American options

1. American options Recall that an American option allows exercise at any time up to the expiry date. For a call option with no dividend payments then the following argument shows that it is never advantageous to exercise it early, so its value is identical to the corresponding European option. If the price is S and the option is exercised then it will yield an amount S-K. However, suppose that instead of exercising the option you sell the stock short (ie sell it without owning it, though you have to deliver it or buy it back eventually). You will then receive S immediately and when the option expires you either pay K for the stock and deliver the stock you have sold short, or you pay the current price if it is below K. So, the choice is between getting S-K now and getting S now and paying at most K later. The latter is obviously preferable. The cost of such an option equals the European equivalent.

2. For a put option the situation is different. The Black-Scholes formula for the European put shows that there are conditions before the expiry date for which P(S,t)<max(K-S, 0). If this were the case for the American option then we could buy the asset for S, at the same time buy the option for P. Then if we immediately exercise the option we sell the share for K and make an instant risk-free profit K-P-S. For any American option the value must always be greater than the payoff, since otherwise we can buy the option, exercise it immediately and make a profit. From the plot in Lectures 6 showing the value of the call option with dividend as a function of S and t it is apparent that the value can fall below the payoff for a European option. So, in this case, the value of the American and European options cannot be the same.

3. To see that we can have P<K-S for the European put recall the put- call parity relation so that Now, by taking S/K small enough we can make the d’ s as small as we like and hence the the N’ s as near to zero as we want. So, in this limit if t<T.

4. The diagram below shows some values of P from the B-S formula. The volatility is 10%, the interest rate 5% and the strike price 100. The full (red) curve is with 1 year to expiry and the dotted (blue) curve with 6 months to expiry. The dashed (green) curve shows the payoff at expiry.

5. To value American options we need to have some criterion to determine when it should be exercised. Typically there is a boundary in (S, t) space on one side of which the option should be exercised, while on the other side it is retained. This gives us a free boundary problem. Similar problems occur in other areas, for example if we calculate the progress of melting of a lump of ice.

6. The American put option We suppose that there is a boundary S=F(t) and that the option should be exercised if S<F(t). (Remember thatthe lower the share price the more advantageous the put offer becomes.) We need to find conditions sufficient to determine F(t). If K is the strike price, we can assume that F(t)<K , since the option is worthless if the share price is above K. Now, the slope of the payoff function max(K-S, 0) is thus -1 where it meets S=F(t). We begin by showing that the slope of the option value as a function of S, ie =∂P/∂Sis also equal to -1 at S=F(t)

7. Suppose first the gradient is <-1, so we have the situation shown below. As S increases from the point of intersection at S=F(t) ( the left hand intersection) then P(S,t) falls below the payoff (the dotted line) which is not allowed, as shown already.

8. Making the option value tangent to the payoff curve maximises the value of the option given that it cannot go below the payoff curve when S>F. The option holder has the right to exercise the option or not, and so is the party to the deal who sets the strategy so as to maximise the option value. The value of the American option is thus found by solving the B-S equation for S>F(t). The value is found in the usual way at the expiry time and the curve F(t) determined by the fact that on the boundary. A moving boundary value problem like this usually needs numerical methods of solution.

9. Implied Volatility The way we have presented the B-S theory we take interest rates, present price of the stock etc and calculate the value of the derivative. Most of the parameters are easily found or, like the strike price and date of expiry, are written into the contract. The one parameter which is difficult to measure in any satisfactory way is the volatility, which does not appear to be constant over sufficiently long periods of time to give an accurate estimate. Nevertheless, options and other derivatives are traded in the market. One approach is to look at the prices on offer for, say, call options and work back to the so-called implied volatility. According to B-S theory there should be a single value for the volatility of a share.

10. What is generally found is that the implied volatility varies with the strike price. Usually it goes through a minimum around the current value of the asset, producing a curve like that illustrated below, known as a “smile”.

11. Various ideas like stochastic volatility have been explored to investigate this. However, it should be borne in mind that B_S theory is based on very idealised notions of geometric Brownian motion for stocks and hedging strategies which are impossible to follow exactly. It is not too surprising if real life is not always in complete agreement with the theory.