Lie Symmetry Analysis of a First-Order Feedback Model of Option Pricing

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 themarket.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.


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][2][3][4][5][6][7][8][9][10][11][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][14][15][16].In this paper we consider one such extension, the first-order feedback model [17].

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 have where (  , ) is an arbitrary function that models the nature of price impact and liquidity [17].Applying Itô's formula on the function (  , ), we obtain Advances in Mathematical Physics which when substituted into (2) leads to By simply squaring both sides of (4) and applying the usual rules,  ⋅  =  ⋅   =   ⋅  = 0 = 0 and   ⋅   =  [18], we obtain 2 +  () (as  → 0) .( 5) Substituting ( 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 that where  is the strike price.Consistent with standard Black-Scholes arguments, the drift of the modified process μ(  , ) 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  BS 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  BS 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][21][22][23][24].The primary objective of the present study is to determine general solutions (as invariant solutions) of the first-order feedback model (11).

The Admitted Lie Point Symmetries
For convenience, we let , V, and  represent  BS , , 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 by respectively, with  =  and  =  = .The constants , , and  are to be considered nonzero.
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 (, , , V)space given by where  is the group parameter, with infinitesimal generator For ( 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  7 of infinitesimal generators of system ( 15)-( 16) is spanned by the seven operators: Table 1 is the commutator table for the operators (19).

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][8][9].Each of the operators in ( 19) is of the form and 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) is where  is an arbitrary function and are functionally independent solutions of the characteristic system The invariants  1 ,  2 , and  3 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 .

Invariant Solutions of an Arbitrary
Element of  7 .In principle every element from  7 can be used to construct an invariant solution of system ( 15)-( 16).Take, for example,  taken as in which case the infinitesimal coefficients are The invariants of ( 25) are obtained as solutions of the characteristic equation (24), with the infinitesimals given in (26).We obtain The 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 form where  and  are arbitrary functions.When V 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 obtain where  1 and  2 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  7 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  7 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  7 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  7 .We set out to construct an optimal system of one-dimensional subalgebras of  7 .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 Ad(exp(  )) defined by the Lie series The meaning of (34) is that the generator Ad(exp(  ))  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 Ad(exp(  ))  .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 vector we 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.

𝑎
We can make the coefficient of  1 vanish by setting With this choice of  1 , we obtain Similarly   in (39) is simplified further by using Ad(  2  2 ) with We obtain Further simplification of   is achieved with Ad(  3  3 ) followed by Ad(  4  4 ), with  3 = 2 3 /(2 6 − 1) and  4 = 2 4 .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  7 ̸ = 0 is equivalent to the subalgebra spanned by   =  5  5 +  6  6 +  7 . (43) The remaining subalgebras are spanned by vectors of the form (35) with  7 = 0.
4.4. 6 ̸ = 0.In this case the general element can be written in the form The 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  4 to obtain ) ,  4 > 0.

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.

𝑋 = 𝛼𝑋
Solving the characteristic equation, (24), corresponding to  =  1 +  2 +  3 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 solution  (, ) =  − (57) of ( 16), (15) reduces to the ordinary differential equation: the solution of which is where  1 is an arbitrary constant.Hence one family of invariant solutions of system ( 15)-( 16) arising from  =  1 +  2 +  3 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 as which arises from the following symmetry of (15): 5.2. =  4 +  5 +  6 ,  ̸ = 0. Invariant solutions of system (15)-( 16) arising from  in this case are found to be of the form  (, ) =  +  ()  , V (, ) =  /  () .
(63) Substituting (63) in system ( 15)-( 16) leads to the following set of ordinary differential equations: Solving (64) for , we obtain where  1 and  2 are arbitrary constants.The solution of (65) taking into account the (66) completes the process.It is however not easy to solve (65) analytically.
(67) Substituting (67) into system ( 15)-( 16), we obtain the following pair of coupled ODEs: Equation ( 68) is a linear ODE and is solved to give where  1 and  2 are arbitrary constants.Equation ( 69) is to be solved for .
5.4. =  5 + 6 + 7 ,  ̸ = 1/.Invariant solutions arising from this element of the optimal system are such that V and  have the forms where  and  are arbitrary functions , and Substituting (71) into system ( 15)-( 16), we obtain the following coupled system of ODEs: where Equation ( 73) is a linear second-order ODE with constant coefficients.The solution is where  1 and  2 are arbitrary constants.This leaves (74) to be solved for .
Remark 3. We remark that (65), (69), and (74) are variable coefficient and linear second-order ODEs, which are of the form with  () = 0. (77) The analysis of such equations can be found in many standard books on ordinary differential equations such as [26,27].Also numerical methods are available for solution of differential equations of the form (77).

Interpretation of the Invariant Solutions.
Analytic solutions of the first-order feedback problem obtained as group invariant solutions in this study have various uses.They may be used to benchmark numerical schemes developed for solution of the first-order feedback problem.They may also be used to differentiate between types of auxiliary conditions which lead to qualitatively different forms of option prices.Perhaps more importantly, the solutions can be used to provide insight into valuation dynamics of the firstorder feedback model.A phenomenon of particular interest which can be investigated using the invariant solutions is the relationship between the value of the option and the liquidity parameter .However, not all the invariant solutions generated have direct relevance to the context of the firstorder feedback problem.Among the solutions constructed, we identify two solutions (corresponding to  =  4 +  5 +  6 and  =  3 +  4 +  5 ) that depict an important feature of the first-order feedback model.In Figure 1 we illustrate the option value in these cases for the liquidity parameters  = 1/4,  = 1/2,  = 3/4, and  = 1, with time to maturity  = 1, risk-free rate  = 0.04, and volatility  = 0.2.There are other free parameters in the solution that ought to be assigned suitable values.The graphs in the two cases considered were generated with the help of Mathematica's NDSolve command [28].In both cases the option value exhibits the important feature whereby as  is increased, the option value is eroded monotonically.

Concluding Remarks
In this paper we have used Lie symmetry analysis to study a nonlinear equation that arises as a generalisation of the Black-Scholes equation when liquidity is incorporated into the market.The generalised Black-Scholes equation is called a first-order feedback model and is coupled with the standard Black-Scholes equation as a result of the assumption that a hedger holds the number of stocks dictated by the standard Black-Scholes equation delta rather than the delta from the modified option price in the market with liquidity.We have exploited the Lie point symmetries admitted by the coupled system to construct particular solutions that represent option prices in the modified Black-Scholes market.
As explained in Section 2, an improvement to the firstorder feedback model is the full feedback (13), a fully nonlinear PDE.Existence and uniqueness of the solution to (13) are established by Frey [14], who has shown that options in such a market can be perfectly replicated, an attestation to the completeness of the market.Unfortunately ( 13) is poorly endowed with Lie point symmetries and, therefore, does not lend itself to any interesting analysis via Lie point symmetries.Perhaps analysis of this equation would benefit from the use of more general symmetries.