Robustness of the Black and Scholes Formula

Abstract

Consider an option on a stock whose volatility is unknown and stochastic. An agent assumes this volatility to be a specific function of time and the stock price, knowing that this assumption may result in a misspecification of the volatility. However, if the misspecified volatility dominates the true volatility, then the misspecified price of the option dominates its true price. Moreover, the option hedging strategy computed under the assumption of the misspecified volatility provides an almost sure one-sided hedge for the option under the true volatility. Analogous results hold if the true volatility dominates the misspecified volatility. These comparisons can fail, however, if the misspecified volatility is not assumed to be a function of time and the stock price. The positive results, which apply to both European and American options, are used to obtain a bound and hedge for Asian options.

Model Risk in Incomplete Markets with Jumps

“…It is worth noting that in a complete market setup, the loss process corresponds to the tracking error of [15].…”

(S,s)-adjustment Strategies and Hedging under Markovian Dynamics

“…In the presence of feedback effects, Black-Scholes strategies based on the assumption of a constant volatility produce a tracking error that is almost surely non-zero. El Karoui et al (1998) show how to derive a formula for the tracking error, which measures the difference between the actual and the theoretical value of a self-financing hedge portfolio for a European call calculated from the Black-Scholes formula with constant volatility. Proposition 2 gives us an insight about the behaviour of the tracking error: clearly, for rðc; =Þ sufficiently small, as required in Proposition 2, the tracking error vanishes.…”

Superreplication of European multiasset derivatives with bounded stochastic volatility

“…As already outlined in [1], [9], [17] and [20], natural candidates to be respectively superreplication capital and a Markov superstrategy in this context are…”

“…This payoff can be used as a lower bound for options on the sum of 2 assets (see [9] for details). We suppose that we superhedge this option using a model in which Σ is as in Example 12.…”