Discussion Papers 2016

CIRJE-F-1013	"Derivatives Pricing with Market Impact and Limit Order Book"
Author Name	Saito, Taiga and Akihiko Takahashi
Date	May 2016
Full Paper	PDF File
Remarks	Revised in August 2017; "Derivatives pricing with market impact and limit order book", 2017, Automatica, 86, 154-165.

Abstract
This paper investigates derivatives pricing under existence of liquidity costs and market impact for the underlying asset in continuous time. Firstly, we formulate the charge for the liquidity costs and the market impact on the derivatives prices through a stochastic control problem that aims to maximize the mark-to-market value of the portfolio less the quadratic variation multiplied by a risk aversion parameter during the hedging period and the liquidation cost at maturity. Then, we obtain the derivatives price by reduction of this charge from the premium in the Bachelier model. Secondly, we consider a second order semilinear partial differential equation (PDE) of parabolic type associated with the control problem, which is analytically solved or approximated by an asymptotic expansion around a solution to an explicitly solvable nonlinear PDE. Finally, we present numerical examples of the pricing for a variance option and a European call option, and show comparative static analyses.
Keywords: Stochastic control; Application in nance; Derivatives pricing.

Abstract

This paper investigates derivatives pricing under existence of liquidity costs and market impact for the underlying asset in continuous time. Firstly, we formulate the charge for the liquidity costs and the market impact on the derivatives prices through a stochastic control problem that aims to maximize the mark-to-market value of the portfolio less the quadratic variation multiplied by a risk aversion parameter during the hedging period and the liquidation cost at maturity. Then, we obtain the derivatives price by reduction of this charge from the premium in the Bachelier model. Secondly, we consider a second order semilinear partial differential equation (PDE) of parabolic type associated with the control problem, which is analytically solved or approximated by an asymptotic expansion around a solution to an explicitly solvable nonlinear PDE. Finally, we present numerical examples of the pricing for a variance option and a European call option, and show comparative static analyses.

Keywords: Stochastic control; Application in nance; Derivatives pricing.