Adaptive mixture for a controlled smile: the LT model

Abstract

We build here an arbitrage-free model for an equity-type asset that allows the forward volatility surface to be uncertain today and change profiles over time with some stationarity. Spot and forward distributions are both expressed in the same format through convenient lognormal mixtures ensuring consistent recombinations. The underlying spot process is obtained as the product of a lognormal and a jump variable that can be easily simulated. Closed-form solutions are derived for standard and forward-start European options. The model addresses the drawbacks of local volatilities and is very tractable with respect to jump-diffusion and stochastic vol models. It allows to calibrate, as needed, on both cliquet and vanilla options and offers a convenient framework to observe and control volatility surface evolution.

Acknowledgements

The author would like to thank anonymous referees for helpful comments and suggestions. The ideas presented here do not necessarily reflect the view of Natexis Banques Populaires.

Notes

 The smile usually refers to a non-uniform implied Black–Scholes [3] volatility that varies with respect to option strike and maturity.

 Local volatility [12] can be defined as the instantaneous volatility seen over a very short period of time at a given date and spot level.

 This density is seen from time T_l , spot level S_l and state i.

 This density is seen from time T_l , spot level S_l and state i, which is assumed to switch to j at time T_m .

 In particular, if we set

, the distribution of S at T_m conditional to evolving from state i ₀ to state j between T ₀ and T_m is log-normal with an expected level of

.

 σ _{l, m} is different here from the usual market volatility data. Related variance term-structure v_l will be determined later on.

 We suggest here to respect this order of magnitude and avoid very small or large values that would create a redundant or irregular coverage of the density distribution.

 Using the relationship between

and the constraints on Q(t ₁, t ₂), it is easy to obtain

, which ensures the constraint v(t ₁) ≤ v(t ₂) for

 More generally if t ₁ and t ₂ belong to

respectively with

.

 

can be obtained in general as

The instantaneous transition matrix is given by :

 The forward volatility of Y between T_l and T_m is given by