**1. Introduction**

30 Risk Management – Current Issues and Challenges

Insurance: Math. and Economics. 2010; 47(2): 190-197.

[28] Christiansen, M. Biometric worst-case scenarios for multi-state life insurance policies.

This work deals with the Dirichlet problem for some PDEs of second order with non-negative characteristic form. One main motivation is to study some boundary-value problems for PDEs of Black-Scholes type arising in the pricing problem for financial options of barrier type. Barrier options on stocks have been traded since the end of the Sixties and the market for these options has been dramatically expanding, making barrier options the most popular ones among the exotic. The class of standard barrier options includes 'in' barriers and 'out' barriers, which are activated (knocked in) and, respectively, extinguished (knocked out) if the underlying asset price crosses the barrier before expiration. Moreover, each class includes 'down' or 'up' options, depending on whether the barrier is below or above the current asset price and thus can be breached from above or below. Therefore there are eight types of standard barrier options, depending on their 'in' or 'out', 'down' or 'up', and 'call' or 'put' attributes. It is possible to include a cash rebate, which is paid out at option expiration if an 'in' ('out') option has not been knocked in (has been knocked out, respectively) during its lifetime. One can consider barrier options with rebates of several types, terminal payoffs of different forms (e.g. power options), more than one underlying assets and/or barriers, and allow for time-dependent barriers, thus enriching this class still further. On the other hand, a large variety of new exotic barriers have been designed to accommodate investors' preferences. Another motivation for the study of such options is related to credit risk theory. Several credit-risk models build on the barrier option formalism, since the default event can be modeled throughout a signalling variable hitting a pre-specified boundary value (See [3],[8] among others). As a consequence, a substantial body of academic literature provides pricing methods for valuating barrier options, starting from the seminal work of [18], where an exact formula is offered for a down-and-out European call with zero rebate. Further extensions

©2012 Slavova et al., 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 Slavova et al., 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 32 Risk Management – Current Issues and Challenges Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach <sup>3</sup>

are provided - among others - in [22] for the different types of standard barrier options, in [16] for simultaneous 'down' and 'up' barriers with exponential dependence on time, in [10] for two boundaries via Laplace transform, in [12] and [7] for partial barrier and rainbow options, in [17] for multi-asset options with an outside barrier, in [5] in a most comprehensive setting employing the image solution method. Many analytical formulas for barrier options are collected also in handbooks (see [11], for example).

where 0 ≤ *t* ≤ *T* and *xj* ≥ 0, 1 ≤ *j* ≤ *n*.

and this is the boundary value problem:

In (1) *aij* = *aji* = *const*, *bi* = *const*, *c* = *const* and

, *<sup>∂</sup>*2*<sup>u</sup> <sup>∂</sup>xi∂xj* <sup>=</sup> *<sup>e</sup>*−*yi*−*yj* [ *<sup>∂</sup>*2*<sup>u</sup> <sup>∂</sup>yi∂yj* <sup>−</sup> *<sup>δ</sup>ij*

*∂τ* <sup>+</sup>

*n* ∑ *i*,*j*=1 *aij*

> ⎧ ⎪⎨

> ⎪⎩

*Lu*˜ <sup>=</sup> <sup>−</sup> *<sup>∂</sup><sup>u</sup>*

*∂u ∂τ* <sup>=</sup> � *Lu* = *f*(*t*, *x*),

0 ≤ *t* ≤ *T*

*u*|*xj*=*aj* = *gj*(*t*, *x*)|*xj*=*aj*

0 ≤ *t* ≤ *T*

1 ≤ *j* ≤ *n*

Our first step is to make in the non-hypoelliptic PDE *L* the change of the space variables:

*∂u ∂yi*

*∂*2*u ∂yi∂yj*

> + *n* ∑ *i*=1 ˜ *bi ∂u ∂yi*

*<sup>u</sup>*|*τ*=<sup>0</sup> <sup>=</sup> *<sup>u</sup>*�0(*y*) *u*|*yj*=*a*˜*<sup>j</sup>* = *gj*|*yj*=*a*˜*<sup>j</sup>*

+ *n* ∑ *i*=1

� *Lu*˜ <sup>=</sup> *<sup>f</sup>* , 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>T</sup>*

*Lu*˜ <sup>=</sup> *<sup>f</sup>* , 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>T</sup>*

*aijξiξ<sup>j</sup>* ≥ *c*0|*ξ*|

0 ≤ *xj* ≤ *aj*, *aj* > 0

*xj* ≥ 0, 1 ≤ *j* ≤ *n*

�

*Lu* = *f*

*u*|*t*=*<sup>T</sup>* = *u*0(*x*)

⎧ ⎨ ⎩

*yj* = *lnxj*, 1 ≤ *j* ≤ *n*, *τ* = *T* − *t* ⇒

*n* ∑ *i*,*j*=1 *aij*

> *∂*2*u ∂yi∂yj*

⎧ ⎨ ⎩

> *n* ∑ *i*,*j*=1

*<sup>u</sup>*|*t*=*<sup>T</sup>* <sup>=</sup> *<sup>u</sup>*0(*x*) , (2)

Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach

2, *c*<sup>0</sup> = *const* > 0. (4)

*∂τ* , *yj* <sup>∈</sup> **<sup>R</sup>**<sup>1</sup> (5)

(*bi* − *aii*) + *cu* = *f* , (6)

*bi* = *bi* − *aii*

(3)

33

(8)

, 1 ≤ *j* ≤ *n*,

*∂u <sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup> *<sup>∂</sup><sup>u</sup>*

*∂u ∂yi*

<sup>+</sup> *cu* <sup>−</sup> *<sup>f</sup>* ; ˜

*<sup>u</sup>*|*τ*=<sup>0</sup> <sup>=</sup> *<sup>u</sup>*�0(*y*) = *<sup>u</sup>*0(*ey*<sup>1</sup> ,...*eyn* ), *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***n*, (7)

], *δij* being the Kronecker symbol.

This is the Cauchy problem:

*∂u*

i.e.

*<sup>∂</sup>xi* <sup>=</sup> *<sup>e</sup>*−*yi <sup>∂</sup><sup>u</sup>*

*∂yi*

Thus, (1) takes the form:

In the case (2) we have

while in the case (3)

For analytical tractability most literature assumes that the barrier hitting is monitored in continuous time. However there exist some works dealing with the discrete version, i.e. barrier crossing is allowed only at some specific dates -typically at daily closings. (See [1] and [15], for a survey). Furthermore, a recent literature relaxes the Brownian motion assumption and considers a more general Lévy framework. For example, [4] study barrier options of European type assuming that the returns of the underlying asset follows a Lévy process from a wide class. They employ the Wiener-Hopf factorization method and elements of pseudodifferential calculus to solve the related boundary problem. This book chapter adopts a classical Black-Scholes framework. The problem of pricing barrier options is reducible to boundary value problems for a PDE of Black-Scholes type and with pre-specified boundaries. The value at the terminal time *T* is assigned, specifying the terminal payoff which is paid provided that an 'in' option is knocked in or an 'out' option is not knocked out during its lifetime. The option holder may be entitled or not to a rebate. From a mathematical point of view, the boundary condition can be inhomogeneous or homogeneous. While there are several types of barrier options, in this work we will focus on 'up' barriers in view of the relationships between the prices of different types of vanilla options (see [25]). Moreover, the case of floating barriers of exponential form can be easily accommodated by substitution of the relevant parameters (see [25], Chapter 11), thus we confine ourselves to the case of constant barriers. On the other hand, we work within a general framework that allows for multi-asset options, a generic payoff and rebate. Furthermore, we tackle some regularity questions and the problem of existence of generalized solutions. In Section 2 the (initial) boundary value problem is studied in a multidimensional framework generalizing the Black-Scholes equation and analytical solutions are obtained, while a comparison principle is provided in Section 4. Section 3 presents some applications in Finance: our general setting incorporates several known pricing expressions and, at the same time, allows to generate new valuation formulas. Section 5 and the Appendix study the existence and regularity of generalized solutions to the boundary value problems for a class of PDEs incorporating the Black-Scholes type. We build on the approach of Oleinik and Radkeviˇc and adapt the method to the PDEs of interest in the financial applications.
