Risk Management for Derivatives in Illiquid Markets: A Simulation Study

Abstract

Summary. In this paper we study the hedging of derivatives in illiquid markets. More specifically we consider a model where the implementation of a hedging strategy affects the price of the underlying security. Following earlier work we characterize perfect hedging strategies by a nonlinear version of the Black-Scholes PDE. The core of the paper consists of a simulation study. We present numerical results on the impact of market illiquidity on hedge cost and Greeks of derivatives. We go on and offer a new explanation of the smile pattern of implied volatility related to the lack of market liquidity. Finally we present simulations on the performance of different hedging strategies in illiquid markets.

On Option-Valuation in Illiquid Markets: Invariant Solutions to a Nonlinear Model

“…Later on, in [Frey and Patie, 2002] Frey and Patie followed another approach and examined the feedback effect of the option replication strategy of the large trader on the asset price process. They obtain a new model by introduction of a liquidity coefficient which depends on the current stock price.…”

“…The Feynman-Kac theorem together with a fixed point argument [Frey and Patie, 2002] leads in case of (3) to a partial differential equation which is similar to the equation (2) where the constant ρ is replaced by ρλ (S). The values of ρ and λ(S) may be estimated from the observed option prices and depend on the payoff h(S).…”

Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function

“…The linear Black-Scholes equation with a constant volatility σ has been derived under several restrictive assumptions like e.g., frictionless, liquid and complete markets, etc. Such assumptions have been relaxed in order to model the presence of transaction costs (see e.g., Leland [25], Hoggard et al [19], Avellaneda and Paras [2]), feedback and illiquid market effects due to large traders choosing given stocktrading strategies (Frey [13], Frey and Patie [14], Frey and Stremme [15], Schönbucher and Wilmott [28]), imperfect replication and investor's preferences (Barles and Soner [4]), risk from unprotected portfolio (Kratka [22], Jandačka andŠevčovič [21] or [30]). …”

“…We also present results of numerical computations of the free boundary position, option price and their dependence on model parameters. In the recent paper [17] we investigated the case when the volatility function may depend on S and ∂ 2 S V including other models proposed by Barles and Soner [4], Frey and Patie [14], Frey and Stremme [15]. However, for these models there is no single implicit equation for the free boundary position and numerical methods have to be adopted.…”

Symmetries and exact solutions of a nonlinear pricing options equation

“…As it was shown in works [11,14,15], the model of the cost of the hedge stategy on a illiquid market with taking into consideration the influence of the large trader operations can be represented as…”