Consider a market which consists of one stock and a continuum of options on that stock (indexed by strikes and maturities). Assume that the volatility of the stock is stochastic. Under the arbitrage free distribution the discounted prices of the stock and all the options must be martingales. We show how this condition can be used to model the dynamics of the stochastic volatility. Finally we apply this idea to the modelling of LIBOR rates with stochastic volatility.

References

• Balland P and Hughston LP (2000), 'Pricing and hedging with a sticky-delta smile', April Risk Conference, Paris.
• Carr P (2000), 'A survey of preference free option valuation with stochastic volatility', April Risk Conference, Paris.
• Reiner E (2000), 'Weak skew limits of implied processes', Working paper.
• Reiner E (2000), 'Understanding skew and smile behaviour in the context of jump processes and applying these results to the pricing and hedging of exotic options', Working paper.
• Schonbucher PJ (1998), 'A market model for stochastic implied volatility', Working paper, University Bonn.

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