The Financial Market Consequences of Growing Awareness: The Case of Implied Volatility Skew
34 Pages Posted: 7 Jan 2014
Date Written: January 1, 2014
The belief that the essence of the Black Scholes model is correct implies that one is unaware that a delta-hedged portfolio is risky, while believing that the proposition, a delta-hedged portfolio is risk-free, is true. Such partial awareness is equivalent to restricted awareness in which one is unaware of the states in which a delta-hedged portfolio is risky. In the continuous limit, two types of restricted awareness are distinguished. 1) Strongly restricted awareness in which one is unaware of the type of the true stochastic process. 2) Weakly restricted awareness, in which one is aware of the type of the true stochastic process, but is unaware of the true parameter values. We apply the generalized principle of no-arbitrage (analogy making) to derive alternatives to the Black Scholes model in each case. If the Black Scholes model represents strongly restricted awareness, then the alternative formula is a generalization of Merton’s jump diffusion formula. If the Black Scholes formula represents weakly restricted awareness, then the alternative formula, first derived in Siddiqi (2013), is a generalization of the Black Scholes formula. Both alternatives generate implied volatility skew. Hence, the sudden appearance of the skew after the crash of 1987 can be understood as the consequence of growing awareness, as investors realized that a delta-hedged portfolio is risky after suffering huge losses in their portfolio-insurance delta-hedges. The different implications of strongly restricted awareness vs. weakly restricted awareness for option pricing are discussed.
Keywords: Partial Awareness, Restricted Awareness, Black Scholes Model, Analogy Making, Generalized Principle of No-Arbitrage, Implied Volatility Skew, Implied Volatility Smile, Portfolio Insurance Delta-Hedge
JEL Classification: G13, G12
Suggested Citation: Suggested Citation