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Pricing rule based on non-arbitrage arguments for random volatility and volatility smile

Nikolai Dokuchaev
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Abstract

We consider a generic market model with a single stock and with random volatility. We assume that there is a number of tradable options for that stock with different strike prices. The paper states the problem of finding a pricing rule that gives Black-Scholes price for at-money options and such that the market is arbitrage free for any number of tradable options, even if there are two Brownian motions only: one drives the stock price, the other drives the volatility process. This problem is reduced to solving a parabolic equation.

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Nikolai Dokuchaev. 2002-05-10. Pricing rule based on non-arbitrage arguments for random volatility and volatility smile. https://arxiv.org/abs/math/0205120

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