SWIFT Calibration of the Heston Model

Abstract

In the present work, the SWIFT method for pricing European options is extended to Heston model calibration. The computation of the option price gradient is simplified thanks to the knowledge of the characteristic function in closed form. The proposed calibration machinery appears to be extremely fast, in particular for a single expiry and multiple strikes, outperforming the state-of-the-art method we compare it with. Further, the a priori knowledge of SWIFT parameters makes a reliable and practical implementation of the presented calibration method possible. A wide range of stress, speed and convergence numerical experiments is carried out, with deep in-the-money, at-the-money and deep out-of-the-money options for very short and very long maturities.

On the Class of Risk Neutral Densities under Heston’s Stochastic Volatility Model for Option Valuation

“…Clearly, we have c µ (0) = µ. Note that c µ (•) is merely the undiscounted version of C S (•) in (7), so that with µ = S • e rt , as in (6), we have c µ (K) ≡ e rt • C S (K).…”

“…In this section, we identify the class of distributions (and, therefore, the class of possible RNDs) that admit the presentation in (12) for the price of a European call option. Specifically, we show that any RND candidate that satisfies ( 6) and (7) with a scale parameter µ = S • e rt will admit the presentation in (12) and hence, in light of the results in (11), will equivalently satisfy Heston's option pricing model in (5). While the development given below could be seen as straightforward (and perhaps even trivial), it is instrumental for the theoretical characterization of Heston's RNDs as a scale family of distributions.…”

“…Clearly, Theorem 2 asserts that any RND for Heston's stochastic volatility model must be a member of the class of scale parameter distributions that satisfiesAssumption 1, with the mean being the forward spot price. Indeed, any member of this class of scale parameter distributions could serve as a possible RND for the direct and risk neutral calculation of an option price, as in (7) or Corollary 1. In the next section, we identify and list several one-parameter versions of some well-known distributions that could serve as RNDs under Heston's SV model.…”

“…The problem of 'calibrating' or estimating the parameters of Heston's model has naturally received much attention in the literature. Various approaches have been utilized, from standard (weighted or penalized) MSE approaches (as in [6]) to fast SWIFT method approaches (as in [7]) and direct maximum likelihood approaches (as in [8]), as well as models with implied stochastic volatility approaches (as in [9]), to name a few. On the other hand, the utility and usefulness of the risk neutral probability measure Q in option valuation, both in general and in determining specific solutions in (5) (or in (3)), cannot be overstated (in the 'risk neutral' world).…”

The Generalized Gamma Distribution as a Useful RND under Heston’s Stochastic Volatility Model

“…The components of ϑ have particular meaning in the context of Heston's SV model: ρ is the correlation between the random components of the spot's price and volatility processes, θ is the long-run average volatility, κ is the mean-reversion speed for the volatility dynamics and η 2 is the variance of the volatility V. It should be noted that different choices of ϑ will lead to different values C S (K) in (3) and hence, the value ϑ = (κ, θ, η, ρ) must be appropriately 'calibrated' first for C S (K) to actually match the option market data. However, this calibration process typically involves substantial numerical challenges (largely resulting from numerical issues involved in the required multi-dimensional optimization, see for example Bin 2007, or Section 2.1 in Romo and Ortiz-Gracia 2021). These challenges are an obvious hindrance to the retail option traders who do not have the numerical tools or the know-how to finely calibrate the Heston (1993) SV model, as needed in the evaluation of C S (K) in (3).…”