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2.6 OPTION VALUATION: A SYNTHETIC OPTION APPROACH

An alternative approach to valuing the option is to replicate the option using the stock and bond. This replicating portfolio is called a synthetic option. The value of this replicating portfolio gives you the arbitrage-free value of the option directly. You can contrast this approach to the riskless hedge approach. The riskless hedge portfolio is also a synthetic security. This portfolio gives you the value of a synthetic riskless bond in terms of a stock and an option. Given that you know the value of the stock and the arbitrage-free value of the synthetic riskless bond, you can derive the value of the option.

Clearly, the two approaches must give you the same answer but the way that you apply the two approaches is different. In the riskless hedge portfolio you may recall that the portfolio consists of the ratio of +1 stock and -k call options (or +1 stock and +k put options) where k is the hedge ratio. For a call option, the hedge ratio is determined from

S - kC = Constant Payoff

and for a put option the hedge ratio is determined from:

S + kP = Constant Payoff

The hedge ratio comes out to be

for the call, and

for the put.

You can interpret the constant payoff as the end-of-period value of a riskless bond B. As a result, you can also determine the option price by using the stock and the riskless bond to replicate the cash flows of the call option (put option). By rearranging the call option example and substituting B for constant payoff, you can verify that:

Constructing a Synthetic Option Directly

Let us now consider how a portfolio that replicates one call option is constructed. Suppose you buy d stocks and borrow $b today, and you choose d and $b so that at the end of the period:

and

Given that we have two equations and two unknowns, we can choose d and b so that:

d is the change in the call value (numerator) divided by the change in the stock price. Therefore, if we choose d and b according to these equations, then our portfolio of d stocks and $b of the risk-free asset has exactly the same cash flows as the call option. But then the price of the call option must equal the price of this (equivalent) portfolio, otherwise one could make a pure arbitrage profit. This means that the cost today is:

Now let us consider an example where X = 30, Su = 40, Sd = 20, and one plus risk-free interest rate r = 1, so

which gives d = 1/2. Therefore, to replicate the call option payoffs, b = -10. We again find that the cost of the synthetic call option is C = (S - 20)/2.

In the synthetic option equation, observe that

d measures the sensitivity of the call price with respect to the price of the underlying stock. It is commonly referred to as the "delta" of the option.

We can also value a European put option in a similar manner. Now, we require

leading to

and the cost of the put is

Note that the delta of a put option is negative since

From either this synthetic option approach or the riskless hedge portfolio approach, we can deduce a very general and important valuation principle known as the Risk-Neutral Valuation Principle.

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