Q 7.6: Why is it impossible to model an American-style option using a one- period binomial option pricing model?
7.3 OPTION VALUATION, ONE-PERIOD BINOMIAL OPTION PRICING MODEL
The purpose of the binomial option pricing model is to solve for the option value at initiation. Let's explore call option value in the context of the one-period model.
To obtain an equation for call value we form a portfolio consisting of the following two positions:
A portion of the underlying asset A short call
We form this portfolio, as through doing so it will be possible to structure the portfolio so that it has the same payoff whether the underlying asset price increases or decreases. If the portfolio has identical payoffs in both scenarios it is a risk-free investment as its future payoff is known with certainty. If the payoff is risk-free, we can discount it to the present using the risk-free interest rate. Once we do so, we can solve for the option's value. To summarize, the steps through which we will identify the option value are as follows:
Step 1: Structure a portfolio so that it has the same payoff whether the underlying asset price increases or decreases.
Step 2: Discount the portfolio's payoff to the present using the risk-free rate.
Step 3: Solve for the option value.
Let's explore each of these steps.
Step 1: Structure a portfolio so that it has the same payoff whether the underlying asset price increases or decreases.
The cost at initiation and payoff at expiration associated with a portfolio consisting of some portion α of the underlying asset and a short position in the call is illustrated in Figure 7.6.
Figure 7.6 Cost and payoffs associated with a portfolio consisting of α of the underlying asset and a short call
We identify the value of α for which the two payoffs are equal as follows:
Solving, we find:
Step 2: Discount the portfolio's payoff to the present using the risk-free rate.
Since both payoffs are identical, the portfolio is risk-free and we can discount it to the present using the risk-free rate. Let's discount the payoff when the underlying asset price increases:2
where:
= the continuously compounded risk-free interest rate
= the number of periods associated with the given model. Our model is a one-period model; hence, n = 1.
Step 3: Solve for the option value.
We know that the cost of the portfolio consisting of α of the underlying asset and a short position in the call is:
In words, this indicates that the portfolio cost is equal to the cost associated with
acquiring portion α of the underlying asset minus the revenue received through writing the short call. We also know that the present value of its payoff is:
Since the portfolio value should be equal to the present value of its payoff, it follows:
Rearranging this equation we obtain :
With reorganization, one can show that this expression is equivalent to the following:
where:
Let's explore this approach using the example introduced earlier, where:
= $50
= 1.1
= K = $49
T = 1 year n = 1-period
r = 5%
In this example:
As illustrated in Figure 7.7 the payoff is $28.57 in both scenarios.
Figure 7.7 Cost and payoffs example associated with a portfolio consisting of α of the underlying asset and a short call
The value of is:
The value of this call is:
This analysis uses as inputs the factors and , where is the factor by which the underlying asset increases and , which is equal to , is the factor by which the underlying asset decreases. U can be identified as follows, where = underlying asset volatility:
In our example, the value for U was given as 1.1. This value for U is due to the fact that the underlying asset volatility is 9.531%, as the following demonstrates:
The methodology through which we price a call can be used to price any position that has payoffs that are a function of whether the underlying asset price increases or decreases.
In each case, multiply the payoff when the underlying asset increases by and the payoff when the underlying asset decreases by . For example, a put option has payoffs of:
and
when the underlying asset prices increase or decrease, respectively. The expression for a put's value is therefore:
Knowledge check
Q 7.7: What are the steps through which we obtain an equation for call value?
Q 7.8: Why do we form a portfolio of a portion of the underlying asset and a short call?
Q 7.9: Why do we structure a portfolio so that it has the same payoff whether the underlying price increases or decreases?
Q 7.10: What is the expression for call value in the one-period binomial option pricing model?
Q 7.11: What is the relationship between U and volatility?
Q 7.12: What is the expression for put value in the one-period binomial option pricing model?