23.1.1 European Put

We price a two-period European put with using RNV. The tree for the stock price has , , , and the risk-neutral probability . First we calculate the payoffs at maturity (Figure 23.1) , and . We then move backwards through the tree:

(23.1a)

(23.1b)

The (European) put premium is:

(23.2)

23.1.2 American Put

For an American put option the payoffs at maturity are the same as for the European put. But early exercise may be profitable at nodes and at . To price the American put we see if the intrinsic value (which is the cash received for early exercise) at any of the nodes is greater than the ‘recursive values’¹ for the put, or (that is, the value of the put if you do not exercise it at that node). If > ‘recursive value’ then early exercise is profitable and we replace the recursive value, ‘ or ’ with the at that node. Expressed mathematically the payoff to an American put at node-U is:

At node-U, and so early exercise is not profitable and we retain in the tree. At node-D, and so early exercise is profitable and we replace in the tree with . The ‘recursive value’ at now becomes:

Finally, we compare and , which indicates that early exercise is not profitable at and hence the American put premium is . Notice that the American put is worth more than the equivalent European put (which has ) – the American put has extra ‘optionality’ as it allows for the possibility that early exercise may be profitable.

This general principle of working backwards through the tree and seeing if early exercise is worthwhile applies when pricing all types of American options: on stock indices, currencies, futures contracts and commodities (e.g. corn, oil, gas etc.).

23.2.1 Continuous Dividend Yield

If a stock has a continuous annual dividend yield (decimal), then the (total) return on the stock consists of the capital gain plus the dividend yield , where is the growth (or proportionate change) in the stock price. In a risk-neutral world, the (total) return on a stock must equal the risk-free rate and hence, the expected growth in the stock price equals and . In a risk-neutral world, the expected stock price at time is given by:

Hence:

(23.4)

where .

When using the binomial recursion formula to price an option on a stock which pays continuous dividends at a (constant) rate , the only change required is that ‘’ in the definition of is replaced by . The values and remain unchanged.

23.2.2 Options on Foreign Currency and Futures

To price options on a foreign currency, the equivalent to is the foreign risk-free rate . For options on a futures contract the underlying futures price grows at the rate , hence the definitions for become:

• Option on foreign currency:
• Option on futures:

When setting up the tree, and remain unchanged and is the real world historical volatility of the underlying asset (i.e. volatility of the stock return, volatility of the return on the exchange rate or the volatility of the return of the futures price, depending on the types of option being considered). Hence, the binomial tree for the underlying asset is constructed in the usual way. Hence, the only difference in the BOPM is the definition of ‘’ (and hence in Equation (23.4)).

The maturity of the option is divided into or more time steps, with each step – this gives a reasonably accurate value for the option's price. For a European option on a stock we only require the payoffs at maturity to price the option. For time steps there are terminal stock prices – which is manageable. But for there are or about a billion alternative stock price paths (e.g. even for there are 8 possible paths). Since many exotic options are path dependent, the BOPM with takes considerable computing power and it may take quite a while for the computer to ‘grind out’ a value for the option price. Hence various methods to speed up the calculations are used, such as trinomial trees and control variate techniques.

23.2.3 Control Variate Techniques

Control variate techniques can be used in the BOPM framework to obtain more accurate values for option premia (for any fixed number of nodes, in the tree). To illustrate this approach suppose we wish to price an American option – which is path dependent. If we value the American option using the standard BOPM with , this should give a good estimate of its true price. Assume this ‘true BOPM price’ is – but computational time will be considerable.

To save on computational time, suppose we decide to price an American option using only steps in the BOPM and this gives . In order to get closer to the true price , the control variate technique adjusts in the following way. First, we use the standard BOPM with to calculate the price of a European option (on the same underlying, with the same strike and time to maturity). Both the American and European binomial prices with n = 5 are subject to error. However, we know the ‘exact’ Black–Scholes price for a European option (say). Using the control variate technique, the price of the American option is:

(23.5)

The control variate technique adjusts the American binomial price (obtained using ), by the error in the BOPM estimate when pricing the (equivalent) European option, . If the BOPM overpriced the European option by 0.2 (with ) we assume it will overprice the American option by the same amount. The control variate price is much closer to the ‘true’ price of the American option (using n = 100), than the ‘unadjusted’ binomial estimate with , and the control variate approach takes much less computing time.

23.3.1 American Futures Option

What about pricing an American futures option where we have the possibility of early exercise? To keep things simple, suppose we hold a two-period American call option on a futures contract. Early exercise at node-U is worthwhile if the intrinsic value , where the recursive formula gives . If this is the case, we replace in the tree by . This calculation is repeated for node-D (and at ) – that is comparing , and and then and – and taking the maximum value in each case.

23.3.2 Numerical Example

Suppose an American call futures option has a time to maturity year, strike price and the current futures price is , with the volatility of the futures (return) p.a. We divide the time period into periods so each step in the tree is (which represents 1 month). We set and .

For a futures option so (and hence the option price is independent of the risk-free rate). The intrinsic value of the option at each node is . Let each node be denoted where and .

In Figure 23.2 the upper cells show the stock price and the lower cells the option value (price). At , the option is exercised at the two upper nodes, only. For and (2,2), the intrinsic value of the option exceeds its binomial recursive value – hence here the option is assumed to be exercised and the lower cells at these nodes show the option's intrinsic value , rather than its binomial recursive value. The s at these nodes are then used in calculating the next recursive values as we move backwards through the tree and the American call premium is 6.387. (As we increase the number of nodes in the tree, so that , we obtain a more accurate estimate of the call premium.)

For American options on stocks paying continuous dividends at a rate and options on FX, the procedure is the same as above. The tree is constructed using and where is the real world (estimated annual) standard deviation of the stock return or FX return. Also for the option on the stock and for the FX-option (where and ).

The BOPM can also be used to price options where the underlying asset is a stochastic interest rate (e.g. options on T-bonds, on T-bond futures or options on interest rates, known as caps and floors) but this requires a tree where interest rates are allowed to vary at each node. This is dealt with in Chapter 41.

23.4.1 Dividends and the BOPM

We have already seen that for a stock (or stock index) that pays a constant continuous dividend yield , we simply use as the risk-neutral probability and proceed in the usual fashion to price European or American options on the dividend paying stock (index). In practice, the continuously compounded dividend rate has to be estimated and clearly while the assumption of a constant dividend rate is not unreasonable for a stock index (which contains many stocks), it is not plausible for individual stocks, where dividend payments tend to be bunched in certain months of the year. The BOPM gets a little tricky when dividends are discrete.

23.4.2 Single Known Dividend Yield

Assume the option matures in 30 days, , so that . We now apply the BOPM to a European call option where the underlying stock (index) pays a single dividend at time . If the time is prior to the stock going ex-dividend, the nodes on the tree correspond to stock prices:

If time is after the stock goes ex-dividend the nodes would have values

where is the known single dividend yield. For example, given a single dividend payment prior to the 2nd node, the binomial tree is shown in Figure 23.3. If there are several known dividend yields over the life of the option, then the nodes after the ex-dividend dates would be .

23.4.3 Known Dollar Dividend

First note that when a dividend is paid, the stock price falls by the amount of the dividend payment D. (We ignore any tax issues here.)² If we let then unfortunately with discrete dividends, the binomial tree for does not recombine and there are a very large number of nodes to evaluate. To avoid this problem and obtain a recombining tree we proceed as follows. We let and apply to the stock price minus the present value of all known future dividends (over the life of the option), which we denote . Suppose a single ex-dividend date is at τ and the dividend paid is . Then the values for at times are:

The tree for is constructed using , where .³ This gives us a recombining tree for . To obtain a ‘new’ tree, we now add back the PV of future dividends, at each node. Suppose we have calculated at . Then a ‘new’ tree for at times is:

The option is then priced off this ‘new’ tree using as the risk-neutral probability.⁴

For example, suppose , there is one dividend of $10 with an ex-dividend date at the end the second month (0.1667 years), then . If then . We calculate the tree for using the above equations and then work backwards through the tree (from the maturity date of the option) to give the European call (or put) premium.

To value an American call with strike on a dividend paying stock we would calculate the intrinsic value at each node and the early exercise decision is based on (not ). For example, if the stock has just gone ex-dividend and at the next ‘upper node’ and the dividend at τ is then (since the present value of the dividend at τ is the dividend itself of $3). For an instant, and then ‘immediately’ it falls to $110 as it goes ex-dividend. But just before the stock goes ex-dividend the call has an intrinsic value of . If at this node and this is greater than the recursive value in the tree, then we replace with . We proceed in this way at each node, to see if the intrinsic value exceeds the recursive value. So, apart from the construction of the stock price tree, an American option on a stock which pays discrete dividends is priced in the usual way.

As we see in Chapters 47 and 48, continuous time models of stock prices can be represented in terms of continuously compounded (‘log’) returns or proportionate changes over a short period of time, . The ‘up’ and ‘down’ movements in the binomial tree for stock prices are an approximation to these continuous time processes and are designed to produce an outcome for the stock price at , which is (approximately) lognormal. This requires movements of the stock price in the tree to replicate the ‘real world’ volatility of the stock price. In addition, when pricing options (on a non-dividend paying stock) using the BOPM under risk-neutral valuation (RNV), we must set the growth rate of the stock price equal to the risk-free rate.

In the BOPM we divide the time to maturity of the option (years) into -periods of equal length, . Over a small interval of time , the expected return of the stock is measured as (where = continuously compounded annual growth rate, decimal). Over a small time interval, the variance of the stock return is (where = annual standard deviation (decimal) of the continuously compounded stock return and is calculated from historical data). Hence:

We price an option (on a non-dividend paying stock) using the BOPM under RNV. Therefore we calibrate , and , so the stock price in the tree satisfies two conditions (over the time period ):

1. Expected return equals the risk-free rate (RNV)
2. Variance of the stock price, (‘real world’ volatility)

Hence, RNV and replicating the ‘real world’ volatility gives two equations and three unknowns (U, D, q):

(23.A.1)

(23.A.2)⁵

From (23.A.1):

(23.A.3)

Multiplying (23.A.3) by and simplifying:

(23.A.4)

(23.A.5)

Simplifying (23.A.2):

(23.A.6)

Substituting in (23.A.6) from (23.A.5) and (23.A.3):

(23.A.7)

We have three unknowns , and and only two equations – the RNV equation (23.A.1) and the (simplified) volatility equation (23.A.7). We arbitrarily use our one ‘degree of freedom’ by setting . Equation (23.A.1) or equivalently (23.A.3) gives directly:

(23.A.8)

If higher order terms than are ignored, a solution to the volatility equation (23.A.7) (with ) is:

(23.A.9)

In a risk-neutral world and are independent of the expected growth rate of the stock (i.e. the expected ‘real world’ stock return ), and therefore so is the option price. From (23.A.9) we have so U/D is determined by the ‘real world’ volatility of the stock return and hence so are the option premia.

As we move from the ‘real world’ to our equations in a ‘risk-neutral’ world, the expected return on the stock changes from to (see 23.A.1) but the volatility of the stock return is the same as in the real world – this is a manifestation of Girsanov's theorem. It is easy to see that (23.A.9) satisfies the volatility equation (23.A.7) by substituting (the Taylor series approximations up to order ):

in the left-hand side of (23.A.7) (and ignoring terms in or higher).

The above analysis can be repeated for an option on an asset that pays a continuous yield (e.g. dividend yield) at a rate . The return on a stock equals the capital gain plus the (continuously compounded, dividend) yield . In a risk-neutral world the asset (stock) return equals the risk-free rate and hence the expected value of the asset price . Therefore, to price an option on an asset that pays a continuous yield, the only change in the above analysis is in (23.A.1) where we replace with which results in in (23.A.8).

Other Risk-neutral Probabilities

In the above derivation we found we had ‘one degree of freedom’ and imposed (Cox, Ross and Rubinstein 1979). This gives unique values for , and with which to construct the binomial tree, which is then used to price the option. But we could have used another ‘trick’ in the derivation of , and , which results in being the same for options with different underlying assets, S (e.g. options on stocks that pay no dividends, on stocks that pay continuous dividends, options on FX-spot rates or commodities or futures contracts).

This seems a little counter-intuitive but it is to do with how we ‘allocate’ our one degree of freedom. Given our two equations to determine the three ‘unknowns’ , and we can arbitrarily set. Then solving our two equations (23.A.3) and (23.A.7) for and we obtain (when terms of higher order than are ignored):

(23.A.10a)

(23.A.10b)

Clearly, using these values of and would give a different tree for the stock price than if we use the Cox, Ross and Rubinstein formulas but the value of the option premium from backward recursion using the BOPM under RNV is the same using either approach. The different values for in the two trees would exactly offset the different values for , and the price of the option using backward recursion turns out to be the same. (After all we can only have one ‘correct’ or ‘no-arbitrage’ price for the option.)

For a stock (index) paying a continuous dividend at a rate , we replace by in Equations (23.A.10a) and (23.A.10b):

(23.A.11a)

(23.A.11b)

In addition, for currency options the foreign interest rate and for options on futures contracts , so is omitted from the above equations. Hence, Equations (23.A.11a) and (23.A.11b) enable construction of a tree for the underlying asset and hence price options on dividend paying stocks, currencies and futures contracts when using .

Note that the size of and D (and the value of ) have all been derived assuming a risk-neutral world. So, when pricing options, the tree for the stock price does not represent actual movements in the stock price but it still correctly prices the option because of the equivalence of backward recursion using RNV and the ‘no-arbitrage’ approach.

Trinomial Tree

When pricing an option, the use of a trinomial tree rather than a binomial tree can reduce computational time. The tree is set up so that at each node there is an up, middle, and down step. For example, for a non-dividend paying stock, the tree mimics the ‘real world’ volatility and has the stock price growing at the risk-free rate if:

where , , are the risk-neutral probabilities for the down move, up move and for the ‘middle’ path. We then use backward recursion on the trinomial tree to calculate the option premium. For assets paying a continuous dividend yield at a rate , we replace by in the above equations. Also, for currency options the foreign interest rate and for options on futures , so is omitted from the above equations. The trinomial tree is equivalent to the explicit finite difference method, discussed in Chapter 48.