Option prices as probabilities

Dilip Madan (1) , Marc Yor (2) , Bernard Roynette (3, 4)

1 University of Maryland [College Park]
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
3 IECN - Institut Élie Cartan de Nancy
4 TOSCA - Simuler et calibrer des modèles stochastiques

Four distribution functions are associated with call and put prices seen as functions of their strike and maturity. The random variables associated with these distributions are identified when the process for moneyness defined as the stock price relative to the forward price is a positive local martingale with no positive jumps that tends to zero at infinity. Results on calls require moneyness to be a continuous martingale as well. It is shown that for puts the distributions in the strike are those for the remaining supremum while for calls, they relate to the remaining infimum. In maturity we see the distribution functions for the last passage times of moneyness to strike.

Mots clés: Calendar spreads; Spectrally negative processes; Black–Scholes formula

Domaines: Probabilités [math.PR]

Dilip Madan, Marc Yor, Bernard Roynette. Option prices as probabilities. Finance Research Letters, 2008, 5 (2), pp.79-87. ⟨10.1016/j.frl.2008.02.002⟩. ⟨hal-00292135⟩