We have solved the problem of finding (HARA) fair option price under a general stochastic volatility model. For any HARA utility, the "risk premium," i.e., the "market price of volatility risk" is determined via a solution of a certain nonlinear PDE. Equivalently, the fair option price is determined as a solution of an uncoupled system of a non-linear PDE and a Black–Scholes type PDE. Résumé Pr...

This paper examines the behaviour of European option price (Duan (1995)) and the Black-Scholes model bias when stock returns follow a GARCH (1,1) process. The GARCH option price is not preferenceneutral and depends on the unit risk premium (λ) as well as the two GARCH (1,1) process parameters (α1 , β1). In general, the GARCH option price does not seem overly sensitive to these parameters. Deep-...

We introduce a reduced basis method for the efficient numerical solution of partial integro-differential equations which arise in option pricing theory. Our method uses a basis of functions constructed from a sequence of Black-Scholes solutions with different volatilities. We show that this choice of basis leads to a sparse representation of option pricing functions, yielding an approximation w...

We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynamical programming approach leads to an investigation of the Hamilton Jacobi Bellman (HJB) equation which is a highly non linear partia...

We investigate a method for pricing the generic spread option beyond the classical two-factor Black-Scholes framework by extending the fast Fourier Transform technique introduced by Carr & Madan (1999) to a multi-factor setting. The method is applicable to models in which the joint characteristic function of the underlying assets forming the spread is known analytically. This enables us to inco...

Computing the current value of an American call option requires solving the BlackScholes PDE. The asset may be exercised at any time before its expiration date. The numerical computation involves solving this free boundary problem. A finite difference approximation leads to a linear complementarity problem with a tridiagonal matrix. What is new in this paper is an efficient technique for solvin...

Using the Malliavin calculus in time inhomogeneous jump-diffusion models, we obtain an expression for the sensitivity Theta of an option price (with respect to maturity) as the expectation of the option payoff multiplied by a stochastic weight. This expression is used to design efficient numerical algorithms that are compared to traditional finite difference methods for the computation of Theta...

(a) The link between risk-neutral expectations and PDE’s. We have discussed two apparently different approaches to the valuation of a European option: (i) take the discounted risk-neutral expected payoff, or (ii) solve the Black-Scholes PDE. Let’s show now that these two approaches are equivalent. First, consider options on a forward price. We saw long ago that in the discrete time setting, the...

We propose, formalise and analyse the deep parametric PDE method to solve high-dimensional partial differential equations with a focus on financial applications. A single neural network approximates solution of whole family PDEs after being trained without need sample solutions. As practical application, we compute option prices Greeks in multivariate Black–Scholes model as there is an urgent f...