﻿ 6.8 Application to a Futures Example

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2.8   Reconciliation of Risk-Neutral Valuation with CAPM

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ow can you reconcile risk-neutral valuation with a model of asset price determination such as CAPM, where you solve a relatively complex problem to value assets?

For the underlying stock price, S, there is no conflict because S can come about from any asset pricing model.  In fact, we have simply assumed that we know S, and we do not "determine" it in any way.  For the option price, the answer lies in the observation that we can form a synthetic put or call option from a portfolio of the stock and the bond.  As a result, all the risk from holding say a call option can be diversified away using the stock and the bond.

This is the essence of why an option is a vehicle for transferring risk from one party to another, but it does not change the total risk.   Any premium the market may be paying in terms of additional return for risk is captured by the current stock price S.  Hence, risk considerations enter the option price through S, since the option contract does not create any additional problem of pricing risk.

The risk-neutral valuation principle applies for certain models of stock price movements, but must be modified for others.  For example, it does not generally apply if the stock can move up, down, or stay the same.

To see this, consider the payoffs to a one-period call option with Sd<X<S<Su.  The call has payoffs Su-X, S-X, and 0.  If we try to replicate this by holding n stocks and  m bonds, we would have to satisfy

n Su + m Br  =  Su - X

n S   + m Br  =  S   - X

and

n Sd  + m Br  =  0,

where B is the dollar amount we invest in the risk-free bond.

This gives us three equations with two unknowns, which generally has no solution.

The problem is that we can no longer diversify all the risk associated with the option using only the stock and the bond.  As a result, there is now a source of risk associated with the option contract that must be priced by the capital market.

However, if there is a third "independent" asset, whose payoffs are correlated with the stock, then we could again diversify all the risk from the option.  This would let us obtain the price of the call option in terms of the stock price, the risk-free interest rate, and the price of this third asset, without any reference to the market price of risk.

Any portfolio that creates a synthetic riskless bond can be valued without any reference to the market price of risk.  In the next topic, Put-Call Parity: European Options,  we analyze a synthetic riskless security that is constructed from the stock and its underlying options.