Abstract

A first-order feedback model of option pricing consisting of a coupled system of two PDEs, a nonliner generalised Black-Scholes equation and the classical Black-Scholes equation, is studied using Lie symmetry analysis. This model arises as an extension of the classical Black-Scholes model when liquidity is incorporated into the market. We compute the admitted Lie point symmetries of the system and construct an optimal system of the associated one-dimensional subalgebras. We also construct some invariant solutions of the model.

1. Introduction

Extensions of the Black-Scholes equation typically lead to nonlinear PDEs that are often tackled with numerical methods. In instances when analytic solutions exist, Lie symmetry methods [1–12] may be employed to construct the solutions. The model we consider in this paper arises as an extension of the Black-Scholes equation when liquidity is incorporated into the market. We start by introducing the Black-Scholes equation to provide a context. The Black-Scholes equation has played a very significant role in the development of modern derivative asset analysis. Based on a small set of assumptions on the price behavior of the underlying asset, the Black-Scholes equation allows investors to calculate the “fair” price of a derivative security. In its simplest form, the Black-Scholes equation requires the estimation of only one parameter that cannot be observed in the market, namely, the market volatility of the underlying asset price, which is assumed to be constant. Another assumption of the Black-Scholes model is that the market is perfectly liquid. However, increases in market volatility of asset prices have been observed in recent years and it seems they are caused by the extensive usage of the Black-Scholes model and the associated hedging strategies for pricing derivative securities [13]. This observed feedback effect of Black-Scholes pricing on the underlying’s price and consequently on the price of the derivatives has led to extensions to the Black-Scholes model aimed at accounting for this phenomenon [13–16]. In this paper we consider one such extension, the first-order feedback model [17].

2. The First-Order Feedback Model

In the classical Black-Scholes model the price process of the underlying asset is assumed to be governed by a geometric Brownian motion:where is the price of the underlying stock, and (assumed to be nonnegative constants) are the drift and volatility, respectively, and is a standard Brownian motion. To incorporate liquidity into the Black-Scholes we modify the underlying asset price process. We introduce a forcing term, , dependent on the stock price and time so that in place of (1) we havewhere is an arbitrary function that models the nature of price impact and liquidity [17].

Applying Itô’s formula on the function , we obtain which when substituted into (2) leads toBy simply squaring both sides of (4) and applying the usual rules, and [18], we obtainSubstituting (5) into (4) and rearranging, we arrive at a stochastic process analogous to (1):where Under this setting, where the stock price follows the modified stochastic process (6), it is deduced via standard arguments that , the derivative price, satisfies the generalised Black-Scholes equation (see Duffie [19] for more details):where is the risk-free interest rate, assumed to be a positive constant. When (8) is used as a model for the value of a European call option, for example, then at expiry time we have thatwhere is the strike price.

Consistent with standard Black-Scholes arguments, the drift of the modified process does not appear in the option pricing PDE. In the context of markets with finite elasticity, can be interpreted as the number of extra shares that should be held due to some deterministic hedging/trading strategy and hence specifies the number of shares needed to be bought or sold at time and price due to such a strategy. If we are interested in the price impact due to delta hedging then we can identify as the delta of the option being replicated; that is,This leads to the question of what strategy the hedgers are assumed to follow. Either they use the Black-Scholes option delta to hedge, ignoring price impact, or they try to incorporate price impact into the hedging strategy by using the modified delta. The first case is called the first-order feedback, and in this case (8) becomes a linear PDE:where is the solution to the standard Black-Scholes equation:Both (11) and (12) are subject to the same terminal condition (9), with being replaced by in the case of (12).

The other scenario, in which the hedger is assumed to be aware of the feedback effect and so would change the hedging strategy accordingly, corresponds to the case when . In this case the trading strategy adopted has to be found as part of the problem. This is called the full feedback and leads to the nonlinear PDE:We will apply Lie symmetry analysis to the first-order feedback model (11), which is coupled with the standard Black-Scholes equation (12). Investigation of solutions of differential equations via Lie symmetry analysis has been done to many problems in financial mathematics, for example, [20–24]. The primary objective of the present study is to determine general solutions (as invariant solutions) of the first-order feedback model (11).

3. The Admitted Lie Point Symmetries

For convenience, we let , , and represent , , and , respectively. Further, we use the standard substitution representing time to expiry of the option (and rename as ), so that (11) and (12) are represented byrespectively, with and . The constants , , and are to be considered nonzero.

Remark 1. System (15)-(16) is a generalisation somewhat of (11) and (12). The symmetries reported in (19) and Tables 1 and 2 are based on this generalised system.

We will apply the standard machinery of symmetry analysis to find particular solutions of (15)-(16). Let us consider the one-parameter Lie group of transformations in the -space given bywhere is the group parameter, with infinitesimal generatorFor (17) to be admitted by (15)-(16), one requires that the transformation leaves invariant the set of solutions of system (15)-(16). Applying program Lie [25], we obtain that the Lie algebra of infinitesimal generators of system (15)-(16) is spanned by the seven operators:Table 1 is the commutator table for the operators (19).

4. Determination of Invariant Solutions

An invariant solution is a solution that is mapped into itself by a group transformation. Such solutions are very important in that the majority of exact solutions that have important real world applications are reportedly invariant solutions [3, p. 29]. The algorithm for constructing invariant solutions of a given system of differential equations using admitted symmetries is well documented in many standard books on Lie symmetry analysis [1, 5, 7–9]. Each of the operators in (19) is of the formand generates a one-dimensional subalgebra of the algebra spanned by (19). We consider invariant solutions associated with . Variables in which (15)-(16) reduces to a coupled system of ordinary differential equations are determined by the invariants of , which are obtained by solving the quasilinear PDE:where is some function of the independent and dependent variables. The general solution of (21) iswhere is an arbitrary function andare functionally independent solutions of the characteristic systemThe invariants , , and in (23) are now used to determine new variables in which (15)-(16) should be written so that it is reduced to a second-order system of ODEs, the solution of which leads to a family of invariant solutions of (15)-(16) associated with .

4.1. Invariant Solutions of an Arbitrary Element of

In principle every element from can be used to construct an invariant solution of system (15)-(16). Take, for example, taken asin which case the infinitesimal coefficients areThe invariants of (25) are obtained as solutions of the characteristic equation (24), with the infinitesimals given in (26). We obtainThe functional form of the invariant solution is now constructed from these invariants. Taking the solution in the form we obtain that invariant solutions of (15)-(16) arising from (25) have the formwhere and are arbitrary functions. When and as prescribed in (29) are substituted in (15)-(16) we obtain the following pair of ODEs:These equations are easily solved as linear first-order ODEs, and we obtainwhere and are arbitrary constants. Therefore, invariant solutions of (15)-(16) arising from (25) are of the form where

4.2. Optimal System of One-Dimensional Subalgebras

How every one-dimensional subalgebra of the algebra may be used to find a family of invariant solutions of (15)-(16) has just been illustrated. The exercise of finding all such families from all possible elements of is reduced to that of finding families of invariant solutions associated with only a small number of inequivalent symmetries, called an optimal system of one-dimensional subalgebras [6, 8]. Two elements of are equivalent if the family of invariant solutions associated with one element can be transformed into a family of invariant solutions associated with the other by one of the Lie point symmetries of . We set out to construct an optimal system of one-dimensional subalgebras of .

Following Olver’s approach [8] we start by constructing the adjoint presentation of the Lie group generated by (19) on its Lie algebra. Each of the basis symmetries (19) generates an adjoint representation defined by the Lie seriesThe meaning of (34) is that the generator is equivalent to under the Lie group generated by . All the adjoint actions for the operators (19) are given in Table 2, with the th entry indicating .

We construct the optimal system via the naive approach of taking a general element and subjecting it to various adjoint transformations to simplify it as much as possible [8]. Given a nonzero vectorwe endeavour to simplify as many of the coefficients as possible through judicious applications of adjoint maps to . We deal with particular cases depending on which of the coefficients are nonzero.

4.3.

Scaling away from (35), since , we can write (35) equivalently asIf we now act on (36) by we obtainWe can make the coefficient of vanish by setting With this choice of , we obtainSimilarly in (39) is simplified further by using with We obtainFurther simplification of is achieved with followed by , with and . We obtain the following simplifications: No further simplification is possible through the action of adjoint maps. This means that every one-dimensional subalgebra generated by a general element with is equivalent to the subalgebra spanned byThe remaining subalgebras are spanned by vectors of the form (35) with .

4.4.

In this case the general element can be written in the formThe simplification of (44) is achieved as follows:where Note that none of the ’s in (47) can be eliminated by adjoint transformations, but we can scale away the coefficient of to obtain

4.5.

The general element has the form and is simplified as follows:

4.6.

Simplification of the general elementthrough the action of adjoint maps is not possible in this case!

4.7.

In this case, the general element is simplified through the adjoint map with , . The resulting operator is then divided by . We obtain

4.8.

For , we can only scale away the coefficient :We have therefore constructed an optimal system, (54), of one-dimensional subalgebras for the first-order feedback model (15)-(16):

5. Invariant Solutions of Elements of the Optimal System

In this section we construct invariant solutions associated with elements of the constructed optimal system. We follow the procedure outlined and illustrated in Section 4.

5.1. ,

Solving the characteristic equation, (24), corresponding to we obtain the invariants from which we deduce the form of the invariant solution,where is any solution to the Black-Scholes equation, (16), and is an arbitrary function. Taking, for example, the simple solutionof (16), (15) reduces to the ordinary differential equation:the solution of which is where is an arbitrary constant. Hence one family of invariant solutions of system (15)-(16) arising from have the form

Remark 2. We remark here that in place of (57) one could use any solution of (16). One could, for example, use an invariant solution of (16) such aswhich arises from the following symmetry of (15):

5.2. ,

Invariant solutions of system (15)-(16) arising from in this case are found to be of the form Substituting (63) in system (15)-(16) leads to the following set of ordinary differential equations:Solving (64) for , we obtainwhere and are arbitrary constants. The solution of (65) taking into account the (66) completes the process. It is however not easy to solve (65) analytically.

5.3.

In this case dictates that and that solve (15)-(16) take the formSubstituting (67) into system (15)-(16), we obtain the following pair of coupled ODEs:Equation (68) is a linear ODE and is solved to givewhere and are arbitrary constants. Equation (69) is to be solved for .