Binomial option pricing and the conditions for early exercise: An example using foreign exchange options

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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.

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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 .

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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)

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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)

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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)

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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.

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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* = ujdn - 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.

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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

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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 .

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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.

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