**Acknowledgement**

I like to acknowledge my Supervisor, Professor J.O. Afolayan, for his guidance, teaching, patience, encouragement, suggestions and continued support, particularly for arousing my interest in Risk Analysis (Reliability Analysis of Structures).

**Chapter 0**

**Chapter 6**

**Singular Stochastic Control in Option Hedging**

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

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

*dSt* = *αSt dt* + *σSt dWt*, *S*<sup>0</sup> > 0, (1)

©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

©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.

**with Transaction Costs**

Tze Leung Lai and Tiong Wee Lim

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

**1. Introduction**

equation

Additional information is available at the end of the chapter

cited.
