We consider a pure endowment contract whose life contingent payout is linked to the performance of a risky stock or index. Because of the additional mortality risk, the market is incomplete; thus, a fundamental assumption of the Black-Scholes theory is violated. We price this contract via the principle of equivalent utility and demonstrate that, under the assumption of exponential utility, the ...

In this paper we derive analytic expressions for the value of European Put and Call options when the stock process follows an exponential Lévy-Stable process. It is shown that the generalised Black-Scholes operator for the Lévy-Stable case can be obtained as an asymptotic approximation of a process where the random variable follows a DampedLévy process. Finally, it is also shown that option pri...

The aim of this paper is to present a stochastic model that accounts for the effects of a long-memory in volatility on option pricing. The starting point is the stochastic Black-Scholes equation involving volatility with long-range dependence. We consider the option price as a sum of classical Black-Scholes price and random deviation describing the risk from the random volatility. By using the ...

We study properties of solutions to fully nonlinear versions of the standard Black– Scholes partial differential equation. These equations have been introduced in financial mathematics in order to deal with illiquid markets or with stochastic volatility. We show that typical nonlinear Black–Scholes equations can be viewed as dynamic programming equation of an associated control problem. We esta...

This paper develops a general stochastic model of a frictionless security market with continuous trading. The vector price process is given by a semimartingale of a certain clr;zss, and the general stochastic integral is used to represent capital gains. Within the framework of this model, we discuss the modern theory of contingent claim valuation, including the celebrated option pricing formula...

This paper reviews the Black-Scholes theory and studies a number of extensions. In particular, we focus on logarithmic stochastic volatility (SV) models and square-root SV models with jumps. Inference on those models, using Markov chain Monte Carlo methods, is summarized. Generalized Black-Scholes (GBS) pricing formulas are presented as an important step in calibration of SV models using both r...

This paper presents an empirical test of Dupire's (1993) option price inversion approach using FT–SE 100 index options. The performance of option deltas determined using the Dupire approach is compared to the performance of a pair of Black-Scholes (1973) based deltas. The study finds that Black-Scholes based deltas out-perform the Dupire-deltas, which is consistent with the results in synthesiz...

We justify and give error estimates for binomial approximations of game (Israeli) options in the Black–Scholes market with Lipschitz continuous path dependent payoffs which are new also for usual American style options. We show also that rational (optimal) exercise times and hedging self-financing portfolios of binomial approximations yield for game options in the Black–Scholes market “nearly” ...

This study applies fuzzy set theory to the vulnerable Black-Scholes (1973) or Merton (1973) formula. Expectations of heterogeneity mean option prices are expected to be imprecise, thus making it natural to consider fuzziness to handle this. This article presents a fuzzy approach to value Black-Scholes options subject to non-identical rationality and correlated credit risk. Although no analytica...