Retrieval of Black–Scholes and generalized Erlang models by perturbed observations at a fixed time

Abstract

S-stable laws on the real line (more generally on Hilbert spaces), associated with some non-linear transformations (so-called “shrinking operations”), were introduced in [Jurek, Z.J., 1977. Limit distributions for truncated random variables. In: Proc. 2nd Vilnius Conference on Probability and Statistics, June 28–July 3, 1977. In: Abstracts of Communications, vol. 3, pp. 95–96; Jurek, Z.J., 1979. Properties of s-stable distribution functions. Bull. Acad. Polon. Sci. Sér. Math. XXVII (1), 135–141; Jurek, Z.J., 1981. Limit distributions for sums of shrunken random variables. Dissertationes Math. vol. CLXXXV]. In [Jurek, Z.J., Neuenschwander, D., 1999. S-stable laws in insurance and finance and generalization to nilpotent Lie groups. J. Theoret. Probab. 12 (4), 1089–1107], the authors interpreted s-stable motions on the real line as limits of total amount of claims processes (up to a deterministic premium) of a portfolio of excess-of-loss reinsurance contracts and showed that they led to Erlang’s model or to Brownian motion. In [Neuenschwander, D., 2000b. On option pricing in models driven by iterated integrals of Brownian motion. In: Mitt. SAV 2000, Heft 1, pp. 35–39], we considered stochastic integrals whose integrand and integrator are both independent Brownian motions, thus modelling a stochastic volatility; as a result we got an analogue of the Black–Scholes formula in this model, confirming a result of Hull and White [Hull, J., White, A., 1987. The pricing of options on assets with stochastic volatility. J. Finance XLII (2), 281–300]. In the present paper, we will look at a common generalization of these processes, namely s-stable motions on the real line perturbed by a stochastic integral whose integrand and integrator are both (not necessarily independent) s-stable motions. The main result will be that if we can observe the distribution of such so-perturbed s-stable motions (together with the values of the perturbing processes) at time t = 1 , then we can identify the whole model (including the perturbation) among all models with Lévy processes perturbed by an iterated stochastic integral of two Lévy processes (in the gaussian case) resp. among all models with a compound Poisson process with drift perturbed by an iterated stochastic integral of two compound Poisson processes (in the completely non-gaussian case if the perturbing processes have no drift) without knowing anything about the history or about its distribution during 0 ≤ t < 1 . This applies, e.g., to a situation where several assets obey the same model and one can estimate the distribution at time one by looking at the values of all these assets at time t = 1 . Interestingly enough, it will be convenient to treat the whole matter in the algebraic framework of the so-called Heisenberg group. This is a concept coming in fact from quantum mechanics and is in a certain sense the simplest non-commutative Lie group.