Asymptotic and non asymptotic approximations for option valuation (Q2849673)

The authors give a broad overview of approximation methods to derive analytical formulas for accurate and fast evaluation of option prices. They compare different approaches, from the theoretical point of view regarding the tools they require, and also from the numerical point of view regarding their performances. In the case of local volatility models with general time-dependency, they derive new formulas using the local volatility function at the mid-point between strike and spot. Approximations of the option delta are also provided.NEWLINENEWLINEThroughout the paper a vanilla call option is considered, written on an underlier which follows the \(\text{d}S_{t} = S_{t}\sigma(t,S_{t})\text{d}W_{t}\) local volatility model under the pricing measure. In Section 2 the authors collect some formulae describing the asymptotic behavior of call prices for extreme strikes and/or short/long maturities; while for non-extreme strikes and maturities, limiting cases of the above model are studied where the (stochastic) volatility term is either small, or slowly varying, or so fast that it is effectively constant on the time scale of the option pricing problem. The normal and \(\log\)-normal approximations to the price distribution of the underlier are also introduced: expanding these proxies up to third order in various settings are the main results of the rest of the paper.NEWLINENEWLINEIn Section 3 the \(\log\)-normal proxy using the local volatility at the spot, at the strike, and at a midpoint between spot and strike is expanded up to second order. The second-order expansions of the implied volatility for the normal and log-normal proxies are also given.NEWLINENEWLINESection 4 presents a comparative discussion on how the results of Section 3 may be obtained by stochastic analysis or partial differential equation (PDE) techniques, or mixing the two. The authors conclude that by using stochastic analysis -- their preferred choice -- the derivation of expansion coefficients is direct and easy, and the results apply to general payoffs under mild non-degeneracy conditions; however, the error analysis relies on delicate Malliavin calculus estimates, which in certain cases may be simpler using the PDE approach. Using both approaches is further promoted by the observation that the PDE techniques outlined by the authors break down for digital options, while their Malliavin calculus fails for barrier options.NEWLINENEWLINEIn Section 5 the results of Section 3 are complemented with expansions up to third order, and in Section 6 second-order approximations of the call option delta are given using local volatility at strike or at spot and midpoint.NEWLINENEWLINESection 7 contains the results of numerical experiments for the CEV model with constant parameters \(\sigma(t,S) = \nu S^{\beta - 1}\) for \(\nu = 0.25\) and \(\beta = 0.2, 0.8\); and \(\nu = 0.4\), \(\beta = 0.5\). The accuracy of the different approximations outlined above is tested for maturities from 3 months up to 10 years, and a wide range of strikes. Some implied volatility approximations from [\textit{P. S. Hagan} and \textit{D. E. Woodward}, Appl. Math. Finance 6, No. 3, 147--157 (1999; Zbl 1009.91033); \textit{P. Henry-Labordère}, Analysis, geometry, and modeling in finance. Advanced methods in option pricing. Boca Raton, FL: CRC Press (2009; Zbl 1151.91006)] serve as external benchmarks. The authors observe that {\parindent=0,5cm\begin{itemize}\item[-] for all approximations, errors increase with the maturity; for at-the-money options all approximations perform well, while the log-normal proxy is good for not too extreme strikes, as well;\item[-]approximations based on log-normal proxy perform better than approximations based on normal proxy; \item[-] third-order approximations are more accurate than second-order ones; yet it is noted that for the log-normal proxy case, second-order approximations in spot or strike often underestimate the true implied volatility values while third-order approximations in spot overestimate the true values for out-of-the-money options and underestimate it for in-the-money options; \item[-] methods using the local volatility at mid-point systematically give the best results; \item[-] implied volatility expansions are more precise and stable than price expansions; for extreme strikes, a second-order implied volatility expansion may be more accurate than the corresponding third-order price expansion; \item[-] the error of the log-normal proxy increases for smaller \(\beta\) values where the price dynamics of the underlier is more like normal;NEWLINENEWLINE\item[-] increasing \(\nu\) also leads to an increase of approximation errors.NEWLINENEWLINE\end{itemize}}NEWLINEThe authors conclude that the third-order expansion of the log-normal proxy is the best method, and it is more accurate and robust than the other approximations or either of the external benchmarks.NEWLINENEWLINEFor the entire collection see [Zbl 1272.91008].