The purpose of this paper is to analyze and compute the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility which can be a function of the second derivative of the option price itself. A motivation for studying the nonlinear Black--Scholes equation with a nonlinear volatility arises from option pricing models taking into account e.g. nontrivial transaction costs, investor's preferences, feedback and illiquid markets effects and risk from a volatile (unprotected) portfolio. We present a new method how to transform the free boundary problem for the early exercise boundary position into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. We furthermore propose an iterative numerical scheme that can be used to find an approximation of the free boundary. We present results of numerical approximation of the early exercise boundary for various types of nonlinear Black--Scholes equations and we discuss dependence of the free boundary on various model parameters.
In the past years, the Black-Scholes equation for pricing derivatives has attracted a lot of attention from both theoretical as well as practical point of view. Recall that a European Call (Put) option is the right but not obligation to purchase (sell) an underlying asset at the expiration price E at the expiration time T . In an idealized financial market the price of an option can be computed from the well-known Black-Scholes equation derived by Black and Scholes in [4], and, independently by Merton (see also [21], Dewynne et al. [7], Hull [16]). Assuming that the underlying asset follows a geometric Brownian motion one can derive a governing partial differential equation for the price of an option. We remind ourselves that the equation for option's price V (S, t) is the following parabolic PDE:
where σ is the volatility of the underlying asset price process, r > 0 is the interest rate of a zerocoupon bond, q ≥ 0 is the dividend yield rate. A solution V = V (S, t) represents the price of an option at time t ∈ [0, T ] if the price of an underlying asset is S > 0. In this paper we shall focus our attention to the case when the diffusion coefficient σ 2 may depend on the time T -t to expiry, the asset price S and the second derivative ∂ 2 S V of the option price (referred to as Γ), i.e.
A motivation for studying the nonlinear Black-Scholes equation ( 1) with a volatility σ having a general form (2) arises from option pricing models taking into account nontrivial transaction costs, market feedbacks and/or risk from a volatile (unprotected) portfolio. Recall that the linear Black-Scholes equation with σ constant has been derived under several restrictive assumptions like e.g. frictionless, liquid, complete markets, etc. We also recall that the linear Black-Scholes equation provides a perfectly replicated hedging portfolio. In recent years, some of these assumptions have been relaxed in order to model, for instance, the presence of transaction costs (see e.g. Leland [22], Hoggard et al. [17], Avellaneda and Paras [2]), feedback and illiquid market effects due to large traders choosing given stock-trading strategies (Frey and Patie [12], Frey and Stremme [13], During et al. [8], Schönbucher and Wilmott [29]), imperfect replication and investor’s preferences (Barles and Soner [5]), risk from unprotected portfolio (Kratka [20], Jandačka and Ševčovič [18]). One of the first nonlinear models is the so-called Leland model (c.f. [22]) for pricing Call and Put options under the presence of transaction costs. It has been generalized for more complex options by Hoggard, Whaley and Wilmott in [17]. In this model the volatility σ is given by σ 2 (S 2 ∂ 2 S V, S, τ ) = σ2 (1 + Le sgn(∂ 2 S V )) where σ > 0 is a constant historical volatility of the underlying asset price process and Le > 0 is the so-called Leland constant given by Le = 2/πC/(σ √ ∆t) where C > 0 is a constant round trip transaction cost per unit dollar of transaction in the assets market and ∆t > 0 is the time-lag between portfolio adjustments.
If transaction costs are taken into account perfect replication of the contingent claim is no longer possible and further restrictions are needed in the model. By assuming that investor’s preferences are characterized by an exponential utility function Barles and Soner (c.f. [5]) derived a nonlinear Black-Scholes equation with the volatility σ given by
where Ψ is a solution to the ODE:
Another popular model has been derived for the case when the asset dynamics takes into account the presence of feedback and illiquid market effects. Frey and Stremme (c.f. [13,12]) introduced directly the asset price dynamics in the case when a large trader chooses a given stocktrading strategy (see also [29]). The diffusion coefficient σ is again nonconstant and it can be expressed as:
where σ2 , ̺ > 0 are constants and λ(S) is a strictly convex function, λ(S) ≥ 1.
The last example of the Black-Scholes equation with a nonlinearly depending volatility is the so-called Risk Adjusted Pricing Methodology model proposed by Kratka in [20] and revisited by Jandačka and Ševčovič in [18]. In order to maintain (imperfect) replication of a portfolio by the delta hedge one has to make frequent portfolio adjustments leading to a substantial increase in transaction costs. On the other hand, rare portfolio adjustments may lead to an increase of the risk arising from a volatile (unprotected) portfolio. In the RAPM model the aim is to optimize the time-lag ∆t between consecutive portfolio adjustments. By choosing ∆t > 0 in such way that the sum of the rate of transaction costs and the rate of a risk from unprotected portfolio is minimal one can find the optimal time lag ∆t > 0 (see [18] for details). In the RAPM model, it turns out that the volatility is again nonconstant and it has the following form:
.
(
Here σ2 > 0 is a constant and µ = 3(C 2 R/2π)
where C, R ≥ 0 are nonnegative constant representing the transaction cost measure and the r
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