t*, *S*) and *G*(*S*, *x*) be sufficiently smooth functions, with *F* and *G* representing the running and terminal reward, respectively, and *κ*± the costs of exerting control. The goal of the controller is to maximize an objective function of the form:

with Transaction Costs 5

 �<sup>+</sup> <sup>−</sup> *∂V<sup>k</sup>*

Assuming the value function to be an increasing function of *x*, the optimal control is obtained by considering the following three possible cases (all the other permutations of inequalities

(i) If *<sup>∂</sup>Vk*(*t*, *<sup>S</sup>*, *<sup>x</sup>*)/*∂<sup>x</sup>* <sup>−</sup> *<sup>κ</sup>*+(*t*, *<sup>S</sup>*) <sup>≥</sup> 0 and *<sup>∂</sup>Vk*(*t*, *<sup>S</sup>*, *<sup>x</sup>*)/*∂<sup>x</sup>* <sup>+</sup> *<sup>κ</sup>*−(*t*, *<sup>S</sup>*) <sup>&</sup>gt; 0, then the maximum is achieved by �<sup>−</sup> = 0 and exerting control *ξ*<sup>+</sup> at the maximum possible rate �<sup>+</sup> = *k*.

(ii) If *<sup>∂</sup>Vk*(*t*, *<sup>S</sup>*, *<sup>x</sup>*)/*∂<sup>x</sup>* <sup>−</sup> *<sup>κ</sup>*+(*t*, *<sup>S</sup>*) <sup>&</sup>lt; 0 and *<sup>∂</sup>Vk*(*t*, *<sup>S</sup>*, *<sup>x</sup>*)/*∂<sup>x</sup>* <sup>+</sup> *<sup>κ</sup>*−(*t*, *<sup>S</sup>*) <sup>≤</sup> 0, then the maximum is achieved by �<sup>+</sup> = 0 and exerting control *ξ*<sup>−</sup> at the maximum possible rate �<sup>−</sup> = *k*.

(iii) If *<sup>∂</sup>Vk*(*t*, *<sup>S</sup>*, *<sup>x</sup>*)/*∂<sup>x</sup>* <sup>−</sup> *<sup>κ</sup>*+(*t*, *<sup>S</sup>*) <sup>≤</sup> 0 and *<sup>∂</sup>Vk*(*t*, *<sup>S</sup>*, *<sup>x</sup>*)/*∂<sup>x</sup>* <sup>+</sup> *<sup>κ</sup>*−(*t*, *<sup>S</sup>*) <sup>≥</sup> 0, then the maximum is achieved by not exerting any control, i.e., �<sup>+</sup> = �<sup>−</sup> = 0, and *Vk*(*t*, *S*, *x*) satisfies the partial

Thus, the state space [0, *T*] × (0, ∞) × I is partitioned into three regions, which we denote by <sup>N</sup> (corresponding to no control), <sup>B</sup> (control *<sup>ξ</sup>*<sup>+</sup> is exerted), and <sup>S</sup> (control *<sup>ξ</sup>*<sup>−</sup> is exerted). The boundaries between the no-control region N and the regions B and S are denoted by *∂*B and

state space remains partitioned into the regions N , B and S. If (*t*, *S*, *x*) ∈ B (resp. S), then the control *<sup>ξ</sup>*<sup>+</sup> (resp. *<sup>ξ</sup>*−) must be instantaneously exerted to bring the state to the boundary *<sup>∂</sup>*<sup>B</sup> (resp. *∂*S). Thus, besides an initial jump from B or S to the boundary *∂*B or *∂*S (if necessary), the optimal control process acts thereafter only when (*t*, *S*, *x*) ∈ *∂*B or *∂*S so as to keep the state in N ∪ *∂*B ∪ *∂*S. Because the optimal process behaves like the *local time* of the (optimally controlled) state process at the boundaries, such a control is termed *singular*. In B, since the optimal control is to increase *x* by a positive amount *δx* (up to that required to take the state

to *<sup>∂</sup>*B) at the cost of *<sup>κ</sup>*+(*t*, *<sup>S</sup>*) per unit increase, the value function satisfies the equation

*<sup>V</sup>*(*t*, *<sup>S</sup>*, *<sup>x</sup>*) = *<sup>V</sup>*(*t*, *<sup>S</sup>*, *<sup>x</sup>* <sup>+</sup> *<sup>δ</sup>x*) <sup>−</sup> *<sup>κ</sup>*+(*t*, *<sup>S</sup>*)*δ<sup>x</sup>* (in <sup>B</sup>).

Similarly, since the optimal control in S is to decrease *x* by a positive amount *δx* (up to that required to take the state to *∂*S) at the cost of *κ*−(*t*, *S*) per unit decrease, the value function

*V*(*t*, *S*, *x*) = *V*(*t*, *S*, *x* − *δx*) − *κ*−(*t*, *S*)*δx* (in S).

Letting *δx* → 0 leads to gradient constraints for the value function in B and S. In N , *V*(*t*, *S*, *x*) continues to satisfy the PDE given in (iii) above. From these observations, we obtain the

following free boundary problem (FBP) for the value function *V*(*t*, *S*, *x*):

*<sup>t</sup>*,*<sup>x</sup>* of admissible controls becomes the set A*t*,*<sup>x</sup>* of problem (4) and the

<sup>+</sup> <sup>L</sup>*t*,*SV<sup>k</sup>*(*t*, *<sup>S</sup>*, *<sup>x</sup>*) + *<sup>F</sup>*(*t*, *<sup>S</sup>*, *<sup>x</sup>*) = 0, (*t*, *<sup>S</sup>*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> (0, <sup>∞</sup>) <sup>×</sup> <sup>I</sup>.

*<sup>∂</sup><sup>x</sup>* (*t*, *<sup>S</sup>*, *<sup>x</sup>*) + *<sup>κ</sup>*−(*t*, *<sup>S</sup>*)

Singular Stochastic Control in Option Hedging with Transaction Costs 107

 �− 

or equivalently,

are impossible):

*∂*S.

As *<sup>k</sup>* <sup>→</sup> <sup>∞</sup>, the set <sup>A</sup>*<sup>k</sup>*

satisfies the equation

max <sup>0</sup>≤�+,�−≤*<sup>k</sup>*

*∂V<sup>k</sup>*

*<sup>∂</sup><sup>x</sup>* (*t*, *<sup>S</sup>*, *<sup>x</sup>*) <sup>−</sup> *<sup>κ</sup>*+(*t*, *<sup>S</sup>*)

differential equation (PDE) <sup>L</sup>*t*,*SV<sup>k</sup>*(*t*, *<sup>S</sup>*, *<sup>x</sup>*) + *<sup>F</sup>*(*t*, *<sup>S</sup>*, *<sup>x</sup>*) = 0.

$$\begin{aligned} J\_{t,S,x}(\xi^+, \xi^-) &= E\_{t,S,x} \left\{ \int\_t^T e^{-r(u-t)} F(u, S\_{u'}, \mathbf{x}\_u) \, du - \int\_{[t,T)} e^{-r(u-t)} \kappa^+(u, S\_u) \, d\xi^+\_u \right\} \\ &- \int\_{[t,T)} e^{-r(u-t)} \kappa^-(u, S\_u) \, d\xi^-\_u + e^{-r(T-t)} G(S\_T, \mathbf{x}\_T) \right\}, \end{aligned}$$

where *Et*,*S*,*<sup>x</sup>* denotes conditional expectation given *St* = *S* and *xt* = *x*. The value function of the stochastic control problem is

$$V(t, \mathbf{S}, \mathbf{x}) = \sup\_{(\xi^+, \xi^-) \in \mathcal{A}\_{t, \mathbf{x}}} I\_{t, \mathbf{S}, \mathbf{x}}(\xi^+, \xi^-), \quad (t, \mathbf{S}, \mathbf{x}) \in [0, T] \times (0, \infty) \times \overline{\mathcal{Z}}, \tag{4}$$

where A*t*,*<sup>x</sup>* denotes the set of all admissible controls which satisfy *xt* = *x*.

We derive formally the Hamilton-Jacobi-Bellman equation associated with the stochastic control problem (4), which provides key insights into the nature of the optimal control. Consider, for now, a smaller set <sup>A</sup>*<sup>k</sup> <sup>t</sup>*,*<sup>x</sup>* of admissible controls such that *ξ*<sup>±</sup> are absolutely continuous processes, i.e., *dξ*± *<sup>t</sup>* = �<sup>±</sup> *<sup>t</sup> dt* with 0 <sup>≤</sup> �<sup>+</sup> *<sup>t</sup>* , �<sup>−</sup> *<sup>t</sup>* ≤ *k* < ∞. Under this restriction, the value function (4) becomes

$$V^k(t, S, \mathbf{x}) = \sup\_{(\ell^+, \ell^-) \in \mathcal{A}^k\_{l, \mathbf{x}}} J^k\_{l, S, \mathbf{x}}(\ell^+, \ell^-), \quad (t, S, \mathbf{x}) \in [0, T] \times (0, \infty) \times \overline{\mathcal{Z}}\_{\ell^+} $$

where

$$\begin{split} &f\_{t,\mathcal{S},\mathbf{x}}^{\mathbf{k}}(\ell^{+},\ell^{-}) \\ &= E\_{t,\mathcal{S},\mathbf{x}}\left\{\int\_{t}^{T}e^{-r(\boldsymbol{u}-t)}\left[F(\boldsymbol{u},\mathcal{S}\_{\boldsymbol{u}\prime}\mathbf{x}\_{\boldsymbol{u}})-\kappa^{+}(\boldsymbol{u},\mathcal{S}\_{\boldsymbol{u}\prime})\ell^{+}\_{\boldsymbol{u}}-\kappa^{-}(\boldsymbol{u},\mathcal{S}\_{\boldsymbol{u}\prime})\ell^{-}\_{\boldsymbol{u}}\right]d\boldsymbol{u}+e^{-r(T-t)}G(\mathcal{S}\_{T},\mathbf{x}\_{T})\right\}. \end{split}$$

Since the infinitesimal generator of the stochastic system comprising (1) and *dxt* = (�<sup>+</sup> *t* − �− *<sup>t</sup>* ) *dt* (corresponding to (3) for absolutely continuous *ξ*±) is

$$\varkappa S \frac{\partial}{\partial S} + \frac{\sigma^2 S^2}{2} \frac{\partial^2}{\partial S^2} + (\ell^+ - \ell^-) \frac{\partial}{\partial \mathbf{x}^\prime}$$

the Bellman equation for *Vk*(*t*, *S*, *x*) is

$$\max\_{0 \le \ell^+, \ell^- \le k} \left\{ \left[ \mathcal{L}\_{t,S} + (\ell^+ - \ell^-) \frac{\partial}{\partial \mathbf{x}} \right] V^k(t, S, \mathbf{x}) + F(t, S, \mathbf{x}) - \kappa^+(t, S)\ell^+ - \kappa^-(t, S)\ell^- \right\} = 0, \ell$$

where

$$\mathcal{L}\_{t,S} = \frac{\partial}{\partial t} + \mathfrak{a}S\frac{\partial}{\partial S} + \frac{\sigma^2 S^2}{2} \frac{\partial^2}{\partial S^2} - r\_\star$$

or equivalently,

4 Will-be-set-by-IN-TECH

Let *F*(*t*, *S*, *x*), *κ*±(*t*, *S*) and *G*(*S*, *x*) be sufficiently smooth functions, with *F* and *G* representing the running and terminal reward, respectively, and *κ*± the costs of exerting control. The goal

<sup>−</sup>*r*(*u*−*t*)*F*(*u*, *Su*, *xu*) *du* <sup>−</sup>

where *Et*,*S*,*<sup>x</sup>* denotes conditional expectation given *St* = *S* and *xt* = *x*. The value function of

We derive formally the Hamilton-Jacobi-Bellman equation associated with the stochastic control problem (4), which provides key insights into the nature of the optimal control.

*<sup>t</sup> dt* with 0 <sup>≤</sup> �<sup>+</sup>

Since the infinitesimal generator of the stochastic system comprising (1) and *dxt* = (�<sup>+</sup>

*∂*2

<sup>+</sup> *<sup>α</sup><sup>S</sup> <sup>∂</sup> <sup>∂</sup><sup>S</sup>* <sup>+</sup>

*κ*−(*u*, *Su*) *dξ*−

*<sup>t</sup>* , �<sup>−</sup>

*<sup>t</sup>*,*S*,*x*(�+, �−), (*t*, *<sup>S</sup>*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*] <sup>×</sup> (0, <sup>∞</sup>) <sup>×</sup> <sup>I</sup>,

*<sup>u</sup>* − *κ*−(*u*, *Su*)�<sup>−</sup>

*<sup>∂</sup>S*<sup>2</sup> + (�<sup>+</sup> <sup>−</sup> �−) *<sup>∂</sup>*

*σ*2*S*<sup>2</sup> 2

*u du* + *e*

*∂x* ,

*<sup>V</sup>k*(*t*, *<sup>S</sup>*, *<sup>x</sup>*) + *<sup>F</sup>*(*t*, *<sup>S</sup>*, *<sup>x</sup>*) <sup>−</sup> *<sup>κ</sup>*+(*t*, *<sup>S</sup>*)�<sup>+</sup> <sup>−</sup> *<sup>κ</sup>*−(*t*, *<sup>S</sup>*)�<sup>−</sup>

*∂*2 *<sup>∂</sup>S*<sup>2</sup> <sup>−</sup> *<sup>r</sup>*,

 [*t*,*T*) *e* −*r*(*u*−*t*)

*<sup>u</sup>* + *e*

−*r*(*T*−*t*)

106 Stochastic Modeling and Control Singular Stochastic Control in Option Hedging

*Jt*,*S*,*x*(*ξ*+, *<sup>ξ</sup>*−), (*t*, *<sup>S</sup>*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*] <sup>×</sup> (0, <sup>∞</sup>) <sup>×</sup> <sup>I</sup>, (4)

*<sup>t</sup>*,*<sup>x</sup>* of admissible controls such that *ξ*<sup>±</sup> are absolutely

*κ*+(*u*, *Su*) *dξ*<sup>+</sup>

 ,

*G*(*ST*, *xT*)

*<sup>t</sup>* ≤ *k* < ∞. Under this restriction,

−*r*(*T*−*t*)

*G*(*ST*, *xT*)

 = 0,

 .

*t* −

*u*

of the controller is to maximize an objective function of the form:

 *T t e*

> − [*t*,*T*) *e* −*r*(*u*−*t*)

(*ξ*+,*ξ*−)∈A*t*,*<sup>x</sup>*

where A*t*,*<sup>x</sup>* denotes the set of all admissible controls which satisfy *xt* = *x*.

*<sup>t</sup>* = �<sup>±</sup>

*t*,*x J k*

*<sup>F</sup>*(*u*, *Su*, *xu*) <sup>−</sup> *<sup>κ</sup>*+(*u*, *Su*)�<sup>+</sup>

*σ*2*S*<sup>2</sup> 2

(�+,�−)∈A*<sup>k</sup>*

*<sup>t</sup>* ) *dt* (corresponding to (3) for absolutely continuous *ξ*±) is

*<sup>α</sup><sup>S</sup> <sup>∂</sup> <sup>∂</sup><sup>S</sup>* <sup>+</sup>

> *∂x*

<sup>L</sup>*t*,*<sup>S</sup>* <sup>=</sup> *<sup>∂</sup> ∂t*

*Jt*,*S*,*x*(*ξ*+, *<sup>ξ</sup>*−) = *Et*,*S*,*<sup>x</sup>*

the stochastic control problem is

*V*(*t*, *S*, *x*) = sup

Consider, for now, a smaller set <sup>A</sup>*<sup>k</sup>*

the Bellman equation for *Vk*(*t*, *S*, *x*) is

<sup>L</sup>*t*,*<sup>S</sup>* + (�<sup>+</sup> <sup>−</sup> �−) *<sup>∂</sup>*

*Vk*(*t*, *S*, *x*) = sup

continuous processes, i.e., *dξ*±

the value function (4) becomes

where

*<sup>t</sup>*,*S*,*x*(�+, �−)

 *T t e* −*r*(*u*−*t*) 

= *Et*,*S*,*<sup>x</sup>*

max <sup>0</sup>≤�+,�−≤*<sup>k</sup>*

where

*J k*

�−

$$\begin{split} \max\_{0 \le \ell^{+}, \ell^{-} \le k} \left\{ \left[ \frac{\partial V^{k}}{\partial \mathbf{x}} (t, \mathsf{S}, \mathbf{x}) - \kappa^{+} (t, \mathsf{S}) \right] \ell^{+} - \left[ \frac{\partial V^{k}}{\partial \mathbf{x}} (t, \mathsf{S}, \mathbf{x}) + \kappa^{-} (t, \mathsf{S}) \right] \ell^{-} \right\} \\ + \mathcal{L}\_{t, \mathsf{S}} V^{k} (t, \mathsf{S}, \mathbf{x}) + F(t, \mathsf{S}, \mathbf{x}) &= \mathbf{0}, \quad (t, \mathsf{S}, \mathbf{x}) \in [0, T) \times (0, \infty) \times \overline{\mathcal{Z}}. \end{split}$$

Assuming the value function to be an increasing function of *x*, the optimal control is obtained by considering the following three possible cases (all the other permutations of inequalities are impossible):


Thus, the state space [0, *T*] × (0, ∞) × I is partitioned into three regions, which we denote by <sup>N</sup> (corresponding to no control), <sup>B</sup> (control *<sup>ξ</sup>*<sup>+</sup> is exerted), and <sup>S</sup> (control *<sup>ξ</sup>*<sup>−</sup> is exerted). The boundaries between the no-control region N and the regions B and S are denoted by *∂*B and *∂*S.

As *<sup>k</sup>* <sup>→</sup> <sup>∞</sup>, the set <sup>A</sup>*<sup>k</sup> <sup>t</sup>*,*<sup>x</sup>* of admissible controls becomes the set A*t*,*<sup>x</sup>* of problem (4) and the state space remains partitioned into the regions N , B and S. If (*t*, *S*, *x*) ∈ B (resp. S), then the control *<sup>ξ</sup>*<sup>+</sup> (resp. *<sup>ξ</sup>*−) must be instantaneously exerted to bring the state to the boundary *<sup>∂</sup>*<sup>B</sup> (resp. *∂*S). Thus, besides an initial jump from B or S to the boundary *∂*B or *∂*S (if necessary), the optimal control process acts thereafter only when (*t*, *S*, *x*) ∈ *∂*B or *∂*S so as to keep the state in N ∪ *∂*B ∪ *∂*S. Because the optimal process behaves like the *local time* of the (optimally controlled) state process at the boundaries, such a control is termed *singular*. In B, since the optimal control is to increase *x* by a positive amount *δx* (up to that required to take the state to *<sup>∂</sup>*B) at the cost of *<sup>κ</sup>*+(*t*, *<sup>S</sup>*) per unit increase, the value function satisfies the equation

$$V(t, \mathcal{S}, \mathbf{x}) = V(t, \mathcal{S}, \mathbf{x} + \delta \mathbf{x}) - \kappa^+(t, \mathcal{S}) \delta \mathbf{x} \quad (\text{in } \mathcal{B}).$$

Similarly, since the optimal control in S is to decrease *x* by a positive amount *δx* (up to that required to take the state to *∂*S) at the cost of *κ*−(*t*, *S*) per unit decrease, the value function satisfies the equation

$$V(t, \mathcal{S}, \mathbf{x}) = V(t, \mathcal{S}, \mathbf{x} - \delta \mathbf{x}) - \kappa^-(t, \mathcal{S}) \delta \mathbf{x} \quad (\text{in } \mathcal{S}).$$

Letting *δx* → 0 leads to gradient constraints for the value function in B and S. In N , *V*(*t*, *S*, *x*) continues to satisfy the PDE given in (iii) above. From these observations, we obtain the following free boundary problem (FBP) for the value function *V*(*t*, *S*, *x*):

$$
\mathcal{L}\_{t,S}V(t,\mathcal{S},\mathbf{x}) + F(t,\mathcal{S},\mathbf{x}) = \mathbf{0} \tag{5a}
\qquad \text{in } \mathcal{N}\_{\prime} \tag{5a}
$$

$$\frac{\partial V}{\partial \mathbf{x}}(t, \mathbf{S}, \mathbf{x}) = \mathbf{x}^+(t, \mathbf{S}) \qquad \text{in } \mathcal{B}\_\prime \tag{5b}$$

with Transaction Costs 7

*w*(0, *z*, *x*) = *g*(*e*

inf *<sup>τ</sup>*+∈T (*s*,0)

*τ*−∈T (*s*,0)

<sup>−</sup> *<sup>κ</sup>*˜−(*τ*−, *<sup>Z</sup>τ*<sup>−</sup> )*I*{*τ*−<*τ*+<0} <sup>+</sup> *<sup>g</sup>*(*eZ*<sup>0</sup> , *<sup>x</sup>*0)*I*{*τ*−=*τ*+=0}

*<sup>φ</sup>*(*u*, *Zu*, *xu*) *du* <sup>+</sup> *<sup>κ</sup>*˜+(*τ*+, *<sup>Z</sup>τ*<sup>+</sup> )*I*{*τ*+<*τ*−<0}

where *<sup>φ</sup>*(*s*, *<sup>z</sup>*, *<sup>x</sup>*) = *<sup>∂</sup>F*˜(*s*, *<sup>z</sup>*, *<sup>x</sup>*)/*∂<sup>x</sup>* and *<sup>g</sup>*(·, *<sup>x</sup>*) = *<sup>∂</sup>G*(·, *<sup>x</sup>*)/*∂x*. The FBP (10) can be restated as an optimal stopping problem associated with a Dynkin game, for which *w*(*s*, *z*, *x*) is the value

*τ*−∈T (*s*,0)

<sup>=</sup> *<sup>w</sup>*(*s*, *<sup>z</sup>*, *<sup>x</sup>*) :<sup>=</sup> inf *<sup>τ</sup>*+∈T (*s*,0) sup

where T (*a*, *b*) denotes the set of stopping times taking values between *a* and *b* (> *a*), and

The Dynkin game is a "stochastic game of timing" in which there are two players P and M, each of whom chooses a stopping time (*τ*<sup>+</sup> and *<sup>τ</sup>*−, respectively) in <sup>T</sup> (*s*, 0). As long as the game is in progress, P keeps paying M at the rate *φ*(*s*, *z*, *x*) per unit of time. The game terminates as soon as one of the players decides to stop, i.e., at *<sup>τ</sup>*<sup>+</sup> <sup>∧</sup> *<sup>τ</sup>*−. If player M stops first, he pays P the amount *κ*˜−(*τ*−, *Zτ*<sup>−</sup> ). If player P stops first, he pays M the amount *κ*˜+(*τ*+, *Zτ*<sup>+</sup> ) (resp. *g*(*eZ*<sup>0</sup> , *x*0)) when the game terminates before (resp. at) the end of the time horizon 0. The objective of P is to minimize his expected total payment to M whereas the objective of M is to

In addition to the relationship (9) between the value functions *v* and *w*, the optimal continuation region of the Dynkin game (11) coincides with the no-control region of the singular stochastic control problem (8) in the sense that if (*ξ*+,∗, *ξ*−,∗) is an optimal control of (8) and we define the stopping times *<sup>τ</sup>*+,<sup>∗</sup> <sup>=</sup> inf{*<sup>u</sup>* <sup>∈</sup> [*s*, 0) : *<sup>ξ</sup>*+,∗(*u*) <sup>&</sup>gt; <sup>0</sup>} and *<sup>τ</sup>*−,<sup>∗</sup> <sup>=</sup> inf{*<sup>u</sup>* <sup>∈</sup> [*s*, 0) : *<sup>ξ</sup>*−,∗(*u*) <sup>&</sup>gt; <sup>0</sup>} (inf <sup>∅</sup> <sup>=</sup> 0), then (*τ*+,∗, *<sup>τ</sup>*−,∗) is a *saddlepoint* of the game with the

(*s*, *<sup>z</sup>*, *<sup>x</sup>*), (*s*, *<sup>z</sup>*, *<sup>x</sup>*) <sup>∈</sup> [−*σ*2*T*, 0] <sup>×</sup> **<sup>R</sup>** <sup>×</sup> <sup>I</sup>. (9)

Singular Stochastic Control in Option Hedging with Transaction Costs 109

*w*(*s*, *z*, *x*) + *φ*(*s*, *z*, *x*) = 0 in N , (10a)

*z*

*<sup>w</sup>*(*s*, *<sup>z</sup>*, *<sup>x</sup>*) = *<sup>κ</sup>*˜+(*s*, *<sup>z</sup>*) in <sup>B</sup>, (10b)

*w*(*s*, *z*, *x*) = −*κ*˜−(*s*, *z*) in S, (10c)

*Is*,*z*,*x*(*τ*+, *τ*−)

, *x*), (10d)

*Is*,*z*,*x*(*τ*+, *τ*−), (11)

 .

We now introduce the change of variables

From (7) it follows that *w* solves the FBP

 *∂ ∂s* + 1 2 *∂*2 *∂z*<sup>2</sup> 

*Is*,*z*,*x*(*τ*+, *τ*−) = *Es*,*z*,*<sup>x</sup>*

property that *w*(*s*, *z*, *x*) = *Is*,*z*,*x*(*τ*+,∗, *τ*−,∗).

function. Specifically,

maximize this quantity.

*<sup>w</sup>*(*s*, *<sup>z</sup>*, *<sup>x</sup>*) = *<sup>∂</sup><sup>v</sup>*

*∂x*

*w*(*s*, *z*, *x*) = *w*(*s*, *z*, *x*) := sup

 *<sup>τ</sup>*+∧*τ*<sup>−</sup> *s*

$$\frac{\partial V}{\partial \mathbf{x}}(t, \mathbf{S}, \mathbf{x}) = -\kappa^{-}(t, \mathbf{S}) \quad \text{in } \mathbf{S}\_{\prime} \tag{5c}$$

$$V(T, \mathcal{S}, \mathfrak{x}) = G(\mathcal{S}, \mathfrak{x}).\tag{5d}$$

It also follows that the Hamilton-Jacobi-Bellman equation associated with (4) is the following variational inequality with gradient constraints:

$$\max \left\{ \mathcal{L}\_{t,S} V(t, \mathbf{S}, \mathbf{x}) + F(t, \mathbf{S}, \mathbf{x}), \ \frac{\partial V}{\partial \mathbf{x}}(t, \mathbf{S}, \mathbf{x}) - \kappa^+(t, \mathbf{S}), \ -\frac{\partial V}{\partial \mathbf{x}}(t, \mathbf{S}, \mathbf{x}) - \kappa^-(t, \mathbf{S}) \right\} = 0, \tag{6}$$

$$(t, \mathcal{S}, \mathfrak{x}) \in [0, T) \times (0, \infty) \times \overline{\mathcal{Z}}.$$

With *ρ* = *r*/*σ*<sup>2</sup> and *β* = *α*/*σ*2, a more parsimonious parameterization of (5) can be obtained by considering the change of variables (2) (without *K* and with *ρ* replaced by *β* here since the state process **S** has rate of return *α* under the "physical" measure rather than *r* under the risk-neutral measure) and *v*(*s*, *z*, *x*) = *e*−*<sup>ρ</sup>sV*(*t*, *S*, *x*). Applying the chain rule of differentiation yields

$$\frac{\partial V}{\partial S} = \frac{e^{\rho s}}{S} \frac{\partial v}{\partial z}, \quad \frac{\partial^2 V}{\partial S^2} = \frac{e^{\rho s}}{S^2} \left(\frac{\partial^2 v}{\partial z^2} - \frac{\partial v}{\partial z}\right), \quad \frac{\partial V}{\partial t} - rV = e^{\rho s} \left[\sigma^2 \frac{\partial v}{\partial s} - \left(a - \frac{\sigma^2}{2}\right) \frac{\partial v}{\partial z}\right].$$

We also define *F*˜(*s*, *z*, *x*) = *er*(*T*−*t*)*F*(*t*, *S*, *x*)/*σ*<sup>2</sup> and *κ*˜±(*s*, *z*) = *e*−*ρsκ*±(*t*, *S*). Upon substitution into (5), we arrive at the FBP

$$\left\{\frac{\partial}{\partial s} + \frac{1}{2}\frac{\partial^2}{\partial \bar{z}^2}\right\} v(s, z, \mathbf{x}) + \tilde{F}(s, z, \mathbf{x}) = 0 \qquad\qquad\text{in } \mathcal{N}\_{\prime} \tag{7a}$$

$$\frac{\partial v}{\partial \mathfrak{x}}(s, z, \mathfrak{x}) = \mathfrak{x}^+(s, z) \qquad \text{in } \mathcal{B}\_\prime \tag{7b}$$

$$\frac{\partial v}{\partial \mathfrak{x}}(s, z, \mathfrak{x}) = -\mathfrak{x}^-(s, z) \quad \text{in } \mathcal{S}\_\prime \tag{7c}$$

$$w(0, z, \mathfrak{x}) = G(e^z, \mathfrak{x}),\tag{7d}$$

Note that *v*(*s*, *z*, *x*) is the value function of the corresponding singular stochastic control problem for the Brownian motion {*Zu*, *u* ≤ 0}:

$$v(s, z, x) = \sup\_{\left(\xi^{+}, \xi^{-}\right) \in \mathcal{A}\_{\mathsf{s}, x}} \mathbb{E}\_{\mathsf{s}, z, x} \left\{ \int\_{\mathsf{s}}^{0} \tilde{\mathsf{F}} \left(\mathsf{u}\_{\prime} \mathsf{Z}\_{\mathsf{u}}, \mathsf{x}\_{\mathsf{u}}\right) du - \int\_{\left[s, 0\right)} \tilde{\mathsf{x}}^{+} \left(\mathsf{u}\_{\prime} \mathsf{Z}\_{\mathsf{u}}\right) d\xi^{+}\_{\mathsf{u}} \right. \\ \left. + \int\_{\left[s, 0\right)} \tilde{\mathsf{x}}^{-} \left(\mathsf{u}\_{\prime} \mathsf{Z}\_{\mathsf{u}}\right) d\xi^{-}\_{\mathsf{u}} + \mathsf{G} \left(e^{\mathsf{Z}\_{0}}, \mathsf{x}\_{0}\right) \right\}, \tag{8}$$

where *Es*,*z*,*<sup>x</sup>* denotes conditional expectation given *Zs* = *z* and *xs* = *x*.

We now introduce the change of variables

6 Will-be-set-by-IN-TECH

It also follows that the Hamilton-Jacobi-Bellman equation associated with (4) is the following

With *ρ* = *r*/*σ*<sup>2</sup> and *β* = *α*/*σ*2, a more parsimonious parameterization of (5) can be obtained by considering the change of variables (2) (without *K* and with *ρ* replaced by *β* here since the state process **S** has rate of return *α* under the "physical" measure rather than *r* under the risk-neutral measure) and *v*(*s*, *z*, *x*) = *e*−*<sup>ρ</sup>sV*(*t*, *S*, *x*). Applying the chain rule of differentiation

We also define *F*˜(*s*, *z*, *x*) = *er*(*T*−*t*)*F*(*t*, *S*, *x*)/*σ*<sup>2</sup> and *κ*˜±(*s*, *z*) = *e*−*ρsκ*±(*t*, *S*). Upon substitution

*∂v ∂x*

*∂v ∂x*

Note that *v*(*s*, *z*, *x*) is the value function of the corresponding singular stochastic control

*v*(0, *z*, *x*) = *G*(*e*

*<sup>F</sup>*˜(*u*, *Zu*, *xu*) *du* <sup>−</sup>

*κ*˜−(*u*, *Zu*) *dξ*−

*<sup>∂</sup><sup>x</sup>* (*t*, *<sup>S</sup>*, *<sup>x</sup>*) <sup>−</sup> *<sup>κ</sup>*+(*t*, *<sup>S</sup>*), <sup>−</sup>*∂<sup>V</sup>*

*<sup>∂</sup><sup>t</sup>* <sup>−</sup> *rV* <sup>=</sup> *<sup>e</sup>*

*ρs <sup>σ</sup>*<sup>2</sup> *<sup>∂</sup><sup>v</sup> <sup>∂</sup><sup>s</sup>* −

*<sup>v</sup>*(*s*, *<sup>z</sup>*, *<sup>x</sup>*) + *<sup>F</sup>*˜(*s*, *<sup>z</sup>*, *<sup>x</sup>*) = 0 in <sup>N</sup> , (7a)

*z*

 [*s*,0)

(*s*, *<sup>z</sup>*, *<sup>x</sup>*) = *<sup>κ</sup>*˜+(*s*, *<sup>z</sup>*) in <sup>B</sup>, (7b)

(*s*, *z*, *x*) = −*κ*˜−(*s*, *z*) in S, (7c)

*<sup>u</sup>* + *<sup>G</sup>*(*eZ*<sup>0</sup> , *<sup>x</sup>*0)

, *x*), (7d)

*u*

, (8)

*κ*˜+(*u*, *Zu*) *dξ*<sup>+</sup>

*∂V*

*∂V*

variational inequality with gradient constraints:

<sup>L</sup>*t*,*SV*(*t*, *<sup>S</sup>*, *<sup>x</sup>*) + *<sup>F</sup>*(*t*, *<sup>S</sup>*, *<sup>x</sup>*), *<sup>∂</sup><sup>V</sup>*

, *<sup>∂</sup>*2*<sup>V</sup>*

 *∂ ∂s* + 1 2 *∂*2 *∂z*<sup>2</sup> 

problem for the Brownian motion {*Zu*, *u* ≤ 0}:

(*ξ*+,*ξ*−)∈A*s*,*<sup>x</sup>*

*Es*,*z*,*x*

where *Es*,*z*,*<sup>x</sup>* denotes conditional expectation given *Zs* = *z* and *xs* = *x*.

 0 *s*

> − [*s*,0)

*v*(*s*, *z*, *x*) = sup

*<sup>∂</sup>S*<sup>2</sup> <sup>=</sup> *<sup>e</sup>ρ<sup>s</sup> S*2

 *∂*2*v <sup>∂</sup>z*<sup>2</sup> <sup>−</sup> *<sup>∂</sup><sup>v</sup> ∂z* , *<sup>∂</sup><sup>V</sup>*

max 

yields

*∂V <sup>∂</sup><sup>S</sup>* <sup>=</sup> *<sup>e</sup>ρ<sup>s</sup> S ∂v ∂z*

(*t*, *S*, *x*) ∈ [0, *T*) × (0, ∞) × I.

into (5), we arrive at the FBP

L*t*,*SV*(*t*, *S*, *x*) + *F*(*t*, *S*, *x*) = 0 in N , (5a)

*<sup>∂</sup><sup>x</sup>* (*t*, *<sup>S</sup>*, *<sup>x</sup>*) = *<sup>κ</sup>*+(*t*, *<sup>S</sup>*) in <sup>B</sup>, (5b)

108 Stochastic Modeling and Control Singular Stochastic Control in Option Hedging

*<sup>∂</sup><sup>x</sup>* (*t*, *<sup>S</sup>*, *<sup>x</sup>*) = <sup>−</sup>*κ*−(*t*, *<sup>S</sup>*) in <sup>S</sup>, (5c) *V*(*T*, *S*, *x*) = *G*(*S*, *x*). (5d)

*<sup>∂</sup><sup>x</sup>* (*t*, *<sup>S</sup>*, *<sup>x</sup>*) <sup>−</sup> *<sup>κ</sup>*−(*t*, *<sup>S</sup>*)

 *<sup>α</sup>* <sup>−</sup> *<sup>σ</sup>*<sup>2</sup> 2

 *∂v ∂z* .

= 0, (6)

$$w(s, z, \mathbf{x}) = \frac{\partial v}{\partial \mathbf{x}}(s, z, \mathbf{x}), \quad (s, z, \mathbf{x}) \in [-\sigma^2 T, 0] \times \mathbb{R} \times \overline{\mathcal{Z}}.\tag{9}$$

From (7) it follows that *w* solves the FBP

$$\left\{\frac{\partial}{\partial s} + \frac{1}{2}\frac{\partial^2}{\partial z^2}\right\} w(s, z, \mathbf{x}) + \phi(s, z, \mathbf{x}) = 0 \qquad\qquad\text{in } \mathcal{N}\_{\prime} \tag{10a}$$

$$w(s, z, \mathfrak{x}) = \mathfrak{x}^+(s, z) \qquad \text{in } \mathcal{B}\_\prime \tag{10b}$$

$$w(s, z, \mathfrak{x}) = -\mathfrak{x}^-(s, z) \quad \text{in } \mathfrak{S}\_\prime \tag{10c}$$

$$w(0, z, \mathbf{x}) = \operatorname{g}(e^z, \mathbf{x}),\tag{10d}$$

where *<sup>φ</sup>*(*s*, *<sup>z</sup>*, *<sup>x</sup>*) = *<sup>∂</sup>F*˜(*s*, *<sup>z</sup>*, *<sup>x</sup>*)/*∂<sup>x</sup>* and *<sup>g</sup>*(·, *<sup>x</sup>*) = *<sup>∂</sup>G*(·, *<sup>x</sup>*)/*∂x*. The FBP (10) can be restated as an optimal stopping problem associated with a Dynkin game, for which *w*(*s*, *z*, *x*) is the value function. Specifically,

$$w(s, z, \mathbf{x}) = \underline{w}(s, z, \mathbf{x}) := \sup\_{\tau^- \in \mathcal{T}(s, 0)} \inf\_{\tau^+ \in \mathcal{T}(s, 0)} I\_{s, z, \mathbf{x}}(\tau^+, \tau^-)$$

$$= \overline{w}(s, z, \mathbf{x}) := \inf\_{\tau^+ \in \mathcal{T}(s, 0)} \sup\_{\tau^- \in \mathcal{T}(s, 0)} I\_{s, z, \mathbf{x}}(\tau^+, \tau^-), \tag{11}$$

where T (*a*, *b*) denotes the set of stopping times taking values between *a* and *b* (> *a*), and

$$\begin{aligned} I\_{\mathsf{S},\mathsf{z},\mathsf{x}}(\tau^+,\tau^-) &= E\_{\mathsf{s},\mathsf{z},\mathsf{x}} \left\{ \int\_{\mathsf{s}}^{\tau^+ \wedge \tau^-} \mathfrak{g}(u, Z\_{\mathsf{u}}, \mathsf{x}\_{\mathsf{u}}) \, du + \mathfrak{x}^+(\tau^+, Z\_{\mathsf{T}^+}) I\_{\{\tau^+ < \tau^- < 0\}} \\ &- \mathfrak{x}^-(\tau^-, Z\_{\mathsf{T}^-}) I\_{\{\tau^- < \tau^+ < 0\}} + \mathfrak{g}(\epsilon^{Z\_{\mathsf{d}}}, \mathsf{x}\_0) I\_{\{\tau^- = \tau^+ = 0\}} \right\}. \end{aligned}$$

The Dynkin game is a "stochastic game of timing" in which there are two players P and M, each of whom chooses a stopping time (*τ*<sup>+</sup> and *<sup>τ</sup>*−, respectively) in <sup>T</sup> (*s*, 0). As long as the game is in progress, P keeps paying M at the rate *φ*(*s*, *z*, *x*) per unit of time. The game terminates as soon as one of the players decides to stop, i.e., at *<sup>τ</sup>*<sup>+</sup> <sup>∧</sup> *<sup>τ</sup>*−. If player M stops first, he pays P the amount *κ*˜−(*τ*−, *Zτ*<sup>−</sup> ). If player P stops first, he pays M the amount *κ*˜+(*τ*+, *Zτ*<sup>+</sup> ) (resp. *g*(*eZ*<sup>0</sup> , *x*0)) when the game terminates before (resp. at) the end of the time horizon 0. The objective of P is to minimize his expected total payment to M whereas the objective of M is to maximize this quantity.

In addition to the relationship (9) between the value functions *v* and *w*, the optimal continuation region of the Dynkin game (11) coincides with the no-control region of the singular stochastic control problem (8) in the sense that if (*ξ*+,∗, *ξ*−,∗) is an optimal control of (8) and we define the stopping times *<sup>τ</sup>*+,<sup>∗</sup> <sup>=</sup> inf{*<sup>u</sup>* <sup>∈</sup> [*s*, 0) : *<sup>ξ</sup>*+,∗(*u*) <sup>&</sup>gt; <sup>0</sup>} and *<sup>τ</sup>*−,<sup>∗</sup> <sup>=</sup> inf{*<sup>u</sup>* <sup>∈</sup> [*s*, 0) : *<sup>ξ</sup>*−,∗(*u*) <sup>&</sup>gt; <sup>0</sup>} (inf <sup>∅</sup> <sup>=</sup> 0), then (*τ*+,∗, *<sup>τ</sup>*−,∗) is a *saddlepoint* of the game with the property that *w*(*s*, *z*, *x*) = *Is*,*z*,*x*(*τ*+,∗, *τ*−,∗).

## **2.2. Example: Reversible investment**

Before we outline the computational algorithm for solving the Dynkin game (11), we give an example in mathematical economics of a stochastic control problem which has the form (4). In the notation of (4), the problem of reversible investment is one in which a company, by adjusting its production capacity *xt* through expansion *ξ*<sup>+</sup> *<sup>t</sup>* and contraction *ξ*<sup>−</sup> *<sup>t</sup>* according to market fluctuations *St*, wishes to maximize its overall expected net profit *Et*,*S*,*<sup>x</sup> Jt*,*S*,*x*(*ξ*+, *ξ*−) over a finite horizon. The net profit of such an investment depends on the running production function *F*(*t*, *S*, *x*) of the actual capacity, the benefits of contraction *κ*−(*t*, *S*) ≡ *K*<sup>−</sup> < 0, and the cost of expansion *<sup>κ</sup>*+(*t*, *<sup>S</sup>*) <sup>≡</sup> *<sup>K</sup>*<sup>+</sup> <sup>&</sup>gt; 0, with *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *<sup>K</sup>*<sup>−</sup> <sup>&</sup>gt; 0. The economic uncertainty about *St* (such as the price or demand for the product) is modeled by geometric Brownian motion (1).

with Transaction Costs 9

the largest terminal date of interest, take small positive *δ* and *�* such that *N* := *σ*2*T*max/*δ* is

with *<sup>s</sup>*<sup>0</sup> = 0, *si* = *si*−<sup>1</sup> − *<sup>δ</sup>*, *<sup>z</sup>* ∈ **<sup>Z</sup>***δ*, *<sup>x</sup>* ∈ **<sup>X</sup>***�*, the continuation value at *si* can be computed using

with *w*(0, *z*, *x*) = *g*(*ez*, *x*). The following algorithm allows us to solve for *X*b(*si*, *z*) and *X*s(*si*, *z*)

**Algorithm 1.** Let *<sup>w</sup>*(0, *<sup>z</sup>*, *<sup>x</sup>*) = *<sup>g</sup>*(*ez*, *<sup>x</sup>*) for *<sup>z</sup>* <sup>∈</sup> **<sup>Z</sup>***<sup>δ</sup>* and *<sup>x</sup>* <sup>∈</sup> **<sup>X</sup>***�*. For *<sup>i</sup>* <sup>=</sup> 1, 2, . . . , *<sup>N</sup>* and *<sup>z</sup>* <sup>∈</sup> **<sup>Z</sup>***δ*: (i) Starting at *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> **<sup>X</sup>***�* with *<sup>w</sup>*˜(*si*, *<sup>z</sup>*, *<sup>x</sup>*0) <sup>&</sup>lt; *<sup>κ</sup>*˜+(*si*, *<sup>z</sup>*), search for the first *<sup>j</sup>* ∈ {1, 2, . . . }

which *w*˜(*si*, *z*, *xj*) ≥ −*κ*˜−(*si*, *z*) and set *X*s(*si*, *z*) = *X*b(*si*, *z*) + *j*

−*κ*˜−(*si*, *z*) according to whether *x* ≤ *X*b(*si*, *z*) or *x* ≥ *X*s(*si*, *z*).

The following backward induction equation summarizes this algorithm:

⎧ ⎪⎨

⎪⎩

*w*(*si*, *z*, *x*) =

(ii) For *j* ∈ {1, 2, . . . }, let *xj* = *X*b(*si*, *z*) + *j�*. Compute, and store for use at *si*+1, *w*(*si*, *z*, *xj*) = *w*˜(*si*, *z*, *xj*) as defined by (12). Search for the first *j* (denoted by *j*

(iii) For *<sup>x</sup>* <sup>∈</sup> **<sup>X</sup>***�* outside the interval [*X*b(*si*, *<sup>z</sup>*), *<sup>X</sup>*s(*si*, *<sup>z</sup>*)], set *<sup>w</sup>*(*si*, *<sup>z</sup>*, *<sup>x</sup>*) = *<sup>κ</sup>*˜+(*si*, *<sup>z</sup>*) or

*w*˜(*si*, *z*, *x*) otherwise.

Lai et al. [17] have shown that under suitable conditions discrete-time random walk approximations to continuous-time optimal stopping problems can approximate the value function with an error of the order *O*(*δ*) and the stopping boundary with an error of the

walk. To prove this result, they approximate the underlying optimal stopping problem by a recursively defined family of "canonical" optimal stopping problems which depend on *δ* and for which the continuation and stopping regions can be completely characterized, and use an induction argument to provide bounds on the absolute difference between the boundaries of the continuous-time and discrete-time stopping problems as well as that between the value functions of the two problems. Since the Dynkin game is also an optimal stopping problem (with two stopping boundaries), their result can be extended to the present setting to establish that (13) is able to approximate the value function (11) with an error of the order *O*(*δ*) and

refinements to the random walk approximations can be made by correcting for the excess over the boundary when stopping cocurs in the discrete-time problem; for details and other

Consider now a risk-averse investor who trades in a risky asset whose price is given by the geometric Brownian motion (1) and a bond which pays a fixed risk-free rate *r* > 0 with the

Algorithm 1 is able to approximate the stopping boundaries *Zi*(*s*, *x*) := *X*−<sup>1</sup>

applications, see Chernoff [4], Chernoff & Petkau [5, 6] and Lai et al. [17].

corresponding to the optimal stopping problem (11) with an error of the order *o*(

**3. Utility-based option theory in the presence of transaction costs**

<sup>√</sup>*δ*), where *<sup>δ</sup>* is the interval width in discretizing time for the approximating random

<sup>∗</sup>) for which *<sup>w</sup>*˜(*si*, *<sup>z</sup>*, *<sup>x</sup>*<sup>0</sup> <sup>+</sup> *<sup>j</sup>�*) <sup>≥</sup> *<sup>κ</sup>*˜+(*si*, *<sup>z</sup>*) and set *<sup>X</sup>*b(*si*, *<sup>z</sup>*) = *<sup>x</sup>*<sup>0</sup> <sup>+</sup> *<sup>j</sup>*

*κ*˜+(*si*, *z*) if *w*˜(*si*, *z*, *x*) < *κ*˜+(*si*, *z*), −*κ*˜−(*si*, *z*) if *w*˜(*si*, *z*, *x*) > −*κ*˜−(*si*, *z*),

<sup>√</sup>*δ*,... } and **<sup>X</sup>***�* <sup>=</sup> {0, <sup>±</sup>*�*, <sup>±</sup>2*�*,... }. For *<sup>i</sup>* <sup>=</sup> 1, 2, . . . , *<sup>N</sup>*,

Singular Stochastic Control in Option Hedging with Transaction Costs 111

*<sup>δ</sup>*, *<sup>x</sup>*) + *<sup>w</sup>*(*si* <sup>+</sup> *<sup>δ</sup>*, *<sup>z</sup>* <sup>−</sup> <sup>√</sup>

∗*�*.

*δ*, *x*)]/2 (12)

∗*�*.

∗) for

(13)

*<sup>i</sup>* (*s*, *z*), *i* = b, s,

<sup>√</sup>*δ*). Further

<sup>√</sup>*δ*, <sup>±</sup><sup>2</sup>

*<sup>w</sup>*˜(*si*, *<sup>z</sup>*, *<sup>x</sup>*) = *δφ*(*si*, *<sup>z</sup>*, *<sup>x</sup>*)+[*w*(*si* <sup>+</sup> *<sup>δ</sup>*, *<sup>z</sup>* <sup>+</sup> <sup>√</sup>

an integer, and let **Z***<sup>δ</sup>* = {0, ±

(denoted by *j*

for *z* ∈ **Z***δ*.

order *o*(

Guo & Tomecek [21] studied the infinite-horizon (*T* = ∞) reversible investment problem and provided an explicit solution to the problem with the so-called Cobb-Douglas production function *<sup>F</sup>*(*t*, *<sup>S</sup>*, *<sup>x</sup>*) = *<sup>S</sup>λxμ*, where *<sup>λ</sup>* <sup>∈</sup> (0, *<sup>n</sup>*), *<sup>n</sup>* = [−(*<sup>α</sup>* <sup>−</sup> *<sup>σ</sup>*2/2) + (*<sup>α</sup>* <sup>−</sup> *<sup>σ</sup>*2/2)<sup>2</sup> <sup>−</sup> <sup>2</sup>*σ*2*r*]/*σ*<sup>2</sup> <sup>&</sup>gt; 0 and *μ* ∈ (0, 1]. The optimal strategy is for the company to increase (resp. decrease) capacity when (*S*, *x*) belongs to the investment (resp. disinvestment) region B (resp. S). Here, <sup>B</sup> <sup>=</sup> {(*S*, *<sup>x</sup>*) : *<sup>x</sup>* <sup>≤</sup> *<sup>X</sup>*b(*S*)} and <sup>S</sup> <sup>=</sup> {(*S*, *<sup>x</sup>*) : *<sup>x</sup>* <sup>≥</sup> *<sup>X</sup>*s(*S*)}, where *Xi*(*S*)=(*S*/*νi*)*λ*/(1−*α*), *i* = b, s, *ν*<sup>b</sup> and *ν*<sup>s</sup> are unique solutions to

$$\begin{aligned} \frac{\alpha}{\lambda - m} (\nu\_{\mathbf{b}}^{\lambda - m} - \nu\_{\mathbf{s}}^{\lambda - m}) &= -\frac{r}{m} (K^+ \nu\_{\mathbf{b}}^{-m} + K^- \nu\_{\mathbf{s}}^{-m}),\\ \frac{\alpha}{n - \lambda} (\nu\_{\mathbf{b}}^{\lambda - n} - \nu\_{\mathbf{s}}^{\lambda - n}) &= \frac{r}{n} (K^+ \nu\_{\mathbf{b}}^{-n} + K^- \nu\_{\mathbf{s}}^{-n}),\end{aligned}$$

and *<sup>m</sup>* = [−(*<sup>α</sup>* <sup>−</sup> *<sup>σ</sup>*2/2) <sup>−</sup> (*<sup>α</sup>* <sup>−</sup> *<sup>σ</sup>*2/2)<sup>2</sup> <sup>−</sup> <sup>2</sup>*σ*2*r*]/*σ*<sup>2</sup> <sup>&</sup>lt; 0.

In the case of finite horizon (*T* < ∞), the investment and disinvestment regions have similar forms but are not stationary in time, i.e., B = {(*t*, *S*, *x*) : *x* ≤ *X*b(*t*, *S*)} and S = {(*t*, *S*, *x*) : *x* ≥ *X*s(*t*, *S*)}. It is not possible to express the boundaries *Xi*(*t*, *S*) explicitly. We can solve for them (after applying the change of variables (*t*, *S*, *x*) �→ (*s*, *z*, *x*) given by (2)) by making use of the backward induction algorithm described in the next section; for details and numerical results, see Lai et al. [16].

### **2.3. Computational algorithm for solving the Dynkin game**

In view of the equivalence between the stochastic control problem (8) and the Dynkin game (11), which is an optimal stopping problem with (disjoint) stopping regions B and S coinciding with the control regions of (8) as well as continuation region N coinciding with the no-control region of (8), it suffices to solve (11) rather than (8) directly. The backward induction algorithm outlined below is similar to the algorithms studied by Lai et al. [17], for which convergence properties have been established.

Suppose the buy and sell boundaries can be expressed as functions *X*b(*s*, *z*) and *X*s(*s*, *z*) such that B = {(*s*, *z*, *x*) : *x* ≤ *X*b(*s*, *z*)} and S = {(*s*, *z*, *x*) : *x* ≥ *X*s(*s*, *z*)}. Whereas *w*(*s*, *z*, *x*) is given by (10b) and (10c) in the buy and sell regions (i.e., the stopping region in the Dynkin game), the continuation value of the Dynkin game is a solution of the partial differential equation (10a) and can be computed using backward induction on a symmetric Bernoulli random walk which approximates standard Brownian motion. Specifically, let *T*max denote the largest terminal date of interest, take small positive *δ* and *�* such that *N* := *σ*2*T*max/*δ* is an integer, and let **Z***<sup>δ</sup>* = {0, ± <sup>√</sup>*δ*, <sup>±</sup><sup>2</sup> <sup>√</sup>*δ*,... } and **<sup>X</sup>***�* <sup>=</sup> {0, <sup>±</sup>*�*, <sup>±</sup>2*�*,... }. For *<sup>i</sup>* <sup>=</sup> 1, 2, . . . , *<sup>N</sup>*, with *<sup>s</sup>*<sup>0</sup> = 0, *si* = *si*−<sup>1</sup> − *<sup>δ</sup>*, *<sup>z</sup>* ∈ **<sup>Z</sup>***δ*, *<sup>x</sup>* ∈ **<sup>X</sup>***�*, the continuation value at *si* can be computed using

$$\tilde{w}(\mathbf{s}\_{l},\mathbf{z},\mathbf{x}) = \delta\phi(\mathbf{s}\_{l},\mathbf{z},\mathbf{x}) + \left[w(\mathbf{s}\_{l}+\delta,\mathbf{z}+\sqrt{\delta},\mathbf{x}) + w(\mathbf{s}\_{l}+\delta,\mathbf{z}-\sqrt{\delta},\mathbf{x})\right]/2\tag{12}$$

with *w*(0, *z*, *x*) = *g*(*ez*, *x*). The following algorithm allows us to solve for *X*b(*si*, *z*) and *X*s(*si*, *z*) for *z* ∈ **Z***δ*.

**Algorithm 1.** Let *<sup>w</sup>*(0, *<sup>z</sup>*, *<sup>x</sup>*) = *<sup>g</sup>*(*ez*, *<sup>x</sup>*) for *<sup>z</sup>* <sup>∈</sup> **<sup>Z</sup>***<sup>δ</sup>* and *<sup>x</sup>* <sup>∈</sup> **<sup>X</sup>***�*. For *<sup>i</sup>* <sup>=</sup> 1, 2, . . . , *<sup>N</sup>* and *<sup>z</sup>* <sup>∈</sup> **<sup>Z</sup>***δ*:


The following backward induction equation summarizes this algorithm:

8 Will-be-set-by-IN-TECH

Before we outline the computational algorithm for solving the Dynkin game (11), we give an example in mathematical economics of a stochastic control problem which has the form (4). In the notation of (4), the problem of reversible investment is one in which a company, by

market fluctuations *St*, wishes to maximize its overall expected net profit *Et*,*S*,*<sup>x</sup> Jt*,*S*,*x*(*ξ*+, *ξ*−) over a finite horizon. The net profit of such an investment depends on the running production function *F*(*t*, *S*, *x*) of the actual capacity, the benefits of contraction *κ*−(*t*, *S*) ≡ *K*<sup>−</sup> < 0, and the cost of expansion *<sup>κ</sup>*+(*t*, *<sup>S</sup>*) <sup>≡</sup> *<sup>K</sup>*<sup>+</sup> <sup>&</sup>gt; 0, with *<sup>K</sup>*<sup>+</sup> <sup>+</sup> *<sup>K</sup>*<sup>−</sup> <sup>&</sup>gt; 0. The economic uncertainty about *St* (such as the price or demand for the product) is modeled by geometric Brownian motion (1). Guo & Tomecek [21] studied the infinite-horizon (*T* = ∞) reversible investment problem and provided an explicit solution to the problem with the so-called Cobb-Douglas production function *<sup>F</sup>*(*t*, *<sup>S</sup>*, *<sup>x</sup>*) = *<sup>S</sup>λxμ*, where *<sup>λ</sup>* <sup>∈</sup> (0, *<sup>n</sup>*), *<sup>n</sup>* = [−(*<sup>α</sup>* <sup>−</sup> *<sup>σ</sup>*2/2) + (*<sup>α</sup>* <sup>−</sup> *<sup>σ</sup>*2/2)<sup>2</sup> <sup>−</sup> <sup>2</sup>*σ*2*r*]/*σ*<sup>2</sup> <sup>&</sup>gt; 0 and *μ* ∈ (0, 1]. The optimal strategy is for the company to increase (resp. decrease) capacity when (*S*, *x*) belongs to the investment (resp. disinvestment) region B (resp. S). Here, <sup>B</sup> <sup>=</sup> {(*S*, *<sup>x</sup>*) : *<sup>x</sup>* <sup>≤</sup> *<sup>X</sup>*b(*S*)} and <sup>S</sup> <sup>=</sup> {(*S*, *<sup>x</sup>*) : *<sup>x</sup>* <sup>≥</sup> *<sup>X</sup>*s(*S*)}, where *Xi*(*S*)=(*S*/*νi*)*λ*/(1−*α*),

<sup>b</sup> <sup>−</sup> *<sup>ν</sup>λ*−*<sup>m</sup>* <sup>s</sup> ) = <sup>−</sup> *<sup>r</sup>*

<sup>b</sup> <sup>−</sup> *<sup>ν</sup>λ*−*<sup>n</sup>* <sup>s</sup> ) = *<sup>r</sup>*

In the case of finite horizon (*T* < ∞), the investment and disinvestment regions have similar forms but are not stationary in time, i.e., B = {(*t*, *S*, *x*) : *x* ≤ *X*b(*t*, *S*)} and S = {(*t*, *S*, *x*) : *x* ≥ *X*s(*t*, *S*)}. It is not possible to express the boundaries *Xi*(*t*, *S*) explicitly. We can solve for them (after applying the change of variables (*t*, *S*, *x*) �→ (*s*, *z*, *x*) given by (2)) by making use of the backward induction algorithm described in the next section; for details and numerical results,

In view of the equivalence between the stochastic control problem (8) and the Dynkin game (11), which is an optimal stopping problem with (disjoint) stopping regions B and S coinciding with the control regions of (8) as well as continuation region N coinciding with the no-control region of (8), it suffices to solve (11) rather than (8) directly. The backward induction algorithm outlined below is similar to the algorithms studied by Lai et al. [17], for

Suppose the buy and sell boundaries can be expressed as functions *X*b(*s*, *z*) and *X*s(*s*, *z*) such that B = {(*s*, *z*, *x*) : *x* ≤ *X*b(*s*, *z*)} and S = {(*s*, *z*, *x*) : *x* ≥ *X*s(*s*, *z*)}. Whereas *w*(*s*, *z*, *x*) is given by (10b) and (10c) in the buy and sell regions (i.e., the stopping region in the Dynkin game), the continuation value of the Dynkin game is a solution of the partial differential equation (10a) and can be computed using backward induction on a symmetric Bernoulli random walk which approximates standard Brownian motion. Specifically, let *T*max denote

*<sup>m</sup>* (*K*+*ν*−*<sup>m</sup>*

*<sup>n</sup>* (*K*+*ν*−*<sup>n</sup>*

<sup>b</sup> <sup>+</sup> *<sup>K</sup>*−*ν*−*m*<sup>s</sup> ),

<sup>b</sup> <sup>+</sup> *<sup>K</sup>*−*ν*−*<sup>n</sup>* <sup>s</sup> ),

*<sup>t</sup>* and contraction *ξ*<sup>−</sup>

110 Stochastic Modeling and Control Singular Stochastic Control in Option Hedging

*<sup>t</sup>* according to

**2.2. Example: Reversible investment**

*i* = b, s, *ν*<sup>b</sup> and *ν*<sup>s</sup> are unique solutions to

see Lai et al. [16].

*α <sup>λ</sup>* <sup>−</sup> *<sup>m</sup>* (*νλ*−*<sup>m</sup>*

*α <sup>n</sup>* <sup>−</sup> *<sup>λ</sup>* (*νλ*−*<sup>n</sup>*

and *<sup>m</sup>* = [−(*<sup>α</sup>* <sup>−</sup> *<sup>σ</sup>*2/2) <sup>−</sup> (*<sup>α</sup>* <sup>−</sup> *<sup>σ</sup>*2/2)<sup>2</sup> <sup>−</sup> <sup>2</sup>*σ*2*r*]/*σ*<sup>2</sup> <sup>&</sup>lt; 0.

which convergence properties have been established.

**2.3. Computational algorithm for solving the Dynkin game**

adjusting its production capacity *xt* through expansion *ξ*<sup>+</sup>

$$w(s\_{i\prime}z, \mathbf{x}) = \begin{cases} \mathfrak{K}^+(s\_{i\prime}z) & \text{if } \tilde{w}(s\_{i\prime}z, \mathbf{x}) < \mathfrak{K}^+(s\_{i\prime}z), \\ -\tilde{\mathfrak{K}}^-(s\_{i\prime}z) & \text{if } \tilde{w}(s\_{i\prime}z, \mathbf{x}) > -\tilde{\mathfrak{K}}^-(s\_{i\prime}z), \\ \tilde{w}(s\_{i\prime}z, \mathbf{x}) & \text{otherwise.} \end{cases} \tag{13}$$

Lai et al. [17] have shown that under suitable conditions discrete-time random walk approximations to continuous-time optimal stopping problems can approximate the value function with an error of the order *O*(*δ*) and the stopping boundary with an error of the order *o*( <sup>√</sup>*δ*), where *<sup>δ</sup>* is the interval width in discretizing time for the approximating random walk. To prove this result, they approximate the underlying optimal stopping problem by a recursively defined family of "canonical" optimal stopping problems which depend on *δ* and for which the continuation and stopping regions can be completely characterized, and use an induction argument to provide bounds on the absolute difference between the boundaries of the continuous-time and discrete-time stopping problems as well as that between the value functions of the two problems. Since the Dynkin game is also an optimal stopping problem (with two stopping boundaries), their result can be extended to the present setting to establish that (13) is able to approximate the value function (11) with an error of the order *O*(*δ*) and Algorithm 1 is able to approximate the stopping boundaries *Zi*(*s*, *x*) := *X*−<sup>1</sup> *<sup>i</sup>* (*s*, *z*), *i* = b, s, corresponding to the optimal stopping problem (11) with an error of the order *o*( <sup>√</sup>*δ*). Further refinements to the random walk approximations can be made by correcting for the excess over the boundary when stopping cocurs in the discrete-time problem; for details and other applications, see Chernoff [4], Chernoff & Petkau [5, 6] and Lai et al. [17].

#### **3. Utility-based option theory in the presence of transaction costs**

Consider now a risk-averse investor who trades in a risky asset whose price is given by the geometric Brownian motion (1) and a bond which pays a fixed risk-free rate *r* > 0 with the

#### 10 Will-be-set-by-IN-TECH 112 Stochastic Modeling and Control Singular Stochastic Control in Option Hedging

objective of maximizing the expected utility of his terminal wealth Ω<sup>0</sup> *<sup>T</sup>*. The number of units *xt* the investor holds in the asset is given by (3), where *x*<sup>0</sup> denotes the initial asset position and *ξ*+ *<sup>t</sup>* (resp. *ξ*<sup>−</sup> *<sup>t</sup>* ) represents the cumulative number of units of the asset bought (resp. sold) within the time interval [0, *t*], 0 ≤ *t* ≤ *T*. If the investor pays fractions 0 < *λ* < 1 and 0 < *μ* < 1 of the dollar value transacted on purchase and sale of the asset, the dollar value *yt* of his investment in bond is given by

$$dy\_t = ry\_t \, dt - aS\_t \, d\xi\_t^+ + bS\_t \, d\xi\_t^- \, d\xi\_t^- $$

with *a* = 1 + *λ* and *b* = 1 − *μ*, or more explicitly,

$$y\_T = y\_t e^{r(T-t)} - a \int\_{[t,T)} e^{r(T-u)} S\_{\underline{u}} \, d\xi\_{\underline{u}}^{\pm} + b \int\_{[t,T)} e^{r(T-u)} S\_{\underline{u}} \, d\xi\_{\underline{u}}^{-}.$$

Let *U* : **R** → **R** be a concave and increasing (hence risk-averse) utility function. We can express the investor's problem in terms of the value function

$$\mathcal{V}^{0}(t, S, \mathbf{x}, y) = \sup\_{(\mathfrak{k}^{+}, \mathfrak{k}^{-}) \in \mathcal{A}\_{t, \mathfrak{k}, y}} \mathcal{E}\_{t, S, \mathbf{x}, y} \left[ \mathcal{U} \left( \Omega\_{T}^{0} \right) \right], \tag{14}$$

with Transaction Costs 11

with *<sup>D</sup>*s(*z*) = *<sup>I</sup>*{*z*>0} (short call) and *<sup>D</sup>*b(*z*) = <sup>−</sup>*I*{*z*>0} (long call). For use in (22d) below, we

settlement. In the sequel, we fix *i* = 0, s, b and drop the superscript *i* in the value functions

The formulation of the option (pricing and) hedging problem as two stochastic control problems of the form (14) goes back to Hodges & Neuberger [15]. Davis et al. [14] derived the Hamilton-Jacobi-Bellman (HJB) equations associated with the control problems *V*(*t*, *S*, *x*, *y*) and *H*(*t*, *S*, *x*) = 1 − *V*(*t*, *S*, *x*, 0) in the same way as we did in Section 2.1. By applying the transformation *h*(*s*, *z*, *x*) = *H*(*t*, *S*, *x*) to their HJB equations, we obtain the following free

*<sup>w</sup>*b(*s*, *<sup>z</sup>*, *<sup>x</sup>*) = <sup>−</sup>*aγKez*+(*β*−*ρ*−1/2)*<sup>s</sup>*

*<sup>w</sup>*s(*s*, *<sup>z</sup>*, *<sup>x</sup>*) = <sup>−</sup>*bγKez*+(*β*−*ρ*−1/2)*<sup>s</sup>*

Associated with FBP (19) are three regions: B = {(*s*, *z*, *x*) : *x* ≤ *X*b(*s*, *z*)} where it is optimal to buy the (risky) asset, S = {(*s*, *z*, *x*) : *x* ≥ *X*s(*s*, *z*)} where it is optimal to sell the asset, and <sup>N</sup> = [−*σ*2*T*, 0] <sup>×</sup> **<sup>R</sup>** <sup>×</sup> **<sup>R</sup>** \ (B∪S) where it is optimal to not transact. Since *<sup>∂</sup>*/*∂<sup>s</sup>* + (1/2)*∂*2/*∂z*<sup>2</sup> is the infinitesimal generator of space-time Brownian motion, this means that while (*s*, *Zs*, *xs*) is inside the no-transaction region, the dynamics of *h*(*s*, *Zs*, *xs*) is driven by the standard Brownian motion {*Zs*, *s* ≤ 0}. In the buy and sell regions, it follows from (19b) and (19c)

[*<sup>x</sup>* <sup>−</sup> *<sup>X</sup>*b(*s*, *<sup>z</sup>*)]

[*<sup>x</sup>* <sup>−</sup> *<sup>X</sup>*s(*s*, *<sup>z</sup>*)]

(*z*)) under asset settlement and *B<sup>i</sup>*

(*z*) for asset settlement, *i* = s, b,

Singular Stochastic Control in Option Hedging with Transaction Costs 113

(*z*) for cash settlement, *i* = s, b,

*<sup>∂</sup>z*<sup>2</sup> <sup>=</sup> 0, *<sup>x</sup>* <sup>∈</sup> [*X*b(*s*, *<sup>z</sup>*), *<sup>X</sup>*s(*s*, *<sup>z</sup>*)], (19a)

(*s*, *z*, *x*) = *w*b(*s*, *z*, *x*), *x* ≤ *X*b(*s*, *z*), (19b)

(*s*, *z*, *x*) = *w*s(*s*, *z*, *x*), *x* ≥ *X*s(*s*, *z*), (19c) *h*(0, *z*, *x*) = exp{−*γKA*(*z*, *x*)}, (19d)

(*z*, *x*) = *B*0(*z*, *x*) under cash

*h*(*s*, *z*, *x*), (20a)

*h*(*s*, *z*, *x*). (20b)

*h*(*s*, *z*, *X*b(*s*, *z*)), *x* ≤ *X*b(*s*, *z*), (21a)

*h*(*s*, *z*, *X*s(*s*, *z*)), *x* ≥ *X*s(*s*, *z*). (21b)

(*z*, *x*)/*∂x*, which are given explicitly by *B*0(*z*, *x*) = *A*0(*z*, *x*)/*x* and, for

(*z*)) + *D<sup>i</sup>*

*<sup>z</sup>* <sup>−</sup> <sup>1</sup>)*D<sup>i</sup>*

definitions (16)–(18) of terminal settlement value are

(*z*, *<sup>x</sup>*) = *<sup>A</sup>*0(*z*, *<sup>x</sup>* <sup>−</sup> *<sup>D</sup><sup>i</sup>*

(*z*, *<sup>x</sup>*) = *<sup>A</sup>*0(*z*, *<sup>x</sup>*) <sup>−</sup> (*<sup>e</sup>*

*Ai*

*Ai*

(*z*, *x*) := *∂A<sup>i</sup>*

(and associated quantities).

(*z*, *<sup>x</sup>*) = *<sup>B</sup>*0(*z*, *<sup>x</sup>* <sup>−</sup> *<sup>D</sup><sup>i</sup>*

boundary problem (FBP) for *h*(*s*, *z*, *x*):

*∂h ∂s* + 1 2 *∂*2*h*

<sup>−</sup>*aγKez*+(*β*−*ρ*−1/2)*<sup>s</sup>*

<sup>−</sup>*bγKez*+(*β*−*ρ*−1/2)*<sup>s</sup>*

*∂h ∂x*

*∂h ∂x*

define *B<sup>i</sup>*

where

that

*<sup>h</sup>*(*s*, *<sup>z</sup>*, *<sup>x</sup>*) = exp

*<sup>h</sup>*(*s*, *<sup>z</sup>*, *<sup>x</sup>*) = exp

*i* = s, b, *B<sup>i</sup>*

*<sup>A</sup>*0(*z*, *<sup>x</sup>*) = *xez*(*aI*{*x*<0} <sup>+</sup> *bI*{*x*≥0}),

**3.1. Associated free boundary problems and their solutions**

where A*t*,*x*,*<sup>y</sup>* denotes the set of all admissible controls which satisfy *xt* = *x* and *yt* = *y*, and *Et*,*S*,*x*,*<sup>y</sup>* denotes conditional expectation given *St* = *S*, *xt* = *x* and *yt* = *y*. For the special case of the negative exponential utility function *<sup>U</sup>*(*z*) = <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*γz*, which has constant absolute risk aversion (CARA) *γ*, we can reduce the number of state variables by one by working with

$$\begin{split} &H^{0}(t,\mathbf{S},\mathbf{x}) = 1 - V^{0}(t,\mathbf{S},\mathbf{x},\mathbf{0}) \\ &= \inf\_{(\tilde{\mathbf{x}}^{+},\tilde{\mathbf{x}}^{-}) \in \mathcal{A}\_{l,\mathbf{x}}} E\_{l,\mathbf{S},\mathbf{x}} \left[ \exp\left\{ \gamma \left( \int\_{[t,T]} e^{r(\mathbf{T}-\boldsymbol{\mu})} \mathbf{S}\_{\mathbf{u}} (\boldsymbol{a} \, d\xi^{+}\_{\mathbf{S}\boldsymbol{u}} - \boldsymbol{b} \, d\xi^{-}\_{\mathbf{u}}) - \mathbf{Z}^{0} (\mathbf{S}\_{\mathbf{T}},\mathbf{x}\_{T}) \right) \right\} \right], \end{split} \tag{15}$$

where

$$\mathbf{x}^{0}(\mathbf{S}, \mathbf{x}) = \mathbf{x}\mathbf{S}(aI\_{\{\mathbf{x} < 0\}} + bI\_{\{\mathbf{x} \ge 0\}}) \tag{16}$$

denotes the liquidated value of the asset by trading *x* units of the asset at price *S* to zero unit.

If the investor is presented with an opportunity to enter into a position in a European call option written on the given asset, with strike price *K* and expiration date *T*, the problem can be formulated in the same way as (14) but with Ω<sup>0</sup> *<sup>T</sup>* replaced by <sup>Ω</sup>*<sup>i</sup> <sup>T</sup>* with *i* = s indicating a short call position and *i* = b indicating a long call position. The corresponding value functions *Vi* (*t*, *S*, *x*, *y*) also admit reductions in dimensionality via *H<sup>i</sup>* (*t*, *<sup>S</sup>*, *<sup>x</sup>*) = <sup>1</sup> <sup>−</sup> *<sup>V</sup><sup>i</sup>* (*t*, *S*, *x*, 0), but with *Z*0(*ST*, *xT*) in (15) replaced by *Z<sup>i</sup>* (*ST*, *xT*). If the option is *asset settled*, then the writer delivers one unit of the asset in return for a payment of *K* when the buyer exercises the option at maturity *T*, so

$$Z^i(\mathbf{S}, \mathbf{x}) = Z^0(\mathbf{S}, \mathbf{x} - \Delta^i(\mathbf{S})) + \mathbf{K} \Delta^i(\mathbf{S}), \quad i = \mathbf{s}, \mathbf{b}, \tag{17}$$

where <sup>Δ</sup>s(*S*) = *<sup>I</sup>*{*S*>*K*} (short call) and <sup>Δ</sup>b(*S*) = <sup>−</sup>*I*{*S*>*K*} (long call). In the case of a *cash settled* option, the writer delivers (*ST* <sup>−</sup> *<sup>K</sup>*)<sup>+</sup> in cash at *<sup>T</sup>*, so

$$Z^{i}(\mathbf{S},\mathbf{x}) = Z^{0}(\mathbf{S},\mathbf{x}) - (\mathbf{S} - \mathbf{K})\boldsymbol{\Delta}^{i}(\mathbf{S}), \quad i = \mathbf{s}\_{i}\mathbf{b}. \tag{18}$$

As in Section 2.1, we apply the change of variables (2) (with *ρ* replaced by *β* = *α*/*σ*2) to (15) and work with the resulting value function *h<sup>i</sup>* (*s*, *z*, *x*) = *H<sup>i</sup>* (*t*, *S*, *x*). Corresponding to the definitions (16)–(18) of terminal settlement value are

$$\begin{aligned} A^0(z, \mathbf{x}) &= \mathbf{x} e^z (a I\_{\{\mathbf{x} < 0\}} + b I\_{\{\mathbf{x} \ge 0\}}), \\ A^i(z, \mathbf{x}) &= A^0(z, \mathbf{x} - D^i(z)) + D^i(z) \quad \text{for asset settlement}, \quad i = \mathbf{s}, \mathbf{b}, \\ A^i(z, \mathbf{x}) &= A^0(z, \mathbf{x}) - (\mathbf{c}^z - 1) D^i(z) \quad \text{for cash settlement}, \quad i = \mathbf{s}, \mathbf{b}, \end{aligned}$$

with *<sup>D</sup>*s(*z*) = *<sup>I</sup>*{*z*>0} (short call) and *<sup>D</sup>*b(*z*) = <sup>−</sup>*I*{*z*>0} (long call). For use in (22d) below, we define *B<sup>i</sup>* (*z*, *x*) := *∂A<sup>i</sup>* (*z*, *x*)/*∂x*, which are given explicitly by *B*0(*z*, *x*) = *A*0(*z*, *x*)/*x* and, for *i* = s, b, *B<sup>i</sup>* (*z*, *<sup>x</sup>*) = *<sup>B</sup>*0(*z*, *<sup>x</sup>* <sup>−</sup> *<sup>D</sup><sup>i</sup>* (*z*)) under asset settlement and *B<sup>i</sup>* (*z*, *x*) = *B*0(*z*, *x*) under cash settlement. In the sequel, we fix *i* = 0, s, b and drop the superscript *i* in the value functions (and associated quantities).

#### **3.1. Associated free boundary problems and their solutions**

The formulation of the option (pricing and) hedging problem as two stochastic control problems of the form (14) goes back to Hodges & Neuberger [15]. Davis et al. [14] derived the Hamilton-Jacobi-Bellman (HJB) equations associated with the control problems *V*(*t*, *S*, *x*, *y*) and *H*(*t*, *S*, *x*) = 1 − *V*(*t*, *S*, *x*, 0) in the same way as we did in Section 2.1. By applying the transformation *h*(*s*, *z*, *x*) = *H*(*t*, *S*, *x*) to their HJB equations, we obtain the following free boundary problem (FBP) for *h*(*s*, *z*, *x*):

$$\frac{\partial h}{\partial \mathbf{s}} + \frac{1}{2} \frac{\partial^2 h}{\partial z^2} = 0, \qquad \qquad \mathbf{x} \in [X\_{\mathbf{b}}(\mathbf{s}, z), X\_{\mathbf{s}}(\mathbf{s}, z)], \tag{19a}$$

$$\frac{\partial \mathbf{b}}{\partial \mathbf{x}}(\mathbf{s}, \mathbf{z}, \mathbf{x}) = w\_{\mathbf{b}}(\mathbf{s}, \mathbf{z}, \mathbf{x}), \tag{19b} \qquad \qquad \mathbf{x} \le \mathbf{X}\_{\mathbf{b}}(\mathbf{s}, \mathbf{z}), \tag{19b}$$

$$\frac{\partial \mathbf{h}}{\partial \mathbf{x}}(\mathbf{s}, z, \mathbf{x}) = w\_{\mathbf{s}}(\mathbf{s}, z, \mathbf{x}), \quad \qquad \qquad \mathbf{x} \ge \mathbf{X}\_{\mathbf{s}}(\mathbf{s}, z), \tag{19c}$$

$$h(0, z, \mathbf{x}) = \exp\{-\gamma K A(z, \mathbf{x})\},\tag{19d}$$

where

10 Will-be-set-by-IN-TECH

*xt* the investor holds in the asset is given by (3), where *x*<sup>0</sup> denotes the initial asset position and

the time interval [0, *t*], 0 ≤ *t* ≤ *T*. If the investor pays fractions 0 < *λ* < 1 and 0 < *μ* < 1 of the dollar value transacted on purchase and sale of the asset, the dollar value *yt* of his investment

*dyt* <sup>=</sup> *ryt dt* <sup>−</sup> *aSt <sup>d</sup>ξ*<sup>+</sup>

*<sup>t</sup>* ) represents the cumulative number of units of the asset bought (resp. sold) within

*Su dξ*<sup>+</sup> *<sup>u</sup>* + *b* [*t*,*T*) *e r*(*T*−*u*)

Let *U* : **R** → **R** be a concave and increasing (hence risk-averse) utility function. We can

(*ξ*+,*ξ*−)∈A*t*,*x*,*<sup>y</sup>*

where A*t*,*x*,*<sup>y</sup>* denotes the set of all admissible controls which satisfy *xt* = *x* and *yt* = *y*, and *Et*,*S*,*x*,*<sup>y</sup>* denotes conditional expectation given *St* = *S*, *xt* = *x* and *yt* = *y*. For the special case of the negative exponential utility function *<sup>U</sup>*(*z*) = <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*γz*, which has constant absolute risk aversion (CARA) *γ*, we can reduce the number of state variables by one by working with

denotes the liquidated value of the asset by trading *x* units of the asset at price *S* to zero unit. If the investor is presented with an opportunity to enter into a position in a European call option written on the given asset, with strike price *K* and expiration date *T*, the problem can

short call position and *i* = b indicating a long call position. The corresponding value functions

delivers one unit of the asset in return for a payment of *K* when the buyer exercises the option

where <sup>Δ</sup>s(*S*) = *<sup>I</sup>*{*S*>*K*} (short call) and <sup>Δ</sup>b(*S*) = <sup>−</sup>*I*{*S*>*K*} (long call). In the case of a *cash settled*

As in Section 2.1, we apply the change of variables (2) (with *ρ* replaced by *β* = *α*/*σ*2) to

(*S*, *<sup>x</sup>*) = *<sup>Z</sup>*0(*S*, *<sup>x</sup>*) <sup>−</sup> (*<sup>S</sup>* <sup>−</sup> *<sup>K</sup>*)Δ*<sup>i</sup>*

(*S*)) + *K*Δ*<sup>i</sup>*

(*s*, *z*, *x*) = *H<sup>i</sup>*

*<sup>t</sup>* + *bSt dξ*<sup>−</sup>

*Et*,*S*,*x*,*<sup>y</sup> U* Ω0 *T* 

*Su*(*a dξ*<sup>+</sup>

*<sup>u</sup>* − *b dξ*<sup>−</sup>

*<sup>Z</sup>*0(*S*, *<sup>x</sup>*) = *xS*(*aI*{*x*<0} <sup>+</sup> *bI*{*x*≥0}) (16)

*<sup>T</sup>* replaced by <sup>Ω</sup>*<sup>i</sup>*

*t* ,

*Su dξ*− *u* .

112 Stochastic Modeling and Control Singular Stochastic Control in Option Hedging

*<sup>u</sup>* ) <sup>−</sup> *<sup>Z</sup>*0(*ST*, *xT*)

(*t*, *<sup>S</sup>*, *<sup>x</sup>*) = <sup>1</sup> <sup>−</sup> *<sup>V</sup><sup>i</sup>*

(*S*), *i* = s, b, (17)

(*S*), *i* = s, b. (18)

(*t*, *S*, *x*). Corresponding to the

(*ST*, *xT*). If the option is *asset settled*, then the writer

*<sup>T</sup>*. The number of units

, (14)

*<sup>T</sup>* with *i* = s indicating a

(*t*, *S*, *x*, 0), but

, (15)

objective of maximizing the expected utility of his terminal wealth Ω<sup>0</sup>

 [*t*,*T*) *e r*(*T*−*u*)

*V*0(*t*, *S*, *x*, *y*) = sup

[*t*,*T*) *e r*(*T*−*u*)

with *a* = 1 + *λ* and *b* = 1 − *μ*, or more explicitly,

*<sup>r</sup>*(*T*−*t*) <sup>−</sup> *<sup>a</sup>*

express the investor's problem in terms of the value function

*yT* = *yte*

*<sup>H</sup>*0(*t*, *<sup>S</sup>*, *<sup>x</sup>*) = <sup>1</sup> <sup>−</sup> *<sup>V</sup>*0(*t*, *<sup>S</sup>*, *<sup>x</sup>*, 0)

with *Z*0(*ST*, *xT*) in (15) replaced by *Z<sup>i</sup>*

*Et*,*S*,*<sup>x</sup>* exp *γ* 

be formulated in the same way as (14) but with Ω<sup>0</sup>

*Zi*

option, the writer delivers (*ST* <sup>−</sup> *<sup>K</sup>*)<sup>+</sup> in cash at *<sup>T</sup>*, so

*Zi*

(15) and work with the resulting value function *h<sup>i</sup>*

(*t*, *S*, *x*, *y*) also admit reductions in dimensionality via *H<sup>i</sup>*

(*S*, *<sup>x</sup>*) = *<sup>Z</sup>*0(*S*, *<sup>x</sup>* <sup>−</sup> <sup>Δ</sup>*<sup>i</sup>*

= inf (*ξ*+,*ξ*−)∈A*t*,*<sup>x</sup>*

where

*Vi*

at maturity *T*, so

*ξ*+

*<sup>t</sup>* (resp. *ξ*<sup>−</sup>

in bond is given by

$$w\_{\mathbf{b}}(\mathbf{s}, z, \mathbf{x}) = -a\gamma \mathbf{K} e^{z + (\theta - \rho - 1/2)s} h(\mathbf{s}, z, \mathbf{x}),\tag{20a}$$

$$w\_8(\mathbf{s}, z, \mathbf{x}) = -b\gamma \mathbf{K} e^{z + (\beta - \rho - 1/2)s} h(\mathbf{s}, z, \mathbf{x}).\tag{20b}$$

Associated with FBP (19) are three regions: B = {(*s*, *z*, *x*) : *x* ≤ *X*b(*s*, *z*)} where it is optimal to buy the (risky) asset, S = {(*s*, *z*, *x*) : *x* ≥ *X*s(*s*, *z*)} where it is optimal to sell the asset, and <sup>N</sup> = [−*σ*2*T*, 0] <sup>×</sup> **<sup>R</sup>** <sup>×</sup> **<sup>R</sup>** \ (B∪S) where it is optimal to not transact. Since *<sup>∂</sup>*/*∂<sup>s</sup>* + (1/2)*∂*2/*∂z*<sup>2</sup> is the infinitesimal generator of space-time Brownian motion, this means that while (*s*, *Zs*, *xs*) is inside the no-transaction region, the dynamics of *h*(*s*, *Zs*, *xs*) is driven by the standard Brownian motion {*Zs*, *s* ≤ 0}. In the buy and sell regions, it follows from (19b) and (19c) that

$$h(\mathbf{s}, z, \mathbf{x}) = \exp\left\{-a\gamma K e^{z + (\theta - \rho - 1/2)\mathbf{s}} [\mathbf{x} - \mathbf{X}\_{\mathbf{b}}(\mathbf{s}, z)] \right\} h(\mathbf{s}, z, \mathbf{X}\_{\mathbf{b}}(\mathbf{s}, z)), \quad \mathbf{x} \le \mathbf{X}\_{\mathbf{b}}(\mathbf{s}, z), \tag{21a}$$

$$h(\mathbf{s}, z, \mathbf{x}) = \exp\left\{-b\gamma \mathbf{K} e^{z + (\beta - \rho - 1/2)\mathbf{s}} [\mathbf{x} - \mathbf{X}\_{\mathbf{s}}(\mathbf{s}, z)] \right\} h(\mathbf{s}, z, \mathbf{X}\_{\mathbf{s}}(\mathbf{s}, z)), \quad \mathbf{x} \ge \mathbf{X}\_{\mathbf{s}}(\mathbf{s}, z). \tag{21b}$$

#### 12 Will-be-set-by-IN-TECH 114 Stochastic Modeling and Control Singular Stochastic Control in Option Hedging

Finally, if we let *w*(*s*, *z*, *x*) = *∂h*(*s*, *z*, *x*)/*∂x*, then *w*(*s*, *z*, *x*) satisfies the FBP

$$\frac{\partial w}{\partial \mathbf{s}} + \frac{1}{2} \frac{\partial^2 w}{\partial z^2} = 0, \qquad \qquad \mathbf{x} \in [\mathbf{X\_b(s, z)}, \mathbf{X\_s(s, z)}], \tag{22a}$$

with Transaction Costs 13

continuous-time control problems with an error of the order *O*(*δ*), and Algorithm 2 is able

an investor with no option position and *i* = s, b being analogs of (14) corresponding to an investor with a short or long call position, respectively, is an optimal trading strategy *x<sup>i</sup>*

> *X*b(*t*, *St*) if *xt*<sup>−</sup> < *X*b(*t*, *St*), *X*s(*t*, *St*) if *xt*<sup>−</sup> > *X*s(*t*, *St*),

The optimal *hedging* strategies for the option writer and buyer are then given by *x*<sup>s</sup>

*α* − *r <sup>σ</sup>*<sup>2</sup> , *<sup>x</sup><sup>i</sup>*

(2)) in Section 1. In the case of *α* = *r* (risk-neutrality), it can be shown that *x*<sup>0</sup>

*xt*<sup>−</sup> if *X*b(*t*, *St*) ≤ *xt*<sup>−</sup> ≤ *X*s(*t*, *St*).

*<sup>t</sup>* = <sup>Δ</sup>*<sup>i</sup>*

*<sup>t</sup>* , respectively. In the case of no transaction costs (*λ* = *μ* = 0), it can be shown that

where <sup>Δ</sup>b(*t*, *<sup>S</sup>*) (resp. <sup>Δ</sup>s(*t*, *<sup>S</sup>*) = <sup>−</sup>Δb(*t*, *<sup>S</sup>*)) denotes the Black-Scholes delta for a long (resp. short) call option, given explicitly (as a function of (*s*, *z*) after applying the change of variable

or not there are transaction costs. In particular, if *α* = *r* and *λ* = *μ* = 0, the optimal hedging strategy is to hold Δ shares of stock at all times (see Section 1). Thus, the Black-Scholes option

Whereas Clewlow & Hodges [11] and Zakamouline [20] made use of discrete-time dynamic programming on an approximating binomial tree for the asset price to solve the control problems directly for the optimal hedge, Lai & Lim [18] made use of the simpler Algorithm 2 outlined in Section 3.1. They provided extensive numerical results for the CARA utility function with *α* = *r*, for which only one pair of boundaries need to be computed (since it is then optimal not to trade in the risky asset when the investor does not have an option position). As an illustration of Algorithm 2, we compute and show in Fig. 1 the optimal buy (lower) and sell (upper) boundaries for a short asset-settled call (solid black lines) with strike price *K* = 20 and for the case of no option (solid red lines) at four different times before expiration (*T* − *t* = 1.5, 0.5, 0.25, 0.1) when proportional transaction costs are incurred at the rate of *<sup>λ</sup>* <sup>=</sup> *<sup>μ</sup>* <sup>=</sup> 0.5%; the dashed lines correspond to *<sup>X</sup>*0(*t*, ·) and <sup>Δ</sup>s(*t*, ·) + *<sup>X</sup>*0(*t*, ·) for the case of no transaction costs. Other parameters are: absolute risk aversion *γ* = 2.0, risk-free rate *r* = 8.5%, asset return rate *α* = 10% and asset volatility *σ* = 5%. Note that the red boundaries are consistent with the intuitive notion of "buy at the low and sell at the high" when investing only in a risky asset (and bond). However, unlike the case of *α* = *r* in which the buy and sell boundaries corresponding to a short asset-settled call always lie between 0 and 1, the black boundaries in this case (where *α* �= *r*) do not necessarily take values in the interval [0, 1].

*<sup>i</sup>* (*s*, *z*), *i* = b, s, corresponding to

*t*

*<sup>t</sup>* <sup>−</sup> *<sup>x</sup>*<sup>0</sup>

*<sup>t</sup>* ≡ 0 whether

*<sup>t</sup>* and

<sup>√</sup>*δ*); see Lai et al. [19] for details as well

(*t*, *S*, *x*, *y*), *i* = 0 given by (14) corresponding to

Singular Stochastic Control in Option Hedging with Transaction Costs 115

(*t*, *S*) + *X*0(*t*, *S*), *i* = s, b,

to approximate the buy and sell boundaries *Zi*(*s*, *x*) := *X*−<sup>1</sup>

as an extension to the problem of optimal investment and consumption.

these control problems with an error of the order *o*(

Associated with each of the problems *V<sup>i</sup>*

*xt* =

*<sup>t</sup>* <sup>=</sup> *<sup>X</sup>*0(*t*, *<sup>S</sup>*) :<sup>=</sup> *<sup>e</sup>*−*r*(*T*−*t*)

⎧ ⎪⎨

⎪⎩

*γS*

theory is a special case of the more general utility-based option theory.

**3.2. Numerical results**

(*i* = 0, s, b) of the form

*x*0

*x*b *<sup>t</sup>* <sup>−</sup> *<sup>x</sup>*<sup>0</sup>

$$w(s, z, \mathbf{x}) = w\_{\mathbf{b}}(s, z, \mathbf{x}), \tag{22b} \quad \text{and} \qquad \mathbf{x} \le \mathbf{X}\_{\mathbf{b}}(s, z), \tag{22b}$$

$$w(\mathbf{s}, z, \mathbf{x}) = w\_{\mathbf{s}}(\mathbf{s}, z, \mathbf{x}), \tag{22c} \tag{22c} \\ \mathbf{x} \ge \mathbf{X}\_{\mathbf{s}}(\mathbf{s}, z), \tag{22c}$$

$$w(0, z, \mathbf{x}) = -\gamma K B(z, \mathbf{x}) h(0, z, \mathbf{x}).\tag{22d}$$

If the function *h*(*s*, *z*, *x*) is known, then by analogy to (10), the FBP (22) is an optimal stopping problem associated with a Dynkin game, and its solution can be computed using the following analog of the backward induction equation (13):

$$w(s\_i, z, \mathbf{x}) = \begin{cases} w\_{\mathbf{b}}(s\_i, z, \mathbf{x}) & \text{if } \tilde{w}(s\_i, z, \mathbf{x}) < w\_{\mathbf{b}}(s\_i, z, \mathbf{x}), \\ w\_{\mathbf{s}}(s\_i, z, \mathbf{x}) & \text{if } \tilde{w}(s\_i, z, \mathbf{x}) > w\_{\mathbf{s}}(s\_i, z, \mathbf{x}), \\ \tilde{w}(s\_i, z, \mathbf{x}) & \text{otherwise}, \end{cases} \tag{23}$$

where *w*˜(*s*, *z*, *x*) is given by (12) with *φ* ≡ 0. On the other hand, if the boundaries *X*b(*s*, *z*) and *X*s(*s*, *z*) are given, then the FBP (19) can also be solved by backward induction: For *z* ∈ **Z***δ*, compute *h*(*si*, *z*, *x*) using (21) (with *s* replaced by *si*) if *x* ∈ **X***�* is outside the interval [*X*b(*si*, *<sup>z</sup>*), *<sup>X</sup>*s(*si*, *<sup>z</sup>*)], and if *<sup>x</sup>* <sup>∈</sup> **<sup>X</sup>***�* <sup>∩</sup> [*X*b(*si*, *<sup>z</sup>*), *<sup>X</sup>*s(*si*, *<sup>z</sup>*)], let *<sup>h</sup>*(*si*, *<sup>z</sup>*, *<sup>x</sup>*) = ˜ *h*(*si*, *z*, *x*) with

$$\tilde{h}(s, z, \mathbf{x}) = \left[ \hbar(s + \delta, z + \sqrt{\delta}, \mathbf{x}) + \hbar(s + \delta, z - \sqrt{\delta}, \mathbf{x}) \right] / 2. \tag{24}$$

By replacing the unknown *h* in (20a) and (20b) by ˜ *h* and redefining them as *w*˜ <sup>b</sup> and *w*˜ s, Lai & Lim [18] have developed the coupled backward induction algorithm described below to solve for *X*b(*si*, *z*) and *X*s(*si*, *z*), as well as to compute values of *h*(*si*, *z*, *x*) for *x* ∈ **X***�* ∩ [*X*b(*si*, *z*), *X*s(*si*, *z*)].

**Algorithm 2.** Let *h*(0, *z*, *x*) = exp{−*γKA*(*z*, *x*)} and *w*(0, *z*, *x*) = −*γKB*(*z*, *x*)*h*(0, *z*, *x*) for *z* ∈ **Z***<sup>δ</sup>* and *x* ∈ **X***�*. For *i* = 1, 2, . . . , *N* and *z* ∈ **Z***δ*:


It can be established that the convergence property of this algorithm is similar to that of Algorithm 1 even though (22) is not a stopping problem like (10) is. Specifically, the backward inductions (23) as well as (24) and (21) applied to {*sN* <sup>=</sup> <sup>−</sup>*σ*2*T*,*sN*−1,...,*s*1,*s*<sup>0</sup> <sup>=</sup> <sup>0</sup>} × **<sup>Z</sup>***<sup>δ</sup>* <sup>×</sup> **<sup>X</sup>***�* are able to approximate the value functions *w*(*s*, *z*, *x*) and *h*(*s*, *z*, *x*) of the corresponding continuous-time control problems with an error of the order *O*(*δ*), and Algorithm 2 is able to approximate the buy and sell boundaries *Zi*(*s*, *x*) := *X*−<sup>1</sup> *<sup>i</sup>* (*s*, *z*), *i* = b, s, corresponding to these control problems with an error of the order *o*( <sup>√</sup>*δ*); see Lai et al. [19] for details as well as an extension to the problem of optimal investment and consumption.

#### **3.2. Numerical results**

12 Will-be-set-by-IN-TECH

If the function *h*(*s*, *z*, *x*) is known, then by analogy to (10), the FBP (22) is an optimal stopping problem associated with a Dynkin game, and its solution can be computed using the following

*w*˜(*si*, *z*, *x*) otherwise,

where *w*˜(*s*, *z*, *x*) is given by (12) with *φ* ≡ 0. On the other hand, if the boundaries *X*b(*s*, *z*) and *X*s(*s*, *z*) are given, then the FBP (19) can also be solved by backward induction: For *z* ∈ **Z***δ*, compute *h*(*si*, *z*, *x*) using (21) (with *s* replaced by *si*) if *x* ∈ **X***�* is outside the interval

Lai & Lim [18] have developed the coupled backward induction algorithm described below to solve for *X*b(*si*, *z*) and *X*s(*si*, *z*), as well as to compute values of *h*(*si*, *z*, *x*) for *x* ∈ **X***�* ∩

**Algorithm 2.** Let *h*(0, *z*, *x*) = exp{−*γKA*(*z*, *x*)} and *w*(0, *z*, *x*) = −*γKB*(*z*, *x*)*h*(0, *z*, *x*) for *z* ∈

(i) Starting at *x*<sup>0</sup> ∈ **X***�* with *w*˜(*si*, *z*, *x*0) < *w*˜ <sup>b</sup>(*si*, *z*, *x*0), search for the first *j* ∈ {1, 2, . . . }

(ii) For *j* ∈ {1, 2, . . . }, let *xj* = *X*b(*si*, *z*) + *j�*. Compute, and store for use at *si*+1, *<sup>w</sup>*(*si*, *<sup>z</sup>*, *xj*) = *<sup>w</sup>*˜(*si*, *<sup>z</sup>*, *xj*) as defined by (12) with *<sup>φ</sup>* <sup>≡</sup> 0 and *<sup>h</sup>*(*si*, *<sup>z</sup>*, *xj*) = ˜

(iii) For *x* ∈ **X***�* outside the interval [*X*b(*si*, *z*), *X*s(*si*, *z*)], compute *h*(*si*, *z*, *x*) using (21) and set *w*(*si*, *z*, *x*) = *w*b(*si*, *z*, *x*) or *w*s(*si*, *z*, *x*) as defined by (20) according to whether *x* ≤

It can be established that the convergence property of this algorithm is similar to that of Algorithm 1 even though (22) is not a stopping problem like (10) is. Specifically, the backward inductions (23) as well as (24) and (21) applied to {*sN* <sup>=</sup> <sup>−</sup>*σ*2*T*,*sN*−1,...,*s*1,*s*<sup>0</sup> <sup>=</sup> <sup>0</sup>} × **<sup>Z</sup>***<sup>δ</sup>* <sup>×</sup> **<sup>X</sup>***�* are able to approximate the value functions *w*(*s*, *z*, *x*) and *h*(*s*, *z*, *x*) of the corresponding

*<sup>∂</sup>z*<sup>2</sup> <sup>=</sup> 0, *<sup>x</sup>* <sup>∈</sup> [*X*b(*s*, *<sup>z</sup>*), *<sup>X</sup>*s(*s*, *<sup>z</sup>*)], (22a)

114 Stochastic Modeling and Control Singular Stochastic Control in Option Hedging

(23)

*h*(*si*, *z*, *xj*)

*h*(*si*, *z*, *x*) with

*h* and redefining them as *w*˜ <sup>b</sup> and *w*˜ s,

<sup>∗</sup>) for which *w*˜(*si*, *z*, *xj*) ≥ *w*˜ <sup>s</sup>(*si*, *z*, *xj*) and set

*δ*, *x*)]/2. (24)

*w*(*s*, *z*, *x*) = *w*b(*s*, *z*, *x*), *x* ≤ *X*b(*s*, *z*), (22b) *w*(*s*, *z*, *x*) = *w*s(*s*, *z*, *x*), *x* ≥ *X*s(*s*, *z*), (22c) *w*(0, *z*, *x*) = −*γKB*(*z*, *x*)*h*(0, *z*, *x*). (22d)

> *w*b(*si*, *z*, *x*) if *w*˜(*si*, *z*, *x*) < *w*b(*si*, *z*, *x*), *w*s(*si*, *z*, *x*) if *w*˜(*si*, *z*, *x*) > *w*s(*si*, *z*, *x*),

> > *<sup>δ</sup>*, *<sup>x</sup>*) + *<sup>h</sup>*(*<sup>s</sup>* <sup>+</sup> *<sup>δ</sup>*, *<sup>z</sup>* <sup>−</sup> <sup>√</sup>

<sup>∗</sup>) for which *w*˜(*si*, *z*, *x*<sup>0</sup> + *j�*) ≥ *w*˜ <sup>b</sup>(*si*, *z*, *x*<sup>0</sup> + *j�*) and set *X*b(*si*, *z*) = *xi* +

Finally, if we let *w*(*s*, *z*, *x*) = *∂h*(*s*, *z*, *x*)/*∂x*, then *w*(*s*, *z*, *x*) satisfies the FBP

*∂w ∂s* + 1 2 *∂*2*w*

analog of the backward induction equation (13):

*w*(*si*, *z*, *x*) =

˜

[*X*b(*si*, *z*), *X*s(*si*, *z*)].

*j* ∗*�*.

(denoted by *j*

*X*s(*si*, *z*) = *X*b(*si*, *z*) + *j*

*X*b(*si*, *z*) or *x* ≥ *X*s(*si*, *z*).

⎧ ⎪⎨

⎪⎩

[*X*b(*si*, *<sup>z</sup>*), *<sup>X</sup>*s(*si*, *<sup>z</sup>*)], and if *<sup>x</sup>* <sup>∈</sup> **<sup>X</sup>***�* <sup>∩</sup> [*X*b(*si*, *<sup>z</sup>*), *<sup>X</sup>*s(*si*, *<sup>z</sup>*)], let *<sup>h</sup>*(*si*, *<sup>z</sup>*, *<sup>x</sup>*) = ˜

*<sup>h</sup>*(*s*, *<sup>z</sup>*, *<sup>x</sup>*)=[*h*(*<sup>s</sup>* <sup>+</sup> *<sup>δ</sup>*, *<sup>z</sup>* <sup>+</sup> <sup>√</sup>

By replacing the unknown *h* in (20a) and (20b) by ˜

**Z***<sup>δ</sup>* and *x* ∈ **X***�*. For *i* = 1, 2, . . . , *N* and *z* ∈ **Z***δ*:

by (24). Search for the first *j* (denoted by *j*

∗*�*.

Associated with each of the problems *V<sup>i</sup>* (*t*, *S*, *x*, *y*), *i* = 0 given by (14) corresponding to an investor with no option position and *i* = s, b being analogs of (14) corresponding to an investor with a short or long call position, respectively, is an optimal trading strategy *x<sup>i</sup> t* (*i* = 0, s, b) of the form

$$\mathbf{x}\_{t} = \begin{cases} \mathbf{X}\_{\mathbf{b}}(t, \mathbf{S}\_{t}) & \text{if } \mathbf{x}\_{t-} < \mathbf{X}\_{\mathbf{b}}(t, \mathbf{S}\_{t}), \\ \mathbf{X}\_{\mathbf{s}}(t, \mathbf{S}\_{t}) & \text{if } \mathbf{x}\_{t-} > \mathbf{X}\_{\mathbf{s}}(t, \mathbf{S}\_{t}), \\ \mathbf{x}\_{t-} & \text{if } \mathbf{X}\_{\mathbf{b}}(t, \mathbf{S}\_{t}) \le \mathbf{x}\_{t-} \le \mathbf{X}\_{\mathbf{s}}(t, \mathbf{S}\_{t}). \end{cases}$$

The optimal *hedging* strategies for the option writer and buyer are then given by *x*<sup>s</sup> *<sup>t</sup>* <sup>−</sup> *<sup>x</sup>*<sup>0</sup> *<sup>t</sup>* and *x*b *<sup>t</sup>* <sup>−</sup> *<sup>x</sup>*<sup>0</sup> *<sup>t</sup>* , respectively. In the case of no transaction costs (*λ* = *μ* = 0), it can be shown that

$$\mathbf{x}\_t^0 = \mathbf{X}\_0(t, \mathbf{S}) := \frac{e^{-r(T-t)}}{\gamma \mathbf{S}} \frac{\mathbf{a} - r}{\sigma^2}, \quad \mathbf{x}\_t^i = \boldsymbol{\Delta}^i(t, \mathbf{S}) + \mathbf{X}\_0(t, \mathbf{S}), \quad i = \mathbf{s}, \mathbf{b}\_{\prime\prime}$$

where <sup>Δ</sup>b(*t*, *<sup>S</sup>*) (resp. <sup>Δ</sup>s(*t*, *<sup>S</sup>*) = <sup>−</sup>Δb(*t*, *<sup>S</sup>*)) denotes the Black-Scholes delta for a long (resp. short) call option, given explicitly (as a function of (*s*, *z*) after applying the change of variable (2)) in Section 1. In the case of *α* = *r* (risk-neutrality), it can be shown that *x*<sup>0</sup> *<sup>t</sup>* ≡ 0 whether or not there are transaction costs. In particular, if *α* = *r* and *λ* = *μ* = 0, the optimal hedging strategy is to hold Δ shares of stock at all times (see Section 1). Thus, the Black-Scholes option theory is a special case of the more general utility-based option theory.

Whereas Clewlow & Hodges [11] and Zakamouline [20] made use of discrete-time dynamic programming on an approximating binomial tree for the asset price to solve the control problems directly for the optimal hedge, Lai & Lim [18] made use of the simpler Algorithm 2 outlined in Section 3.1. They provided extensive numerical results for the CARA utility function with *α* = *r*, for which only one pair of boundaries need to be computed (since it is then optimal not to trade in the risky asset when the investor does not have an option position). As an illustration of Algorithm 2, we compute and show in Fig. 1 the optimal buy (lower) and sell (upper) boundaries for a short asset-settled call (solid black lines) with strike price *K* = 20 and for the case of no option (solid red lines) at four different times before expiration (*T* − *t* = 1.5, 0.5, 0.25, 0.1) when proportional transaction costs are incurred at the rate of *<sup>λ</sup>* <sup>=</sup> *<sup>μ</sup>* <sup>=</sup> 0.5%; the dashed lines correspond to *<sup>X</sup>*0(*t*, ·) and <sup>Δ</sup>s(*t*, ·) + *<sup>X</sup>*0(*t*, ·) for the case of no transaction costs. Other parameters are: absolute risk aversion *γ* = 2.0, risk-free rate *r* = 8.5%, asset return rate *α* = 10% and asset volatility *σ* = 5%. Note that the red boundaries are consistent with the intuitive notion of "buy at the low and sell at the high" when investing only in a risky asset (and bond). However, unlike the case of *α* = *r* in which the buy and sell boundaries corresponding to a short asset-settled call always lie between 0 and 1, the black boundaries in this case (where *α* �= *r*) do not necessarily take values in the interval [0, 1].

with Transaction Costs 15

Singular Stochastic Control in Option Hedging with Transaction Costs 117

used on certain control problems for which there does not exist an equivalent stopping problem. We show how the "standard" algorithm can be modified to provide a coupled backward induction algorithm for solving the utility-based option hedging problem and

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**Author details**

*Department of Statistics Stanford University, U.S.A.*

Tze Leung Lai

Tiong Wee Lim

**5. References**

**Figure 1.** Optimal buy (lower) and sell (upper) boundaries from negative exponential (CARA) utility maximization for a short asset-settled call with strike price *K* = 20 (solid black lines) and for the case of no option (solid red lines), with proportional transaction costs incurred at the rate of *λ* = *μ* = 0.5%, absolute risk aversion *γ* = 2.0, risk-free rate *r* = 8.5%, asset return rate *α* = 10% and asset volatility *σ* = 5%, at 1.5, 0.5, 0.25 and 0.1 period(s) from expiration *T*. For each pair of boundaries, the "buy asset" region is below the buy boundary and the "sell asset" region is above the sell boundary; the no-transaction region is between the two boundaries. The dashed lines correspond to the case of no transaction costs.

#### **4. Conclusion**

For the so-called bounded variation follower problems, the equivalence between singular stochastic control and optimal stopping can be harnessed to provide a much simpler solution to the control problem by solving the corresponding Dynkin game. This approach can be used on certain control problems for which there does not exist an equivalent stopping problem. We show how the "standard" algorithm can be modified to provide a coupled backward induction algorithm for solving the utility-based option hedging problem and provide numerical illustrations on the vanilla call option.
