**The Economic and Social Review, Vol. 21, No. 2, January, 1990, pp. 151-161 **

**Binomial Option Pricing and the Conditions for **

**Early Exercise: An Example using Foreign **

**Exchange Options **

R I C H A R D B R E E N

*The Economic and Social Research Institute, Dublin *

**Abstract: In this paper arc derived simple and general conditions under which the value of an American ****option will exceed that of its European counterpart. These conditions are developed using the binomial **
**option pricing framework. The derivation is motivated and illustrated by the example of foreign exchange **
**options. **

I I N T R O D U C T I O N

**F**

oreign exchange ( F X ) or currency options were first traded i n Phila
delphia i n 1982 and since then have become c o m m o n on all the w o r l d ' s
major options exchanges. I n the details o f their operation, F X options are
similar t o all other o p t i o n s , w i t h calls giving the r i g h t t o b u y the currency at
the agreed price, while puts confer the right to sell.
Garman-Kohlhagen pricing f o r m u l a for a European F X o p t i o n provides a lower b o u n d o n the possible values o f an A m e r i c a n F X o p t i o n .

I n other words the possibility o f "early exercise" may make the A m e r i c a n F X o p t i o n more valuable t h a n the European. I n this paper we address the question: "under w h a t circumstances w i l l an A m e r i c a n F X o p t i o n be more valuable t h a n its European counterpart?" This is equivalent to asking under w h a t circumstances early exercise o f the o p t i o n w i l l be w o r t h w h i l e . M o r e generally we provide a m e t h o d b y w h i c h this question can be answered i n respect o f A m e r i c a n options w r i t t e n on any asset. The original c o n t r i b u t i o n w h i c h this paper makes lies i n deriving these conditions using the framework of the b i n o m i a l o p t i o n pricing m e t h o d . These conditions are, i n consequence, stated i n terms o f the parameters o f this m e t h o d . They thus have the m e r i t of being b o t h simple and w i d e l y applicable.

I I E A R L Y E X E R C I S E

I t is a straightforward matter t o prove that, i f an American o p t i o n is w r i t t e n on a stock w h i c h pays no dividends, then the holder o f an American call o p t i o n w o u l d never wish t o exercise i t before its m a t u r i t y date: hence, i n this case the Black-Scholes (1973) f o r m u l a provides the correct valuation (the p r o o f was originally given b y M e r t o n , 1973). However, this is n o t true o f A m e r i c a n p u t options o n non-dividend paying stock. Here early exercise may be p r o f i t a b l e , and thus the Black-Scholes f o r m u l a w i l l understate the true value o f the p u t .

A l w a y s , o f course, the f o r w a r d value o f a non-dividend paying stock w i l l be at a p r e m i u m to its spot value.1 However, i t is possible t o show that i f the

f o r w a r d value o f such a stock were at a discount to its present value, i t w o u l d not pay t o exercise a p u t early b u t i t w o u l d pay to exercise a call early. I n the case o f foreign currencies, however, the f o r w a r d exchange rate is given by the interest rate p a r i t y theorem, and can therefore be at a p r e m i u m or a dis count t o the spot rate, depending whether the domestic or foreign risk free interest rate is higher. Thus, for foreign exchange options, some calls (where the f o r w a r d rate is at a discount) and some puts (where the f o r w a r d rate is at a p r e m i u m ) may be exercised early. However, there are circumstances i n w h i c h a call o n a currency w h i c h has a forward rate at a p r e m i u m may also be exer cised early. T o clarify the exact circumstances under w h i c h F X calls and puts w i l l be exercised early, we t u r n first to the b i n o m i a l o p t i o n pricing m o d e l .

I I I T H E B I N O M I A L O P T I O N P R I C I N G M O D E L

I n the original Black-Scholes (1973) o p t i o n pricing m o d e l , the stock price was assumed t o f o l l o w a l o g n o r m a l diffusion process i n continuous t i m e . I n the b i n o m i a l o p t i o n pricing m o d e l , i n t r o d u c e d b y C o x , Ross and Rubinstein (1979) and Rendleman and Bartter (1979) the stock price is modelled as f o l l o w ing a m u l t i p l i c a t i v e b i n o m i a l process i n discrete t i m e . Here the t i m e between the w r i t i n g o f the o p t i o n and its m a t u r i t y date is divided into N discrete periods, d u r i n g each o f w h i c h the price o f the u n d e r l y i n g asset is assumed t o make a single move, either up or d o w n . The magnitude o f these movements is given b y the m u l t i p l i c a t i v e parameters u and d. Hence at the end o f the first p e r i o d the share price w i l l have moved f r o m S to either uS or dS. The present value o f the share price at the end o f the first p e r i o d can be obtained by dis c o u n t i n g either o f these values b y the one period risk free rate, w h i c h we denote by q . u , d and q are obtained f r o m the parameters o f the continuous t i m e m o d e l as

*where a is the annualised standard deviation o f the share (or other asset) price, *
T is the t i m e t o m a t u r i t y o f the o p t i o n , and r is the annualised riskless rate
o f interest. The derivations o f u , d , and q are given b y C o x , Ross and R u b i n
stein ( 1 9 7 9 , p p . 2 4 7 - 2 4 9 ) . N must be chosen to ensure that d < q < u .

The o n l y remaining parameter we require is the p r o b a b i l i t y o f an u p w a r d or d o w n w a r d movement (p and 1-p respectively).2 I n the case o f a share, the

Black-Scholes m o d e l sets the d r i f t o f the u n d e r l y i n g Wiener process equal t o the risk free rate, r. A c c o r d i n g l y , the value o f p must be such t h a t , at the end of one period

I n the case o f a foreign exchange o p t i o n , however, the d r i f t o f the process and the discount rate are n o t identical. For a foreign exchange o p t i o n the one p e r i o d discount rate is again

u = exp { a O T / N )1/2 }

d = 1/u

q = rT/N

p.uS + ( l - p ) . d S = qS _{(1) }

i m p l y i n g that

p = ( q - d ) / ( u - d ) . (2)

whereas

p = ( i - d ) / ( u - d ) (3a)

where

i = ( r / r * )T/N (3b)

where r* is the foreign riskless rate.

C o n t i n u i n g w i t h the simpler example o f an equity o p t i o n we see that, for a one period b i n o m i a l m o d e l the value o f a European call o p t i o n w i l l be

^ (max [ 0 , uS- X ] ) + (max [ 0 , dS- X ] )

where X is the exercise price o f the o p t i o n . For a t w o period model the value is

( p / q )2 (max [ 0 , u2S - X ] ) + 2 ( l - p ) ( p / q2) (max [ 0 , S - X ] )

+ ( ( l - p ) / q )2 (max [ 0 , d2S ~ X ] )

This process generalises to the f o l l o w i n g equations for the value o f a European call and p u t respectively:

pi ( l - p )N"j [uJ dN~j S - X ] e "r T (4a)

pi ( l - p )N _ j [ X - u i dN _ i S] e~r T ( 4 b )

where a is the m i n i m u m ( m a x i m u m ) number of upward stock movements such that the call ( p u t ) finishes in the money. These equations give surpris ingly accurate results (when compared w i t h the Black-Scholes formula) for even small values o f N , and converge to the Black-Scholes value for N > 100.

I n the case o f an American o p t i o n early exercise w i l l be a possibility. A c c o r d i n g l y , at any p o i n t in t i m e , t , the option's value w i l l be v.he larger o f (a) its immediate exercise value ( S ( t ) - X for a call); and (b) its value i f held b e y o n d t.. (i.e., not exercised at t ) . I n the b i n o m i a l model the exercise value at the j t h node i n the b i n o m i a l " t r e e " at the end o f the n t h period (we label this B jn)

is given b y ( i n the case o f a call)

Bj n = ui dn~j S - X (5a)

values o f the o p t i o n at the t w o points or nodes to w h i c h i t can move i n the f o l l o w i n g p e r i o d :

**Aj n = ^ (** **maX [ ° >Aj+l , n+l 'Bj+l , n+l ] ) **

+ X i _ E l ( m a x [ 0 , A . n + 1, B . n + 1] ) ( 5 b )

Thus an o p t i o n w i l l o n l y be exercised early i f

V . = B . > A . (6)

**j n j n j n \**** I **

where V jn is the value o f the o p t i o n at the j n t h node o f the process, given by

V . = m a x f O . A . , B. ) (7)

We can w r i t e the necessary and sufficient c o n d i t i o n for early exercise as

Bj n > ( p / q ) (max [ 0 , Bj + 1 > n +, ] ) + ( ( l - p ) / q ) (max [ 0 , B. „+ 1 ] ) (8)

for some j n t h node in the process. T h a t this is a necessary c o n d i t i o n is clear f r o m ( 7 ) , i n so far as the value o f the o p t i o n at the j n t h node w i l l always be at least as great as the RHS o f ( 8 ) . T h a t i t is a sufficient c o n d i t i o n follows from the fact that Aj N = 0 for j = 0 . . . N (that is, an o p t i o n has zero h o l d i n g

value at the end o f the N t h period) and thus

Vj N = m a x ( 0 , Bj N) f o r j = 0 N .

I n fact, this c o n d i t i o n can be simplified. For (8) t o h o l d , B jn must be greater

than zero w h i c h implies t h a t , in the case o f a call o p t i o n , B j+ 1 n + 1 is also greater

than zero. We can therefore rewrite (8) as

Bj n > (P/q) ( Bj+ln+1) + ( ( l - p ) / q ) (max [ 0 , Bj > n + 1] ) (9a)

In the case o f a p u t (and redefining B jn accordingly), B jn greater than zero

implies that B^ n+1 is also greater than zero, allowing us to rewrite (8) as

Bj n > ( p / q ) (max [ 0 , Bj + l i n + 1. ] ) + ( ( l - p ) / q ) ( B . n + 1) . (9b)

N sufficiently l a r g e3) , we can w r i t e the necessary and sufficient c o n d i t i o n for early exercise o f an A m e r i c a n call as

S * - X > ( p / q ) ( u S * - X ) + ( ( l - p ) / q ) ( d S * - X ) (10a)

where S* is any value o f S such that d S * - X > 0.

For a p u t , given a B j+ 1 n+1 greater than zero we can w r i t e the c o n d i t i o n as

X - S * > ( p / q ) ( X - u S * ) + ( ( l - p ) / q ) ( X - d S * ) (10b)

where S* is some value o f S such that X - u S * > 0. These conditions then reduce t o , respectively,

X < X / q (11a)

and

X > X / q ( l i b )

I n passing we note that ( H a ) demonstrates that a call o n a non-dividend pay ing stock w i l l never be exercised early given positive interest rates.

A p p l y i n g these arguments t o American F X options we f i n d the f o l l o w i n g conditions under w h i c h early exercise w i l l have some value.

*1. Calls: (10a) yields the c o n d i t i o n (via 3 ) : a call o p t i o n on a foreign cur*
rency w i l l have early exercise value i f f

^ S * = uJdn _JS(0) for some pair < j , n > where 0 < j < n , 0 < n < N

such that

S * - X > ( i S * - X ) / q (12)

where S is the exchange rate.

This c o n d i t i o n w i l l obviously h o l d i f the f o r w a r d rate is at a discount since then i < l < q . B u t early exercise w i l l also be feasible i f the f o r w a r d rate is at a p r e m i u m but i is sufficiently small and q sufficiently large — in other words, i f the f o r w a r d p r e m i u m is small and the domestic (and thus the foreign) risk free rates are relatively high.

*2. Puts: (10b) yields the c o n d i t i o n : a p u t o p t i o n o n a foreign currency w i l l *
have early exercise value i f f

**3 S* = u**jdn - jS ( 0 ) for some p a i r < j , n > where 0 < j < n , 0 < n < N

such that

X - S * > ( X - i S * ) / q . (13)

I f i < l < q (i.e., a f o r w a r d discount) the o p t i o n w i l l never be exercised early unless b o t h i and q are large. This w i l l occur when the risk free rates are high but differ l i t t l e i n value. I f there is a f o r w a r d p r e m i u m t h e n the o p t i o n w i l l always have early exercise value.

Conversely, A m e r i c a n calls w i l l have no price p r e m i u m over European values i n cases where the f o r w a r d rate is at a large p r e m i u m relative t o the domestic risk free rate. Likewise, the A m e r i c a n F X p u t should n o t be priced above the European p u t i n cases where, for example, the f o r w a r d rate is at a deep discount relative to the domestic risk free rate.

For certain combinations o f parameters, S* as defined i n conditions (12) and (13) w i l l n o t exist (because o n l y a negative S* could ensure the i n e q u a l i t y ) . On the other hand, any positive value o f S c o u l d be generated w i t h i n the b i n o m i a l p r o v i d i n g that the number o f periods, N , i n the b i n o m i a l m o d e l were set large enough. Clearly, however, i f c o n d i t i o n (12) or (13) can o n l y be met by setting N very large then the early exercise value w i l l be negligible. This introduces an i m p o r t a n t d i s t i n c t i o n between, o n the one hand, options o n non-dividend paying stock and, o n the other, c o m m o d i t y options b r o a d l y defined t o include foreign exchange options. Whereas i n the former we can say w i t h certainty that all calls w i l l have zero early exercise value, i n the latter there are cases i n w h i c h we can o n l y say that the early exercise value o f a particular o p t i o n is so small as t o be worthless. As we m i g h t expect, the con tinuous t i m e approach t o o p t i o n pricing leads t o exactly the same result. L e t T~ be the instant before the option's m a t u r i t y date ( T ~ = T - d t ) : then, defin ing i ' t o be the ratio o f the instantaneous domestic and foreign rates and q ' to be the instantaneous domestic risk free rate, a p u t w i l l have early exercise value i f f

prob { ( S ( T - ) ) < X( 1" . l (q') } > 0 . ( i -1 / q )

A g a i n , all F X p u t o p t i o n s w i l l meet this c o n d i t i o n provided that i t does n o t i m p l y a negative value for S ( T ~ ) b u t for many o f them the p r o b a b i l i t y w i l l be t o o small to be w o r t h considering.

proposed b y Bradley and Walsh (1986) for converting European i n t o A m e r i c a n F X o p t i o n values, w i l l be misleading. Bradley and Walsh (1986) attempt t o arrive at an approximate value for an A m e r i c a n F X call o p t i o n i n terms o f a p r e m i u m over the European valuation.

As a starting p o i n t , however, i t is suggested that a premium o f 15-20% should be added to the Garman and Kohlhagen valuation (Bradley and Walsh, 1986, p . 9 6 ) .

While this p r e m i u m level may have been reasonably accurate for the particular sample data on w h i c h Bradley and Walsh performed their analysis, as a general procedure for valuing A m e r i c a n F X call o p t i o n s , this w i l l give inaccurate results. I n particular, i n the situations i n w h i c h we have i d e n t i f i e d the A m e r i c a n o p t i o n as carrying no value for early exercise, their suggestion w i l l severely over-price the o p t i o n .

The conditions for early exercise derived above can be extended t o any A m e r i c a n o p t i o n . The b i n o m i a l model can be used t o value options w r i t t e n on a variety o f financial instruments. This can be accomplished b y setting

p = ( i - d ) / ( u - d )

where i is n o w defined t o be the one period "cost of c a r r y " o f the underlying asset. Clearly the one period cost o f carry o f a non-dividend paying stock is q: for a foreign currency o p t i o n i t is as given i n ( 3 b ) . For an o p t i o n o n a futures contract, i = l and S, i n this case, is the futures price. For a c o m m o d i t y , i w i l l be the one period physical cost of carry associated w i t h h o l d i n g a long spot p o s i t i o n (see Barone-Adesi and Whaley (1987) for a discussion o f the cost o f carry issue). Consequently, Equations (12) and (13) provide general conditions for early exercise o f A m e r i c a n options w i t h i n the b i n o m i a l o p t i o n pricing m o d e l .

I V P R I C I N G A M E R I C A N F X O P T I O N S U S I N G T H E B I N O M I A L M E T H O D

*As an alternative to ad hoc adjustments to the Garman Kohlhagen m o d e l , *
accurate pricing o f A m e r i c a n F X options can be carried o u t using the b i n o m i a l
m o d e l .4 Cox and Rubinstein (1985) provide a clear exposition o f the use o f

the b i n o m i a l t o price share options. T o apply the method t o F X options i t is

o n l y necessary to replace (2) and (3a) as discussed earlier. I n Table 1 we pre sent the values o f a sample o f nine m o n t h A m e r i c a n F X p u t options (esti mated using the above m e t h o d and N = 5 0 ) , together w i t h their European values (using b o t h the Garman and Kohlhagen f o r m u l a and the b i n o m i a l w i t h N = 5 0 ) . We also show the i m p l i c i t forward rate and the domestic risk free rate and the percentage difference between the A m e r i c a n and European values. The exchange rate (and also the o p t i o n ' s exercise price) is set at 1.2 domestic units per foreign u n i t : thus we can t h i n k o f this as being a l i k e l y scenario for Irish p o u n d / p o u n d sterling o p t i o n s , w i t h the o p t i o n price q u o t e d i n Irish pence per p o u n d sterling (although i t should be n o t e d that this example uses a very high h y p o t h e t i c a l exchange rate v o l a t i l i t y ) .

**T a b l e 1: Option Values for a Sample of Foreign Exchange Put Options **

**Forward Rate Domestic ****Option Value ****Premium for **

**(pence per £ sterling) **

**European ****American ****American over ****Risk Free Rate ****Garman ****Binomial ****Binomial ****European option **

**Kohlhagen **

**(a) ****(b) ****(b-a as % of a) ****(a) Forward discount with low risk free rates **

**1 . 1 7 0 .05 2 0 . 8 ** **2 0 . 7 ** **2 0 . 7 ** **0.0 **

**1.176 .05 2 0 . 6 ** **20.5 ** **2 0 . 5 ** **0.0 **

**1.182 .05 2 0 . 4 ** **2 0 . 3 ** **2 0 . 3 ** **0.0 **

**1.188 .05 2 0 . 2 ** **20.1 ** **2 0 . 2 ** **0.0 **

**(b) Forward premium **

**1.224 .05 19.1 ** **19.0 ** **19.3 ** **1.6 **

**1.218 .05 19.3 ** **19.2 ** **19.5 ** **1.6 **

**1.212 .05 19.5 ** **19.4 ** **1 9 . 6 ** **1.0 **

**1.206 .05 19.7 ** **1 9 . 6 ** **19.8 ** **1.0 **

**(c) Forward discount with high risk free rates **

**1.165 .15 19.5 ** **19.3 ** **19.7 ** **2.1 **

**1.170 .15 1 9 . 2 ** **19.1 ** **19.5 ** **2.1 **

**1.176 .15 19.0 ** **18.9 ** **1 9 . 4 ** **2.6 **

**1.182 .15 18.9 ** **18.8 ** **19.3 ** **2.7 **

**A s s u m e d a n n u a l s t a n d a r d deviation o f e x c h a n g e rate = . 5 . **

N o t e t h a t the A m e r i c a n o p t i o n is always more valuable w h e n the f o r w a r d rate is at a p r e m i u m (greater than 1.2), has no p r e m i u m over the European value i f the risk free rates are small and the f o r w a r d discount is large, b u t regains a p r e m i u m i n cases where the f o r w a r d discount occurs together w i t h high risk free rates (as i n the last few examples i n the table).

The variability o f the A m e r i c a n p r e m i u m over the European is clear f r o m the final c o l u m n o f the table, where the percentage difference ranges f r o m zero t o j u s t over 2 per cent i n these particular examples.

V C O N C L U S I O N

We have demonstrated the circumstances under w h i c h the price o f American F X puts and calls w i l l exceed that for their European equivalents. Because i t w i l l n o t always be profitable t o exercise an A m e r i c a n F X o p t i o n early, such options w i l l n o t always be w o r t h more than their European counterparts. As a result, any rule based o n increasing the European price b y a certain percen tage i n order t o approximate the American o p t i o n price w i l l certainly give misleading results i n many situations.

*I n fact, there is no need for the use o f any such ad hoc methods, i n so far *
as i t is a straightforward matter t o value accurately all A m e r i c a n F X options
by an adaptation o f the b i n o m i a l m e t h o d used for equity options. I n this
paper we have demonstrated h o w the same b i n o m i a l framework can be used
to determine the conditions under w h i c h an American o p t i o n w r i t t e n o n any
asset w i l l carry a p r e m i u m for its early exercise.

**REFERENCES **

**B A R O N E - A D E S 1 , G . , and R . E . W H A L E Y , 1 9 8 7 . " E f f i c i e n t A n a l y t i c A p p r o x i m a t i o n o f **
**A m e r i c a n O p t i o n V a l u e s " , The Journal of Finance, 4 2 , 2, p p . 3 0 1 - 3 2 0 . **

**B L A C K , F . , and M . S C H O L E S , 1 9 7 3 . " T h e Pricing of O p t i o n s and C o r p o r a t e L i a b i l i t i e s " , **
**Journal of Political Economy, 8 1 , 3 , pp. 6 3 7 - 6 5 4 . **

**B R A D L E Y , D . , a n d M . W A L S H , 1 9 8 6 . " T h e D e v e l o p m e n t of a T r a d e d I r i s h P o u n d C u r **
**r e n c y O p t i o n s M a r k e t " , Irish Business and Administrative Research, 8, 1, p p . 8 9 - 1 0 1 . ****B R E N N A N , M . , a n d E . S C H W A R T Z , 1 9 7 7 . " T h e V a l u a t i o n of A m e r i c a n Put O p t i o n s " , **

**The Journal of Finance, 3 2 , 3 , p p . 4 4 9 - 4 6 2 . **

**C O X , J . C . , S . A . R O S S , and M . R U B I N S T E I N , 1 9 7 9 . " O p t i o n Pricing: A S i m p l i f i e d **
**A p p r o a c h " , Journal of Financial Economics, 7, p p . 2 2 9 - 2 6 3 . **

**C O X . J . C , and M . R U B I N S T E I N , 1 9 8 5 . Options Markets, E n g l e w o o d Cliffs, N . J . : ****Prentice-H a l l . **

**G A R M A N , M . B . , a n d S.W. K O H L H A G E N , 1 9 8 3 . " F o r e i g n C u r r e n c y O p t i o n V a l u e s " , **
**Journal of International Money and Finance, 2, p p . 2 3 1 - 2 3 7 . **

**M E R T O N , R . C . , 1 9 7 3 . " T h e o r y o f R a t i o n a l O p t i o n P r i c i n g " , Bell Journal of Economics ****and Management Science, 4, S p r i n g , p p . 1 4 1 - 1 8 3 . **

**P A R K I N S O N , M . , 197 7. " O p t i o n Pricing: T h e A m e r i c a n P u t " , Journal of Business, 5 0 , **

**p p . 2 1 - 3 6 . **

**R E N D L E M A N , R . J r . , and B . B A R T T E R , 1 9 7 9 . " T w o - S t a t e O p t i o n P r i c i n g " , The Journal ****of Finance, 3 4 , 5, p p . 1 0 9 3 - 1 1 1 0 . **