Expansion Formulas for European Quanto Options in a Local Volatility Fx-Libor Model

Abstract

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).

Pricing and Rick Analysis in Hyperbolic Local Volatility Model with Quasi‐Monte Carlo

“…Here we extend previous analysis to a more realistic local volatility-type diffusion, namely the hyperbolic local volatility introduced by Jäckel (2008) and widely used in the quantitative finance industry (see e.g. Bompis & Hok, 2014;Hok & Tan, 2019;Hok et al, 2018).…”

“…This model was introduced in Jáckel (2009). It behaves similarly to the constant elasticity of variance (CEV) model, and has been used for numerical experiments in Bompis and Hok (2014), Hok and Tan (2019), and Hok et al (2018). The advantage of this model is that zero is not an attainable boundary, and that allows us to avoid some numerical instabilities present in the CEV model when the underlying asset price is close to zero (see e.g.…”

“…In the HLV model, there is no analytical formula for Eq. (14). We are going to estimate the price by the MC method.…”

Calibration of local volatility model with stochastic interest rates by efficient numerical PDE methods

“…This model introduced in [39] behaves closely to the CEV model and has been used for numerical experiments as in [40,41]. It presents the advantage to avoid zero to be an attainable boundary and then allows to avoid some numerical instabilities as seen in the CEV model when the underlying S is close to 0 (see e.g [42]).…”

Skewing Quanto with Simplicity

“…Recently several other authors have also proposed novel approaches to overcome the extra compute and speed requirement. For example, [13] and [8] have applied perturbation methods to obtain so-called Proxy expansion formulae for European quanto option prices. This paper joins the quest for an efficient methodology that bypass PDE or Monte Carlo methods.…”