Osama Alrefai

American and European Options

Chapter 7 & 8 presented Options and their valuation. I had a hard time understanding how to find the value of an option. In fact, I even struggled in understanding the concept of the Option. Therefore, I went to some websites and fouind a very good way to calculate options. Before calculating options, I looked at the concept itself and here is the way that helped me understand it better.

The key difference between an American and a European options relates to when the option can be exercised:
- A European option may be exercised only at the expiry date of the option, i.e. at a single pre-defined point in time
- An American option on the other hand may be exercised at any time before the expiry date
For both, the pay-off - when it occurs - is:
Max [ (S – K), 0 ], for a call option
Max [ (K – S), 0 ], for a put option
Where K is the Strike Price and S is the Spot Price of the underlying asset

Difference in Value

European options are typically valued using the Black-Scholes or Black model formula. American options can be valued using the Binomial Pricing Model. American options are rarely exercised early. This is because any option has a non-negative time value and is usually worth more unexercised. Owners who wish to realise the full value of their option will mostly prefer to sell it on, rather than exercise it immediately, sacrificing the time value. Where an American and a European option are otherwise identical (having the same strike price, etc.), the American option will be worth at least as much as the European (which it entails). If it is worth more, then the difference is a guide to the likelihood of early exercise. In practice, one can calculate the Black-Scholes price of a European option that is equivalent to the American option (except for the exercise dates of course). The difference between the two prices can then be used to calibrate the more complex American option model.

To account for the American's higher value there must be some situations in which it is optimal to exercise the American option before the expiration date. This can arise in several ways, such as:
- An in the money (ITM) call option on a stock is often exercised just before the stock pays a dividend which would lower its value by more than the option's remaining time value
- A deep ITM currency option (FX option) where the strike currency has a lower interest rate than the currency to be received will often be exercised early because the time value sacrificed is less valuable than the expected depreciation of the received currency against the strike.
- An American bond option on the dirty price of a bond (such as some convertible bonds) may be exercised immediately if ITM and a coupon is due.
- A put option on gold will be exercised early when deep ITM, because gold tends to hold its value whereas the currency used as the strike is often expected to lose value through inflation if the holder waits until final maturity to exercise the option (they will almost certainly exercise a contract deep ITM, minimizing its time value).

Valuing American Options Using Binomial Pricing Model

The binomial pricing model uses a "discrete-time framework" to trace the evolution of the option's key underlying variable via a binomial lattice (tree), for a given number of time steps between valuation date and option expiration. The valuation process is iterative, starting at each final node, and then working backwards through the tree to the first node (valuation date), where the calculated result is the value of the option. Option valuation using this method is, as described, a three step process:
1- price tree generation
2- calculation of option value at each final node
3- progressive calculation of option value at each earlier node; the value at the first node is the value of the option.

I developed a simple way in MS Excel to calculate the option value using Binomial Pricing Model using the following equations assuming an option for 2 years:

A B C D E F G H I J
1 Find Pi u d rf % Change (u) % Change (d)
2 =(1+D2-C2) / (B2-C2) =1+E2 =1-F2 6% 47% 47%
3 Binomial
4 S0 Sd Su rf vu vd Delta B No Aribtrage V V Using Pi
5 144.06 211.77 76.35 6.0% 0 24.64 =(E5 - F5) / (B5-C5) =(E5 - G5*B5) / (1 + D5) =G5*A5 + H5 =(E5*A2 + F5 * (1 - A2))/(1+D5)
6 51.94 76.35 27.53 6.0% 24.65 73.47 =(E6 - F6) / (B6-C6) =(E6 - G6*B6) / (1 + D6) =G6*A6 + H6 =(E6*A2 + F6* (1-A2)) / (1+D6)
7 98 144.06 51.94 6.0% =I5 =I6 =(E7 - F7) / (B7-C7) =(E7 - G7*B7) / (1 + D7) =G7*A7 + H7 =(I5*A2 + I6 * (1 - A2)) / (1+D7)

Citations

Option style, wikipedia http://en.wikipedia.org/wiki/American_options
Binomial options pricing model, wikipedia http://en.wikipedia.org/wiki/Binomial_options_model