Analysis of Asset Allocation

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Options: Standard binomial Model

In the Black and Sholes section, we have seen the different steps we have to follow to compute the value of an European style option. But most listed options (especially on equities) are American style.

The possibility of early exercise adds a substantial complexity to the pricing of options. Rather than evaluating the exercise value of the option at expiration (as we do in Black and Scholes), a check must be made at every time period to see if the option price will be worth more exercised than not, and the option price must take on the greater of these two values. Fortunately, the numerical procedure of the binomial pricing method provides a natural way of taking into account these values.

For American options, we must evaluate each period whether the option is worth more alive than it is dead. The dead value is the intrinsic value; the maximum of 0 or S-E for a call, the maximum of 0 and E-S for a put, with S the security price and E the strike price. The option alive can be computed with the following formula (for a call):

C = [p Cu + (1-p) Cd] / r

with:

r = one plus the risk free interest rate for the period.
The assumption that the current price of the security will change by either a factor u (e.g. 1.08) or d (e.g. 0.96) by the next period.
p = (r-d) / (u-d)
Cu = Max(uS-E)

If we apply the changes d and u several times, we will have several possible prices at the final maturity (S1, S2, ...., Sx) and consequently several values for C1, C2, ..., Cx.

The algorithm to compute it in practice  has three steps:

  • The underlying security price is propagated forward by repeating the up (u) and down (d) factors to the price. These factors are used to match the volatility of the security.

  • The 0 Max formula is applied at each S values at the final maturity date.

  • The above one period model is applied in a recursive fashion to bring the option prices back period by period.

We can conclude from the above that the most important question to solve is the number of time periods that are necessary to get a good approximation of the reality. As the Black and Scholes is the binomial model with an infinite number of periods, it is often considered that maximum 25 periods must be considered with the binomial formula. 

As options are very volatile and risky instruments, some variables (Greeks)  have been developed to measure these lasts.

 

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