Incomplete Markets and Calibrations for Coherence with Hedged Portfolios. Thought of the Day 154.0

 

In complete market models such as the Black-Scholes model, probability does not really matter: the “objective” evolution of the asset is only there to define the set of “impossible” events and serves to specify the class of equivalent measures. Thus, two statistical models P₁ ∼ P₂ with equivalent measures lead to the same option prices in a complete market setting.

This is not true anymore in incomplete markets: probabilities matter and model specification has to be taken seriously since it will affect hedging decisions. This situation is more realistic but also more challenging and calls for an integrated approach between option pricing methods and statistical modeling. In incomplete markets, not only does probability matter but attitudes to risk also matter: utility based methods explicitly incorporate these into the hedging problem via utility functions. While these methods are focused on hedging with the underlying asset, common practice is to use liquid call/put options to hedge exotic options. In incomplete markets, options are not redundant assets; therefore, if options are available as hedging instruments they can and should be used to improve hedging performance.

While the lack of liquidity in the options market prevents in practice from using dynamic hedges involving options, options are commonly used for static hedging: call options are frequently used for dealing with volatility or convexity exposures and for hedging barrier options.

What are the implications of hedging with options for the choice of a pricing rule? Consider a contingent claim H and assume that we have as hedging instruments a set of benchmark options with prices C_i^∗, i = 1 . . . n and terminal payoffs H_i, i = 1 . . . n. A static hedge of H is a portfolio composed from the options H_i, i = 1 . . . n and the numeraire, in order to match as closely as possible the terminal payoff of H:

H = V₀ + ∑_i=1ⁿ x_iH_i + ∫₀^T φdS + ε —– (1)

where ε is an hedging error representing the nonhedgeable risk. Typically H_i are payoffs of call or put options and are not possible to replicate using the underlying so adding them to the hedge portfolio increases the span of hedgeable claims and reduces residual risk.

Consider a pricing rule Q. Assume that E_Q[ε] = 0 (otherwise E_Q[ε] can be added to V₀). Then the claim H is valued under Q as:

e^-rTE_Q[H] = V₀ ∑_i=1ⁿ x_i e^-rTE_Q[H_i] —– (2)

since the stochastic integral term, being a Q-martingale, has zero expectation. On the other hand, the cost of setting up the hedging portfolio is:

V₀ + ∑_i=1ⁿ x_i C_i^∗ —– (3)

So the value of the claim given by the pricing rule Q corresponds to the cost of the hedging portfolio if the model prices of the benchmark options H_i correspond to their market prices C_i^∗:

∀i = 1, …, n

e^-rTE_Q[H_i] = C_i^∗ —– (4)

This condition is called calibration, where a pricing rule verifies the calibration of the option prices C_i^∗, i = 1, . . . , n. This condition is necessary to guarantee the coherence between model prices and the cost of hedging with portfolios and if the model is not calibrated then the model price for a claim H may have no relation with the effective cost of hedging it using the available options H_i. If a pricing rule Q is specified in an ad hoc way, the calibration conditions will not be verified, and thus one way to ensure them is to incorporate them as constraints in the choice of the pricing measure Q.

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