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oBIN( ) - Constant Step BOPM Derivation

Absence of arbitrage opportunities requires that any two portfolios that have the same terminal payoffs must have the same value. Thus, an option can be valued using the known values of securities used to form a replicating portfolio. Determination of potential payoffs to an option requires a specification for the process of the underlying asset. Here, we assume that S follows a binomial random walk.

 

Initially, assume that there is only one period, and that S can take one of two values at the end of the period,

 

If S = 50, u = 15, d = 0.5, then

Equation Template

 

Assume a European call option is written on S with X = 50. Then at the end of the period,

Equation Template

 

Now consider a replicating portfolio of shares of stock and $B in riskless bonds. The potential values are then in general terms given by:


where, r is defined here to be one plus the riskless rate over a fixed time period.

 

If and B are chosen to equate payoffs, then

Equation Template

 

This gives two equations in two unknowns, with the following solutions

Equation Template

 

To prevent arbitrage, we require that

Equation Template

 

Substituting using the previous definitions of and B gives:

Equation Template

 

 

 

(1)

where,

Equation Template

Note that the option value is not dependent on the expected stock return or q. Thus C will be agreed on by all investors, irrespective of their degree of risk aversion or subjective opinions regarding the probabilities of upward or downward moves. Thus, if risk attitudes are not important we can assume that everyone is risk neutral and the expected return for all assets is the riskless rate. In fact, this is implied by the definition of p. If we set the expected return equal to the riskless rate, then:


Equation Template

 


(2)

Also, as 0 < p < 1 and p + (1 - p) = 1, we can interpret p as a probability a probability in a risk neutral world. With this interpretation, C can be regarded as the discounted expected value in a risk neutral world.

 

Now, how can we extend the analysis to a two-period world? (Assuming that the outcome at the second step is independent from that at the first step the probability of an up-jump at time 2 is the same irrespective of whether there was an up- or down-jump at the previous step).

oBIN( ) Derivation 11

 

Given the stock price process, we know Cuu, Cud, and Cdd. We can determine Cu and Cd in the same manner as that used to determine C above this reduces to applying equation (1).

Equation Template

 

And, working back to the present we have,

Equation Template

 

Note how this reflects that for the two step process, we have one path to both Cuu and Cdd (with probabilities p2 and (1 - p)2 and two paths to Cud (with probability p(1 - p). Thus C is simply the expected value of the option at maturity, discounted back to today. With that observation, we can generalize the process to n periods.

 

After n binomial steps, there are n + 1 potential outcomes and 2n potential paths. For n = 3, the outcomes are d3, ud2, u2d and u3, and there are 23 = 8 potential paths.

Equation Template

 

In general, the number of possible paths to a particular terminal outcome (with j up-jumps from a total of n jumps) are computed as:

Equation Template

 

The total set of possible outcomes after n steps isEquation Template for j = 0, 1, 2, .... , n. Recall that call value can be determined as the discounted expected payoff at maturity. That is:

Equation Template

 

If a is the minimum number of up-jumps such that Equation Template, then

Equation Template

 

Note that the second term in square brackets is the complementary (cumulative) binomial distribution function, Equation Template. That is, the cumulative probability that there will be at least a successes from n trials. We can also interpret the first expression in square brackets as a complementary binomial distribution, Equation Template if we define

Equation Template

Thus

Equation Template

where, a is the smallest non-negative integer such that Equation Template, or the smallest integer greater than:

Equation Template

 

For realistic applications of the Binomial model, we need to divide the fixed calendar interval (T) into a 'large enough' number of intervals (n) such that the binomial step in prices over each interval of length Equation Template is a reasonable approximation to reality.

 

As Equation Template and in the limit we have a continuous stochastic process. But, we still need to choose appropriate estimates for u, d, (and therefore p). It makes sense that these be chosen such that the mean and variance of stock price changes over the interval h match those implied by the risk-neutral version of the Geometric Brownian Motion process, i.e Equation Template. Cox, Ross and Rubinstein (1979) show that this requires:

 

Equation Template


where r is now defined as the continuously compounded riskless rate. With these choices, the binomial process for stock prices approaches the GBM process when Equation Template.

 

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