Volatility Curves of Incomplete Markets

Abstract

The graph of the implied volatility of call options as a function of the strike price is called volatility curve. If the options market were perfectly described by the Black-Scholes model, the implied volatility would be independent of the strike price and thus the volatility curve would be a at horizontal line. However the volatility curve of real markets is often found to have recurrent convex shapes called volatility smile and volatility skew. The common approach to explain this phenomena is by assuming that the volatility of the underlying stock is a stochastic process (while in Black-Scholes it is assumed to be a deterministic constant). The main purpose of this project is to propose and explore the idea that the occurrence of non- at volatility curves is the result of market incompleteness. A market is incomplete if it admits more than one risk-neutral probability. In other words, within an incomplete market, investors do not necessarily agree on the market price of risk. The hypothesis that volatility curves are linked to market incompleteness is, at least from a qualitative perspective, reasonable and justified, since the convex shape of volatility curves indicates that investors demand an extra premium for call options which are out of the money, that is to say, they assume that out-of-the money options are more risky than predicted by Black- Scholes. Mathematically this means that investors use a different risk-neutral probability to price call options with different strikes. This hypothesis will be tested quantitatively by using the trinomial model, which is the simplest example of one- dimensional incomplete market.