We consider closed-form expansions for European put option prices within several stochastic volatility frameworks with time-dependent parameters. Our methodology involves writing the put option price as an expectation of a Black-Scholes formula and performing a second-order Taylor expansion around the mean of its argument. The difficulties then faced are computing a number of expectations induced by the Taylor expansion in a closed-form manner. We establish a fast calibration scheme under the assumption that the parameters are piecewise-constant. Furthermore, we perform a sensitivity analysis to investigate the quality of our approximation and show that the errors are well within the acceptable range for application purposes. Lastly, we derive bounds on the remainder term due to the Taylor expansion.

We consider the problem of determining the L\'evy exponent in a L\'evy model for asset prices given the price data of derivatives. The model, formulated under the real-world measure $\mathbb P$, consists of a pricing kernel $\{\pi_t\}_{t\geq0}$ together with one or more non-dividend-paying risky assets driven by the same L\'evy process. If $\{S_t\}_{t\geq0}$ denotes the price process of such an asset then $\{\pi_t S_t\}_{t\geq0}$ is a $\mathbb P$-martingale. The L\'evy process $\{ \xi_t \}_{t\geq0}$ is assumed to have exponential moments, implying the existence of a L\'evy exponent $\psi(\alpha) = t^{-1}\log \mathbb E(\rm e^{\alpha \xi_t})$ for $\alpha$ in an interval $A \subset \mathbb R$ containing the origin as a proper subset. We show that if the initial prices of power-payoff derivatives, for which the payoff is $H_T = (\zeta_T)^q$ for some time $T>0$, are given for a range of values of $q$, where $\{\zeta_t\}_{t\geq0}$ is the so-called benchmark portfolio defined by $\zeta_t = 1/\pi_t$, then the L\'evy exponent is determined up to an irrelevant linear term. In such a setting, derivative prices embody complete information about price jumps: in particular, the spectrum of the price jumps can be worked out from current market prices of derivatives. More generally, if $H_T = (S_T)^q$ for a general non-dividend-paying risky asset driven by a L\'evy process, and if we know that the pricing kernel is driven by the same L\'evy process, up to a factor of proportionality, then from the current prices of power-payoff derivatives we can infer the structure of the L\'evy exponent up to a transformation $\psi(\alpha) \rightarrow \psi(\alpha + \mu) - \psi(\mu) + c \alpha$, where $c$ and $\mu$ are constants.

This paper deals with a fundamental subject that has seldom been addressed in recent years, that of market impact in the options market. Our analysis is based on a proprietary database of metaorders-large orders that are split into smaller pieces before being sent to the market on one of the main Asian markets. In line with our previous work on the equity market [Said et al., 2018], we propose an algorithmic approach to identify metaorders, based on some implied volatility parameters, the at the money forward volatility and at the money forward skew. In both cases, we obtain results similar to the now well understood equity market: Square-root law, Fair Pricing Condition and Market Impact Dynamics.

We investigate the problem of computing a nested expectation of the form $\mathbb{P}[\mathbb{E}[X|Y] \!\geq\!0]\!=\!\mathbb{E}[\textrm{H}(\mathbb{E}[X|Y])]$ where $\textrm{H}$ is the Heaviside function. This nested expectation appears, for example, when estimating the probability of a large loss from a financial portfolio. We present a method that combines the idea of using Multilevel Monte Carlo (MLMC) for nested expectations with the idea of adaptively selecting the number of samples in the approximation of the inner expectation, as proposed by (Broadie et al., 2011). We propose and analyse an algorithm that adaptively selects the number of inner samples on each MLMC level and prove that the resulting MLMC method with adaptive sampling has an $\mathcal{O}\left( \varepsilon^{-2}|\log\varepsilon|^2 \right)$ complexity to achieve a root mean-squared error $\varepsilon$. The theoretical analysis is verified by numerical experiments on a simple model problem. We also present a stochastic root-finding algorithm that, combined with our adaptive methods, can be used to compute other risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), with the latter being achieved with $\mathcal{O}\left(\varepsilon^{-2}\right)$ complexity.

We propose some machine-learning-based algorithms to solve hedging problems in incomplete markets. Sources of incompleteness cover illiquidity, untradable risk factors, discrete hedging dates and transaction costs. The proposed algorithms resulting strategies are compared to classical stochastic control techniques on several payoffs using a variance criterion. One of the proposed algorithm is flexible enough to be used with several existing risk criteria. We furthermore propose a new moment-based risk criteria.