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

 

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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 Ci, i = 1 . . . n and terminal payoffs Hi, i = 1 . . . n. A static hedge of H is a portfolio composed from the options Hi, i = 1 . . . n and the numeraire, in order to match as closely as possible the terminal payoff of H:

H = V0 + ∑i=1n xiHi + ∫0T φdS + ε —– (1)

where ε is an hedging error representing the nonhedgeable risk. Typically Hi 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 EQ[ε] = 0 (otherwise EQ[ε] can be added to V0). Then the claim H is valued under Q as:

e-rTEQ[H] = V0 ∑i=1n xe-rTEQ[Hi] —– (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 xCi —– (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 Hi correspond to their market prices Ci:

∀i = 1, …, n

e-rTEQ[Hi] = Ci∗ —– (4)

This condition is called calibration, where a pricing rule verifies the calibration of the option prices Ci, 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 Hi. 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.

Self-Financing and Dynamically Hedged Portfolio – Robert Merton’s Option Pricing. Thought of the Day 153.0

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As an alternative to the riskless hedging approach, Robert Merton derived the option pricing equation via the construction of a self-financing and dynamically hedged portfolio containing the risky asset, option and riskless asset (in the form of money market account). Let QS(t) and QV(t) denote the number of units of asset and option in the portfolio, respectively, and MS(t) and MV(t) denote the currency value of QS(t) units of asset and QV(t) units of option, respectively. The self-financing portfolio is set up with zero initial net investment cost and no additional funds are added or withdrawn afterwards. The additional units acquired for one security in the portfolio is completely financed by the sale of another security in the same portfolio. The portfolio is said to be dynamic since its composition is allowed to change over time. For notational convenience, dropping the subscript t for the asset price process St, the option value process Vt and the standard Brownian process Zt. The portfolio value at time t can be expressed as

Π(t) = MS(t) + MV(t) + M(t) = QS(t)S + QV(t)V + M(t) —– (1)

where M(t) is the currency value of the riskless asset invested in a riskless money market account. Suppose the asset price process is governed by the stochastic differential equation (1) in here, we apply the Ito lemma to obtain the differential of the option value V as:

dV = ∂V/∂t dt + ∂V/∂S dS + σ2/2 S22V/∂S2 dt = (∂V/∂t + μS ∂V/∂S σ2/2 S22V/∂S2)dt + σS ∂V/∂S dZ —– (2)

If we formally write the stochastic dynamics of V as

dV/V = μV dt + σV dZ —– (3)

then μV and σV are given by

μV = (∂V/∂t + ρS ∂V/∂S + σ2/2 S22V/∂S2)/V —– (4)

and

σV = (σS ∂V/∂S)/V —– (5)

The instantaneous currency return dΠ(t) of the above portfolio is attributed to the differential price changes of asset and option and interest accrued, and the differential changes in the amount of asset, option and money market account held. The differential of Π(t) is computed as:

dΠ(t) = [QS(t) dS + QV(t) dV + rM(t) dt] + [S dQS(t) + V dQV(t) + dM(t)] —– (6)

where rM(t)dt gives the interest amount earned from the money market account over dt and dM(t) represents the change in the money market account held due to net currency gained/lost from the sale of the underlying asset and option in the portfolio. And if the portfolio is self-financing, the sum of the last three terms in the above equation is zero. The instantaneous portfolio return dΠ(t) can then be expressed as:

dΠ(t) = QS(t) dS + QV(t) dV + rM(t) dt = MS(t) dS/S + MV(t) dV/V +  rM(t) dt —– (7)

Eliminating M(t) between (1) and (7) and expressing dS/S and dV/V in terms of their stochastic dynamics, we obtain

dΠ(t) = [(μ − r)MS(t) + (μV − r)MV(t)]dt + [σMS(t) + σV MV(t)]dZ —– (8)

How can we make the above self-financing portfolio instantaneously riskless so that its return is non-stochastic? This can be achieved by choosing an appropriate proportion of asset and option according to

σMS(t) + σV MV(t) = σS QS(t) + σS ∂V/∂S QV(t) = 0

that is, the number of units of asset and option in the self-financing portfolio must be in the ratio

QS(t)/QV(t) = -∂V/∂S —– (9)

at all times. The above ratio is time dependent, so continuous readjustment of the portfolio is necessary. We now have a dynamic replicating portfolio that is riskless and requires zero initial net investment, so the non-stochastic portfolio return dΠ(t) must be zero.

(8) becomes

0 = [(μ − r)MS(t) + (μV − r)MV(t)]dt

substituting the ratio factor in the above equation, we get

(μ − r)S ∂V/∂S = (μV − r)V —– (10)

Now substituting μfrom (4) into the above equation, we get the black-Scholes equation for V,

∂V/∂t + σ2/2 S22V/∂S2 + rS ∂V/∂S – rV = 0

Suppose we take QV(t) = −1 in the above dynamically hedged self-financing portfolio, that is, the portfolio always shorts one unit of the option. By the ratio factor, the number of units of risky asset held is always kept at the level of ∂V/∂S units, which is changing continuously over time. To maintain a self-financing hedged portfolio that constantly keeps shorting one unit of the option, we need to have both the underlying asset and the riskfree asset (money market account) in the portfolio. The net cash flow resulting in the buying/selling of the risky asset in the dynamic procedure of maintaining ∂V/∂S units of the risky asset is siphoned to the money market account.