Renewal equations for option pricing

Abstract

In this paper we will develop a methodology for obtaining pricing expressions for financial instruments whose underlying asset can be described through a simple continuous-time random walk (CTRW) market model. Our approach is very natural to the issue because it is based in the use of renewal equations, and therefore it enhances the potential use of CTRW techniques in finance. We solve these equations for typical contract specifications, in a particular but exemplifying case. We also show how a formal general solution can be found for more exotic derivatives, and we compare prices for alternative models of the underlying. Finally, we recover the celebrated results for the Wiener process under certain limits.PACS. 89.65.Gh Economics; econophysics, financial markets, business and management -05.40.Fb Random walks and Levy flights -02.50.Ey Stochastic processes

Equilibrium Asset and Option Pricing under Jump-Diffusion Model with Stochastic Volatility

“…Due to drawbacks of the Black-Scholes model which cannot explain numerous empirical facts such as large and sudden movements in prices, heavy tails, volatility clustering, the incompleteness of markets, and the concentration of losses in a few large downward moves, many option valuation models have been proposed and tested to fit those empirical facts. Jump-diffusion models with stochastic volatility could overcome these drawbacks of the Black-Scholes model in [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Based on those advantages, in this paper, we focus on studying the jump-diffusion model with stochastic volatility.…”

A note on intraday option pricing

“…In the following, we shall consider intra-day European options written on semi-Markov pure jump models which are compound renewal processes (see also Scalas, 2011;Baleanu et al, 2012). Related papers are Montero (2008) and Cartea (2010). In Montero (2008), the focus is on option prices for derivatives written on compound Poisson processes and in the presence on non-vanishing risk free interest rate, whereas Cartea (2010) extends Lévy option prices to the semi-Markov case by developing suitable approximations.…”

“…Related papers are Montero (2008) and Cartea (2010). In Montero (2008), the focus is on option prices for derivatives written on compound Poisson processes and in the presence on non-vanishing risk free interest rate, whereas Cartea (2010) extends Lévy option prices to the semi-Markov case by developing suitable approximations. Finally, in a recent paper, Shaw and Schofield (2011) consider Laplace transform methods to deal with order and trade renewal flows in an agent-based model where the trade counting process is not necessarily Poisson.…”

A Parsimonious Model for Intraday European Option Pricing

“…In the following, we shall consider intra-day European options written on semi-Markov pure jump models which are compound renewal processes (see also [25] and [1]). Related papers are [20] and [4]. In [20], the focus is on option prices for derivatives written on compound Poisson processes and in the presence on non-vanishing risk-free interest rate, whereas [4] extends Lévy option prices to the semi-Markov case by developing suitable approximations.…”

“…Related papers are [20] and [4]. In [20], the focus is on option prices for derivatives written on compound Poisson processes and in the presence on non-vanishing risk-free interest rate, whereas [4] extends Lévy option prices to the semi-Markov case by developing suitable approximations. Finally, a recent paper by [28] considers Laplace transform methods to deal with order and trade renewal flows in an agent-based model where the trade counting process is not necessarily Poisson.…”