Abstract. We introduce an analytical approximation to efficiently price forward start options on equity in time-dependent local volatility models as the forward start date, the maturity or the volatility coefficient are small. We use a conditional expectation argument to represent the price as an expectation of a Black-Scholes formula computed with a stochastic implied volatility depending on the value of the equity at the forward date. Then we perform a volatility expansion to derive an analytical approximation of the forward implied volatility with a precise error estimate. We also illustrate the accuracy of the formula with some numerical experiments. Some results and tools of this work were presented at the conference SMAI 2013 in the mini-symposium "Méthodes asymptotiques en finance".Résumé. Nous introduisons une approximation analytique afin d'évaluer efficacement les optionsà départ différé dites forward start dans les modèlesà volatilité locale qui dépend du temps quand la date forward, la maturité ou le coefficient de volatilité sont petits. Nous utilisons un argument d'espérance conditionnelle pour représenter le prix comme l'espérance d'une formule de Black-Scholes calculée avec une volatilité implicite stochastique qui dépend de la valeur de l'actionà la date forward. Ensuite, nous effectuons un développement en volatilité pour obtenir une approximation analytique de la volatilité implicite forward avec une estimation précise de l'erreur. Nous illustronségalement la précision de notre formule avec quelques expériences numériques. Certains résultats et outils de ce travail ontété présentés au congrès SMAI 2013 dans le mini-symposium "Méthodes asymptotiques en finance".

We develop an expansion approach for the pricing of European quanto options written on LIBOR rates (of a foreign currency). We derive the dynamics of the system of foreign LIBOR rates under the domestic forward measure and then consider the price of the quanto option. In order to take the skew/smile effect observed in fixed income and FX markets into account, we consider local volatility models for both the LIBOR and the FX rate. Because of the structure of the local volatility function, a closed form solution for quanto option prices does not exist. Using expansions around a proxy related to log-normal dynamics, we derive approximation formulas of Black-Scholes type for the price, that have the benefit of giving very rapid numerical procedures. Our expansion formulas have the major advantage that they allow for an accurate estimation of the error, using Malliavin calculus, which is directly related to the maturity of the option, the payoff, and the level and curvature of the local volatility function. These expansions also illustrate the impact of the quanto drift adjustment, while the numerical experiments show an excellent accuracy.Key words and phrases. European quanto derivatives, convexity adjustment, volatility skew/smile, local volatility FX-LIBOR model, expansion formula, analytical approximations, Malliavin calculus. 1 arXiv:1801.01205v2 [q-fin.PR] 3 Apr 2018The FX forward rate is, per definition, a Q i -martingale, and we assume it follows again a local volatility model of the form 5)where W i is the Q i -Brownian motion and σ i : [0, T i ] × R → R d + , i ∈ I, is a continuous, deterministic function satisfying a suitable linear growth condition, and represents the local volatility of the FX forward rate.The domestic and foreign forward Brownian motions are related viafor all i ∈ I, and this equation together with (2.3) determines also the relations between the domestic forward Brownian motions; see Schlögl (2002) for the details (in particular Fig. 2).

Long maturity options or a wide class of hybrid products are evaluated using a local volatility type modelling for the asset price S(t) with a stochastic interest rate r(t). The calibration of the local volatility function is usually time-consuming because of the multi-dimensional nature of the problem. In this paper, we develop a calibration technique based on a partial differential equation (PDE) approach which allows an efficient implementation. The essential idea is based on solving the derived forward equation satisfied by P (t, S, r)Z(t, S, r), where P (t, S, r) represents the risk neutral probability density of (S(t), r(t)) and Z(t, S, r) the projection of the stochastic discounting factor in the state variables (S(t), r(t)). The solution provides effective and sufficient information for the calibration and pricing. The PDE solver is constructed by using ADI (Alternative Direction Implicit) method based on an extension of the Peaceman-Rachford scheme. Furthermore, an efficient algorithm to compute all the corrective terms in the local volatility function due to the stochastic interest rates is proposed by using the PDE solutions and grid points. Different numerical experiments are examined and compared to demonstrate the results of our theoretical analysis.

Here we develop an option pricing method based on Legendre series expansion of the density function.The key insight, relying on the close relation of the characteristic function with the series coefficients, allows to recover the density function rapidly and accurately. Based on this representation for the density function, approximations formulas for pricing European type options are derived. To obtain highly accurate result for European call option, the implementation involves integrating high degree Legendre polynomials against exponential function. Some numerical instabilities arise because of serious subtractive cancellations in its formulation (96) in proposition 7.1. To overcome this difficulty, we rewrite this quantity as solution of a second-order linear difference equation and solve it using a robust and stable algorithm from Olver. Derivation of the pricing method has been accompanied by an error analysis. Errors bounds have been derived and the study relies more on smoothness properties which are not provided by the payoff functions, but rather by the density function of the underlying stochastic models. This is particularly relevant for options pricing where the payoffs of the contract are generally not smooth functions. The numerical experiments on a class of models widely used in quantitative finance show exponential convergence.

A wide class of hybrid products are evaluated with a model where one of the underlying price follows a local volatility diffusion and the other asset value a log-normal process. Because of the generality for the local volatility function, the numerical pricing is usually much time consuming. Using proxy approximations related to log-normal modeling, we derive approximation formulas of Black–Scholes type for the price, that have the advantage of giving very rapid numerical procedures. This derivation is illustrated with the best-of option between equity and inflation where the stock price follows a local volatility model and the inflation rate a Hull–White process. The approximations possibly account for Gaussian HJM (Heath-Jarrow-Morton) models for interest rates. The experiments show an excellent accuracy.