Fabio Antonelli

Weak Solutions of Forward–Backward SDE's

In this note we study a class of forward–backward stochastic differential equations (FBSDE for short) with functional-type terminal conditions. In the case when the time duration and the coefficients are “compatible” (e.g., the time duration is small), we prove the existence and uniqueness of the strong adapted solution in the usual sense. In the general case we introduce a notion of weak solution for such FBSDEs, as well as two notions of uniqueness. We prove the existence of the weak solution under mild conditions, and we prove that the Yamada–Watanabe Theorem, that is, pathwise uniqueness implies uniqueness in law, as well as the Principle of Causality also hold in this context.

Pricing options under stochastic volatility: a power series approach

In this paper we present a new approach for solving the pricing equations (PDEs) of European call options for very general stochastic volatility models, including the Stein and Stein, the Hull and White, and the Heston models as particular cases. The main idea is to express the price in terms of a power series of the correlation parameter between the processes driving the dynamics of the price and of the volatility. The expansion is done around correlation zero and each term is identified via a probabilistic expression. It is shown that the power series converges with positive radius under some regularity conditions. Besides, we propose (as in Alós in Finance Stoch. 10:353–365, 2006) a further approximation to make the terms of the series easily computable and we estimate the error we commit. Finally we apply our methodology to some well-known financial models.

Asset pricing with a forward-backward stochastic differential utility

Abstract In an intertemporal setting we model the anticipation–disappointment effect through a habit formation process which is a function of past consumption and of past expected utility. We show that in equilibrium the anticipation effect reduces the risk premium, whereas the disappointment effect induces a higher risk premium.

A comparison result for FBSDE with applications to decisions theory

In general, a comparison Lemma for the solutions of Forward-Backward Stochastic Differential Equations (FBSDE) does not hold. Here we prove one for the backward component at the initial time, relying on certain monotonicity conditions on the coefficients of both components. Such a result is useful in applications. Indeed, one can use FBSDEs to define a utility functional able to capture the disappointment-anticipation effect for an agent in an intertemporal setting under risk. Exploiting our comparison result, we prove some “desirable” properties for the utility functional, such as continuity, concavity, monotonicity and risk aversion. Finally, for completeness, in a Markovian setting, we characterize the utility process by means of a degenerate parabolic partial differential equation.

Calibrated American option pricing by stochastic linear programming

Abstract We propose an approach for computing the arbitrage-free interval for the price of an American option in discrete incomplete market models via linear programming. The main idea is built replicating strategies that use both the basic asset and some European derivatives available on the market for trading. This method goes under the name of calibrated option pricing and it has given significant results for European options. Here, we extend the analysis to American options showing that the arbitrage-free interval can be characterized in terms of martingale measures and that it gets significantly reduced with respect to the non-calibrated case.

RATE OF CONVERGENCE OF MONTE CARLO SIMULATIONS FOR THE HOBSON–ROGERS MODEL

The Hobson–Rogers model is used to price derivative securities under the no-arbitrage condition in a stochastic volatility setting, preserving the completeness of the market. Here we are studying the rate of convergence of the Euler/Monte Carlo approximations, when pricing European, Asian and digital type options. The aim of the present work is to express the approximation error in terms of the time step size, denoted by h, used for the Euler scheme. We recover an already known result, obtained by other authors using PDE approximations, for European options. Namely we show that for a Lipschitz coefficient of the driving equations for the asset price and Lipschitz payoffs, we obtain an error of the order of

Backward-Forward Stochastic Differential Equations

\sqrt{h}

Rate of convergence of a particle method to the solution of the Mc Kean-Vlasov's equation

. Moreover, using Malliavin Calculus techniques, we show that with a regular coefficient we may attain an error of the order of h for regular payoffs and of the order of

Exchange option pricing under stochastic volatility: a correlation expansion

\sqrt{h}

Existence of the solutions of backward–forward SDE's with continuous monotone coefficients

for non Lipschitz payoffs. Finally we show some numerical simulations supporting our theoretical results.