Abstract
In option pricing, it is customary to first specify a stochastic underlying model and then extract valuation equations from it. However, it is possible to reverse this paradigm: starting from an arbitrage-free option valuation formula, one could derive a family of risk-neutral probabilities and a corresponding risk-neutral underlying asset process. In this paper, we start from two simple arbitrage-free valuation equations, inspired by the log-sum-exponential function and an ell ^{p} vector norm. Such expressions lead respectively to logistic and Dagum (or “log-skew-logistic”) risk-neutral distributions for the underlying security price. We proceed to exhibit supporting martingale processes of additive type for underlying securities having as time marginals two such distributions. By construction, these processes produce closed-form valuation equations which are even simpler than those of the Bachelier and Samuelson–Black–Scholes models. Additive logistic processes provide parsimonious and simple option pricing models capturing various important stylised facts at the minimum price of a single market observable input.
Highlights
The classic early asset pricing models by Bachelier [2] and Samuelson–Black–Scholes (SBS) (Samuelson [29], Black and Scholes [5]) set at least two paradigms in deriva- Parma, ItalyP
This way of proceeding is rather logical, since it is intrinsic in the fundamental theorem of asset pricing that specifying directly a risk-neutral distribution for the underlying must lead to a no-arbitrage valuation formula
We have demonstrated that simple no-arbitrage valuation formulae can produce risk-neutral distributions fully supported by additive processes
Summary
The classic early asset pricing models by Bachelier [2] and Samuelson–Black–Scholes (SBS) (Samuelson [29], Black and Scholes [5]) set at least two paradigms in deriva-. We present two extremely simple no-arbitrage option valuation formulae that produce, in the modelling approach described above, risk-neutral distributions of logistic type. As it turns out, there exists a class of infinitely divisible distributions, the generalised z-distributions (GZD) introduced by Grigelionis [17], whose associated processes retain a simple and yet rich structure, able to naturally accommodate logistic marginals. After an appropriate measure change, we are able to present physical (non-martingale) dynamics for the involved processes which, not logistic, still belong to the GZD class This ideally concludes our “reverse trek” in stochastic modelling starting from, rather than leading to, no-arbitrage option prices.
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