Marc Decamps

Atomic Implied Volatilities

In this note, we present a novel approach to derive asymptotics for Black implied volatilities under the same generic model as proposed in Antonov and Misirpashaev (2009). We perform a time substitution as used by Duru and Kleinert (1979) to calculate the path integral formulation of the H-atom. We demonstrate that the method provides asymptotic implied volatility formula comparable to the result of Hagan and Woodward (1999) for local volatility models and Hagan et al. (2001) for stochastic volatility models. We also discuss possible application to the pricing of basket options. The method is presented as an alternative to Markov projection as introduced by Piterbarg (2006) and is claimed to be applicable to a wide range of numerical problems arising in finance.

Duru-Kleinert Asymptotic Expansions for Long-Term Foreign Exchange and Swaptions Implied Volatility Smile

In this paper, we develop asymptotic formulas for long-dated Foreign Exchange (FX) and swaptions implied volatilities. We extend the method exposed in Decamps and De Schepper (2009b) to a generic model with time-dependent parameters. Imposing a condition on the skew, we derive averaging formulas for the parameters. The method is applied to the pricing of FX options when the domestic and foreign interest rate curves are driven by Gaussian short-term rate models and to the pricing of swaptions in the Libor market model.

“Geometric Brownian Motion Models for Assets and Liabilities: From Pension Funding to Optimal Dividends”, Hans U. Gerber and Elias S. W. Shiu, January 2003

This discussion relates this excellent paper to the Feynman-Kac theorem. On one hand, standard martingale theory makes it possible to derive a partial differential equation for the moment-generating function of the cash flows at one barrier, as in Gerber and Shiu (2003). On the other hand, we consider a specific Wiener functional for which the calculation of the moment-generating function reduces to solving a partial differential equation. The comparison of both differential equations emphasizes the key role of the intriguing boundary conditions (6.1) to (6.4) imposed by Gerber and Shiu in this paper. As in the last remark of Section 5, payments are only made up to time 0. We define D(t) as the aggregate overflow at the upper barrier from time t to , assuming there is no lower barrier ( 1 0). The authors prove in an elegant way that E[D(t)] K( , t, 2) satisfies the partial differential equation (5.15)

“Pricing Lookback Options and Dynamic Guarantees,” Hans U. Gerber and Elias S. W. Shiu, January 2003

MARC DECAMPS* AND MARC J. GOOVAERTS Actuarial mathematics has been applied in various fields of economics over the years. Among many others, we can cite Hans Bühlmann (1980) who proved that the Esscher transform can be used to describe the Walrasian equilibrium in a pure exchange economy and Hans U. Gerber and Elias S. W. Shiu (1994) who wrote the seminal actuarial paper on option pricing in incomplete markets. In this excellent paper, Gerber and Shiu demonstrate once again that actuarial concepts are suitable in finance. They derive closed-form expressions for the price of lookback options and dynamic guarantees with arguments arising in risk theory. Lookback options have also been investigated in finance and some methods rely on the notion of local time. In this discussion, we aim to relate both approaches and show that the notion of local time can be applicable to actuarial purposes. The Wiener process {X(t) x0 t B(t), t 0} models the log-return of the underlying asset, and the authors are interested in the stoploss premiums E[D(t)] E[(M(t) u) ] of the running maximum M(t) max0 t X( ). As mentioned in the introduction, we can interpret D(t) as the aggregate dividends paid by a company up to time t. The notion of local time was devised by Paul Lévy to measure the time spent by a diffusion in the vicinity of a point. We can bravely define the local time of the diffusion X at the point a as Lt a X