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# Option Pricing Examples

In this section, I explore the implications of theorems 5 and 6 through the use of three option pricing examples. In the first, I consider how options are priced in the example from sections 4 and 5. In the second, I use a binomial option model. In the third, I use a continuous time diffusion model.

Each example illustrates a different facet of the determinants of option prices. The first example highlights the nature of the risk-neutral density function. I constructed this example, so that when traders hold objective beliefs, the risk-netural density is (approximately) log-normal. I then point out that heterogeneity causes the risk-neutral density to be multi-modal and fat tailed, just like the the probability density of the representative trader. I note that this example features a single period and many discrete states. Equation (24) implies that Black-Scholes applies to single period models as well as to continuous time models. I use the binomial example to discuss why heterogeneity causes interest rates and volatility to be stochastic, and how this affects option prices. In the binomial example, both interest rates and volatility are constant under homogeneity, but stochastic under heterogeneity. In the limit, Black-Scholes holds in the homogeneous case, but not in the heterogeneous case. Of course, the limit involves continuous time, which leads to the third example. I use the continuous time example to investigate the character of option smiles.

The continuous time example can be obtained from the binomial model through a suitable limiting procedure. Notably, I demonstrate that heterogeneity introduces smile effects into equilibrium option prices, and leads implied volatilties for call options to differ from implied volatilities for put options, even when both share the same exercise price. Consider the first example. When both traders hold objectively correct beliefs, then by theorem 3 the interest rate will be 5.35%,27 and the representative trader’s density function will be (approximately) lognormal.

Consider a security whose date 0 price is 1, and whose return has the same (lognormal) distribution as the market portfolio. In this case, the Black-Scholes formula can be used to compute the equilibrium price of an option defined on this security. For example, a call option with an exercise price of K = 1.05, expiring at date 1, will have an equilibrium price of 0.0166.28 As I discussed earlier, when traders hold heterogeneous beliefs, the interest rate falls from 5.35% to 4.95%. The equilibrium option price, computed directly using the one-period interest rate and pricing kernel (from equation (24)) is 0.0314. However, the Black-Scholes price is 0.0148, a lower value. I note that the equilibrium price of 0.0314 is higher than the Black-Scholes price of 0.0148 because the bulls drive up the price of the call option.

That is, traders’ beliefs affect option prices, that being the focus of option pricing expression (22). The difference between the equilibrium option price and Black-Scholes price is caused by nonzero sentiment. Nonzero sentiment affects more than the interest rate. Nonzero sentiment also distorts the risk-neutral density function, causing it to depart from lognormality.

See figure 3 which contrasts the risk-neutral density when sentiment is zero (the objective case) with the bimodal, fat-tailed density associated with nonzero sentiment in the example.

I turn my attention to the second example, involving a binomial model. Assume that there is a single physical asset that produces a single consumption good at each date. The amount of the good available for consumption at date 0 is 1 unit. Thereafter, aggregate consumption will grow stochastically from date to date, either at rate u (with probability πu) or at rate d (with probability 1−πu.) The market portfolio is a security that pays the value of aggregate consumption at each date. Let u = 1.05, and πu = 0.7. Let there be two traders in the model, and assume that each initially holds one half of the market portfolio. There is also a risk-free security available for trade at each date.

Because of the binomial character of uncertainty, these two securities will be sufficient to complete the market. Both traders are assumed to have additively separable preferences, logarithmic utility, and discount factors equal to unity (zero impatience). They also hold beliefs about the branch probability in the binomial tree. Trader 1 assumes that the value of the branch probability Pu is P1,u, while trader 2 believes the value to be P2,u. Each trader seeks to maximize subjective expected utility subject to the condition that the present value of lifetime consumption be equal to initial wealth. The single budget constraint here stems from markets being complete.

When P 1,u ≠ P 2,u traders have heterogeneous beliefs. As I mentioned earlier, and discuss in the appendix, equilibrium prices can be characterized through the beliefs of a representative trader R, whose tree probabilities are a convex combination of the tree probabilities of the individual traders, where the weights are given by relative wealth. Because the two traders in this example have the same wealth at date 0, the representative trader attaches probability

to the occurrence of an up-move at the end of date 0. The probability that the representative trader attaches at date 0 to two successive up moves, occuring at the end of date 0 and the end of date 1 respectively, is:

which is the (relative wealth-weighted) average of the two traders’ binomial probabilities attached to the node in question. For general P 1,u and P 2,u, the equilbrium state price vu in this example satisfies: [29]

This equation illustrates the fact that in equilibrium, state prices can be expressed as weighted sums of state prices derived from corresponding homogeneous belief cases. It follows that all security prices can be expressed as weighted sums of security prices derived from corresponding homogeneous belief cases.[30] For ease of reference, I call this the weighted average property.

In the log-utility binomial example, the equation for the equilibrium interest rate is:[31]

This equation implies that in the case of homogeneous beliefs, the short-term interest rate will be constant over time. However, in the case of heterogeneous beliefs about the true value of the binomial branch probability, the equilibrium short-term interest rate will be stochastic. The reason for this involves the weighted average property described earlier. To illustrate the impact of heterogeneity on interest rates, consider four cases. In three of the cases, the two traders agree about the value of P u. In the first case, both correctly believe its value to be 0.7. In the second case, both believe its value to be 0.8. In the third case, both believe its value to be 0.6. And in the fourth case, trader 1 believes its value to be 0.8 while trader 2 believes its value to be 0.6. Some computation shows

The technique used to find the equilibrium interest rate for the heterogeneous case, based on four cases, applies to all securities including options. It is simply a matter of invoking the weighted average property, and taking a weighted average of prices for corresponding homogeneous cases. Here is a brief illustration. Take the first of the four cases, the case when both correctly believe its value to be 0.7, and the equilbrium interest rate is 1.87%. Consider an underlying asset for the option that pays a zero dividend. Define a security Z so that it has the same (dividend) payoff as the market portfolio for the four dates 2 through 5 inclusive, but pays no dividend prior to date 2. By constructing the state prices from vu, it is easily verified that this security has a price of 4.00 at date 0, and that its price either grows by a factor of u or d in every period before the option expiration date.

Figure 4 shows the standard procedure for computing the price of a European call option on this security that expires at t = 2 and has an exercise price K = 3.80. The price of the option at date 0 is 0.355. The same procedure can be employed to compute the option price for the second and third of the four cases.

Note that in each of these cases, a different set of common beliefs gives rise to a different value for the equilibrium interest rate. In addition to causing interest rates to be volatile, heterogeneity alters the the representative trader’s probability density, which in turn alters the return standard deviation of the asset underlying the option. That is, heterogeneity induces both stochastic interest rates and stochastic volatility. I note that these are the primary channels through which heterogeneity would be seen to impact option prices in traditional partial equilibrium reduced form frameworks.

For this example, heterogeneity impacts no impact through the return distribution of the underlying asset. In section 10, I establish that when utility is logarithmic, the equilibrium price of the market portfolio at any node in the uncertainty tree is independent of traders’ underlying beliefs. Therefore the only impact of different beliefs on the option price above occurs through the interest rate and volatility. I turn to the third example. Consider the implications of heterogeneity for option pricing in continuous time. In the standard binomial option pricing model, the interest rate is constant, and under a suitable limiting argument, the binomial option price converges to the Black-Scholes formula. I note that when traders are homogeneous, the standard limiting argument applies to the binomial general equilibrium model discussed above.

However, as I noted, heterogeneity leads interest rates to be stochastic. In turn, the volatility of shortterm interest rates implies that the one-period conditional binomial state prices do not remain invariant over time. Notably, this disrupts the usual limiting argument developed by Cox, Ross, and Rubinstein (1979), where the Black-Scholes pricing equation is achieved as a limiting case of the binomial option pricing formula. Put another way, heterogenity tends to prevent the conditions necessary for Black-Scholes pricing from holding.33 I note that the weighted average property holds in continuous time, just as it does in discrete time. In their continuous time log-utility model, Detemple-Murthy (1994) establish that when markets are complete, “the price of any contingent claim in the heterogeneous beliefs economy is a weighted average of the prices that would prevail in two single agent economies respectively populated by agents of type 1 and 2.” See p. 311 of their article.

This implies that in the continuous time version of my example, the equilibrium option price is a weighted average of Black-Scholes functions. To state this condition formally, consider the Black-Scholes formula CBS for the price of a call option:

where

Note that qz is the initial price, K is the strike price, σ denotes the return standard deviation of the underlying asset, t is the time to expiration, and r is the continuous compounding rate of interest.

Consider a continuous time limiting version of the binomial example above, in which there are two log-utility traders with equal initial wealth. Imagine a European option on a security Z, whose price at t = 0 is qz, and whose return is lognormally distributed with standard deviation . Given that Black-Scholes may fail to hold in equilibrium, how will the option be priced? To answer this question, invoke the weighted average property. That is, consider two situations. In the first situation, all traders agree with trader 1, the equilibrium value of Z is qz, its return standard deviation is σ, and the equilibrium continuously compounded interest rate is r1. In the second situation, all traders agree with trader 2, the equilibrium value of Z is qz, its return standard deviation is σ, and the equilibrium continuously compounded interest rate is r2. Note that because I use a general equilibrium framework, the interest rate is endogenous. The weighted average property implies that:

Consider an example with extreme values to highlight the properties of the previous equation. Specifically, let r1 = 50% and r2 = −50%. Equation (3.8) on p. 302 of Detemple- Murthy implies that the weighted average property applies to the instantaneous interest rate. Hence, req = 0%. Let qz = 4 and σ = 30%.

Figure 5a shows how four call option prices discussed in this example vary as a function of K. The top curve in figure 5a pertains to the case r1 = 50%, while the bottom curve pertains to the case r2 = −50%. The curves in the middle are for the equilibrium option prices (solid curve), and Black-Scholes prices (dashed curve).

Figure 5b provides another view of how the difference between the equilibrium call option price and Black- Scholes price varies as a function of the exercise price K. Notice that the pattern is cyclical, and is negative for low values of K.

The Black-Scholes formula for the price of a put option is:

The equilibrium price of a put option can be obtained in the same manner as a call option, with an analogous expression

figure 6a

figure 6a

Figures 6a and 6b are the counterparts to figures 5a and 5b.Consider what happens when, for an interval of exercise prices, we infer the implied Black-Scholes volatilities from the equilibrium prices of options. To do so, we solve

and

for σ as implicit functions of K. Figure 7 illustrates the nature of the volatility patterns associated with these implicit functions.

Notice several features about the volatility patterns. First, the implied volatilities are different for calls than for puts. Second, neither pattern is flat.34 In a world where Black Scholes holds, both curves would coincide with one another and be flat. Third, the implied volatility lies above the actual volatility for most of the range, including the case when options are at-the-money. Fourth, the implied volatility may be undefined at low exercise prices, particularly in the case of call options.

Bates (1996), Bakshi, Cao, and Chen (1997), and Pan (2001) discuss index option pricing models that feature stochastic volatility and stochastic interest rates, especially as they relate to smile effects. This literature is partial equilibrium-based, and assumes that the kernel only reflects fundamentals.

In contrast, my model is general equilibrium-based, and stresses that the pricing kernel reflects both underlying fundamentals and sentiment. Recall that the second example features stochastic volatility as well as stochastic interest rates. In this respect, I mention that in the binomial model, volatility becomes constant in the limit. This is best seen by considering what happens in the limiting process that leads the binomial process to converge to a continuous time diffusion process. The analysis in Cox, Ross, and Rubinstein (1979) implies that the branch probabilities u and vu both converge to 1/2 in the limit (p. 249).

This implies that there is little room for the representative trader’s branch probabilities to move during a short interval. Hence in the limit, u alone determines volatility. But since the value of u is fixed at each n, where n denotes the number of trials in the binomial process, volatility is virtually constant for large n. Hence volatility is constant in the limit as the binomial process converges to a diffusion process. This means that in the third example, the failure of Black-Scholes stems entirely from stochastic interest rates. This feature is essentially an artifact of starting out with a binomial model. Were we to begin with a model that featured more than two states, then it would be possible for there to be heterogeneous beliefs about volatility in the limit, and therefore stochastic volatility in the limit.

In principle, my heterogeneous-based approach to smiles need not conflict with the approach in the literature focusing on stochastic volatility in a jump diffusion model. To be sure, sentiment can be a primary determinant of both stochastic volatility, and jumps. To the extent that the approaches do conflict, that conflict will revolve around the structure of the pricing kernel. For instance, the pricing kernel described by Pan (2001) features both Brownian shocks and jumps. Is the nature of kernel in this type of setting equivalent to the type of kernel that arises when beliefs are heterogeneous? It seems to me that although the jumps and stochastic volatility give rise to some of the same features that arise because of heterogeneity - multi-modality comes to mind - the two frameworks can feature quite dissimilar kernels.[35]

Prof. Hersh Shefrin

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