Pricing of European options

We recall the equilibrium setting from the previous section 5.4. There is one asset iwith a different volatility than all other N−1 assets. In this section, we derive the price of European options written on this particular underlying asset i.

The payoff C^ei of a call option with strike priceX and the payoff P^ei of a put option with the same strike price X are given by

Cei= maxⁿY^ei−X, 0^o, (5.12) Pei= maxⁿX−Y^ei, 0^o. (5.13)

Given the distribution assumption from the previous section, the payoff of the underlying asset Y^ei can become negative with a positive probability. Hence, a negative strike price would be feasible. Even though the following valuation formulae are valid for any arbitrary strike price, it makes sense to restrict the analysis to positive exercise prices X >0.

We first derive the call price and later use the put-call-parity to price the put option. The fundamental pricing equation (5.4), which is valid for any arbitrary payoff distribution and all well defined utility functions, can be rewritten in terms of prices. (See appendix

5.C.1 for details.) The call price C0i equals

Intuitively, the price of an asset is equal to the expected payoff discounted at the risk-free rate minus an adjustment term. The price is reduced by the adjustment term when the covariance between the asset payoff and the marginal utility is positive, i.e., whenever the beta factor of the asset is positive.

In the next step, we make use of the distribution assumptions for the asset payoffs and correlations from section 5.4.1. In addition, we recall the assumption for the representa-tive investor’s utility function (see equation 5.5). Applying Stein’s lemma results in the following call price. (See appendix 5.C.2 for a detailed derivation.)

C0i= 1 whereϕ denotes the probability density function of the standard normal distribution and Φ denotes the corresponding cumulative distribution function. As in the previous section, vmis a shorthand notation for the market variance,ccabbreviates the covariance between the call payoff and the market portfolio, and em is the weighting term. The shorthand mp denotes the market risk premium in dollar terms.

For comparison, the call price C^b0i in the standard CAPM, i.e., when using the expected correlation for pricing, is given by

Similarly to expression (5.10) for the expected return of the underlying asset, the weighting term em drops out of the formula.

prices in the two models are different. This difference is illustrated in figure 5.6 using the parameters from the base case calibration from section 5.4.3. The plot shows the price difference C0i−C^b0i in percent of the true call price C0i for three different strike prices depending on the volatility δi of the underlying asset. The solid line represents an in-the-money call (withX= 0.9). The dashed line depicts the values for an at-the-money call (i.e., X= 1.0). The dot-dashed line shows an out-of-the-money call (with X= 1.1).

The graphs show that the price of call options on low volatility assets as predicted by the standard CAPM is lower compared to the price in the correlation risk model. The opposite is true for call options on high volatility assets.

We have already stated in proposition 5.6 that low volatility assets have a higher risk-adjusted return compared to the prediction by the standard CAPM. This higher return corresponds to a lower price. Similarly, call options on low volatility assets exhibit prices below the standard CAPM prediction. Evidently, this observation raises the question whether the price deviation of the call option is resulting from the price deviation of the underlying asset or whether the call option itself has a property which causes and potentially magnifies the effect. The answer to this question is given by the following proposition. (See appendix 5.C.3 for proof.)

Proposition 5.7 (Call pricing)

The price of a European call option written on an underlying asset with low volatility (δi< δ⁻i) is lower than predicted by the standard CAPM, i.e., C0i<C^b0i. The European call option on an underlying asset with high volatility (δ_i> δ−i) is more expensive than predicted by the standard CAPM, i.e., C0i>C^b0i. The price deviation stems solely from the different prices of the underlying asset.

To understand this result, we rearrange the expression for the call price, such that it has the following form

C0i=Y0i·Φ−^X_δ⁻_i¹− X

1 +r_f ·Φ−^X_δ⁻_i¹+ δ_i

(1 +r_f)·ϕ^X_δ⁻¹

i

. (5.22)

This expression of the call price resembles similarities to the well known option pricing of Black and Scholes (1973). The call price is a function of only five variables: the underlying asset value Y0i, the underlying asset’s volatility δ_i, the risk free rate r_f, the strike price X and the time to maturity, which is standardized to 1 in this setting for simplicity.

Notably, the weighting termem is only contained in the price of the underlying assetY0i. Hence, equation (5.22) is valid for both the call price in the correlation risk model as well

Figure 5.6: Difference between call prices

The plot shows the price difference C0i−C^b0i in percent of the true call price C0i for three different strike prices X= 0.9(solid line), X= 1.0(dashed line), and X= 1.1 (dot-dashed line) depending on the volatility δi of the underlying asset. The further parameter values are given in table 5.2.

δi

C0i−C^b0i

C0i

0.0 0.1 0.2 0.3 0.4 0.5

+4%

+2%

0

−2%

−4%

−6%

−8%

price deviation of the call originates only from the price deviation of the underlying asset.

Next, we repeat the derivation for put options. The price of the put option can easily be obtained by applying the put-call parity. The resulting put price is

P0i=−Y0i·Φ^X_δ⁻_i¹+ X

1 +r_f ·Φ^X_δ⁻_i¹+ δi

(1 +r_f)·ϕ^X_δ⁻_i¹. (5.23) In analogy to proposition 5.7, we can summarize the findings regarding put options in the following proposition. (See appendix 5.C.3 for proof.)

Proposition 5.8 (Put pricing)

The price of a European put option written on an underlying asset with low volatility (δi< δ⁻i) is higher than predicted by the standard CAPM, i.e., P0i>P^b0i. The European put option on an underlying asset with high volatility (δi> δ⁻i) is cheaper than predicted by the standard CAPM, i.e., P0i<P^b0i. The price deviation stems solely from the different prices of the underlying asset.