Peter Tankov

A New Look at Short‐Term Implied Volatility in Asset Price Models with Jumps

We analyze the behavior of the implied volatility smile for options close to expiry in the exponential Levy class of asset price models with jumps. We introduce a new renormalization of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal–Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short‐end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model‐independent slope. This result gives a theoretical justification for the preference of the infinite variation Levy models over the finite variation ones in the calibration based on short‐maturity option prices.

Tail behavior of sums and differences of log-normal random variables

We present sharp tail asymptotics for the density and the distribution function of linear combinations of correlated log-normal random variables, that is, exponentials of components of a correlated Gaussian vector. The asymptotic behavior turns out to depend on the correlation between the components, and the explicit solution is found by solving a tractable quadratic optimization problem. These results can be used either to approximate the probability of tail events directly, or to construct variance reduction procedures to estimate these probabilities by Monte Carlo methods. In particular, we propose an efficient importance sampling estimator for the left tail of the distribution function of the sum of log-normal variables. As a corollary of the tail asymptotics, we compute the asymptotics of the conditional law of a Gaussian random vector given a linear combination of exponentials of its components. In risk management applications, this finding can be used for the systematic construction of stress tests, which the financial institutions are required to conduct by the regulators. We also characterize the asymptotic behavior of the Value at Risk for log-normal portfolios in the case where the confidence level tends to one.

Swing Options Valuation: a BSDE with Constrained Jumps Approach

We introduce a new probabilistic method for solving a class of impulse control problems based on their representations as Backward Stochastic Differential Equations (BSDEs for short) with constrained jumps. As an example, our method is used for pricing Swing options. We deal with the jump constraint by a penalization procedure and apply a discrete-time backward scheme to the resulting penalized BSDE with jumps. We study the convergence of this numerical method, with respect to the main approximation parameters: the jump intensity

Asymptotic Optimal Tracking: Feedback Strategies

lambda

Hedging under multiple risk constraints

, the penalization parameter

High Order Weak Approximation Schemes for Lévy-Driven SDEs

p > 0

Lévy Copulas: Review of Recent Results

and the time step. In particular, we obtain a convergence rate of the error due to penalization of order

Tails of weakly dependent random vectors

(lambda p)^{alpha - frac{1}{2}}, forall alpha in left(0, frac{1}{2}right)

Implied Volatility of Basket Options at Extreme Strikes

. Combining this approach with Monte Carlo techniques, we then work out the valuation problem of (normalized) Swing options in the Black and Scholes framework. We present numerical tests and compare our results with a classical iteration method.

Numerical Methods for the Quadratic Hedging Problem in Markov Models with Jumps

This is a companion paper to [Cai, Rosenbaum and Tankov, Asymptotic lower bounds for optimal tracking: a linear programming approach, Arxiv: 1510.04295]. We consider a class of strategies of feedback form for the problem of tracking and study their performance under the asymptotic framework of the above reference. The strategies depend only on the current state of the system and keep the deviation from the target inside a time-varying domain. Although the dynamics of the target is non-Markovian, it turns out that such strategies are asympototically optimal for a large list of examples.