The British call option

Quantitative Finance, Vol. 13, No. 1, January 2013, 95–109

The British call option

GORAN PESKIR* and FARMAN SAMEE

School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK

(Received 22 January 2010; in final form 21 May 2012)

Alongside the British put option (Peskir and Samee [Appl. Math. Finance, 2011, 18, 537–563])we present a new call option where the holder enjoys the early exercise feature of Americanoptions whereupon his payoff (deliverable immediately) is the ‘best prediction’ of theEuropean payoff under the hypothesis that the true drift of the stock price equals a contractdrift. Inherent in this is a protection feature which is key to the British call option. Should theoption holder believe the true drift of the stock price to be unfavourable (based upon observedprice movements) he can substitute the true drift with the contract drift and minimise hislosses. The practical implications of this protection feature are most remarkable as not onlycan the option holder exercise at or below the strike price to a substantial reimbursement ofthe original option price (covering the ability to sell in a liquid option market completelyendogenously) but also when the stock price movements are favourable he will generallyreceive high returns. We derive a closed form expression for the arbitrage-free price in terms ofthe rational exercise boundary and show that the rational exercise boundary itself can becharacterised as the unique solution to a nonlinear integral equation. In addition we derive the‘British put–call symmetry’ relations which express the arbitrage-free price and the rationalexercise boundary of the British call option in terms of the arbitrage-free price and the rationalexercise boundary of the British put option where the roles of the contract drift and theinterest rate have been swapped. These relations provide a useful insight into the Britishpayoff mechanism that is of both theoretical and practical interest. Using these resultswe perform a financial analysis of the British call option that leads to the conclusions aboveand shows that with the contract drift properly selected the British call option becomes a veryattractive alternative to the classic European/American call.

Keywords: British call option; European/American call option; Arbitrage-free price; Rationalexercise boundary; British put–call symmetry; Liquid/illiquid market; Geometric Brownianmotion; Optimal stopping; Parabolic free-boundary problem; Nonlinear integral equation;Local time–space calculus; Non-monotone free boundary

JEL Classification: C, C6, C61, G, G1, G13

Mathematics Subject Classification 2000: 91B28, 60G40, 35R35, 45G10, 60J60

1. Introduction

The purpose of the present paper is to introduce a new

call option which endogenously provides its holder with a

protection mechanism against unfavourable stock price

movements. This mechanism is intrinsically built into the

option contract using the concept of optimal prediction

(see, for example, Du Toit and Peskir 2007 and the

references therein) and we refer to such contracts as

‘British’ for the reasons outlined in Peskir and Samee

(2011), where the British put option was introduced.

Similarly to the British put option, most remarkable

about the British call option is not only that it provides a

unique protection against unfavourable stock price move-

ments (endogenously covering the ability of an European

call holder to sell his contract in a liquid option market),

but also when the stock price movements are favourable

it enables its holder to obtain high returns. This reaffirms

the fact noted in Peskir and Samee (2011) that the British

feature of optimal prediction acts as a powerful tool for

generating financial instruments which aim at both

providing protection against unfavourable price move-

ments as well as securing high returns when these

movements are favourable (in effect by enabling the

seller/hedger to ‘milk’ more money out of the stock). We

recall that in view of the recent turbulent events in the

financial industry and equity markets in particular, these*Corresponding author. Email: [email protected]

Quantitative FinanceISSN 1469–7688 print/ISSN 1469–7696 online � 2013 Taylor & Francis

http://www.tandfonline.comhttp://dx.doi.org/10.1080/14697688.2012.696676

combined features appear to be especially appealing asthey address problems of liquidity and return completelyendogenously (reducing the need for exogenousregulation).

The paper is organised as follows. In section 2 wepresent a basic motivation for the British call option. Itshould be emphasised that the full financial scope of theoption goes beyond these initial considerations (especiallyregarding the provision of high returns that appears as anadditional benefit). In section 3 we formally define theBritish call option and present some of its basic proper-ties. This is continued in section 4 where we derivea closed form expression for the arbitrage-free pricein terms of the rational exercise boundary (the early-exercise premium representation) and show that therational exercise boundary itself can be characterised asthe unique solution to a nonlinear integral equation(theorem 1). Many of these arguments and results standin parallel to those of the British put option (Peskir andSamee 2011) and we follow these leads closely in order tomake the present exposition more transparent and self-contained (indicating the parallels explicitly where it isinsightful). In section 5 we derive the ‘British put–callsymmetry’ relations which express the arbitrage-free priceand the rational exercise boundary of the British calloption in terms of the arbitrage-free price and the rationalexercise boundary of the British put option where theroles of the contract drift and the interest rate have beenswapped (theorem 2). These relations provide a usefulinsight into the British payoff mechanism (at the level ofput and call options) that is of both theoretical andpractical interest. We note that similar symmetry relationsare known to be valid for the American put and calloptions written on stocks paying dividends (see Carr andChesney 1996) where the roles of the dividend yield andthe interest rate have been swapped instead. Using theseresults in section 6 we present a financial analysis of theBritish call option (making comparisons with theEuropean/American call option). This analysis providesmore detail/insight into the full scope of the conclusionsbriefly outlined above.

2. Basic motivation for the British call option

We begin our exposition by explaining a basic economicmotivation for the British call option. We remark that thefull financial scope of the option goes beyond these initialconsiderations (see section 6 below for a fuller discussion).

1. Consider the financial market consisting of a riskystock X and a riskless bond B whose prices, respectively,evolve as

dXt ¼ �Xt dtþ �Xt dWt ðX0 ¼ xÞ ð1Þ

dBt ¼ rBt dt ðB0 ¼ 1Þ ð2Þ

where �2 IR is the appreciation rate (drift), �4 0 is thevolatility coefficient, W¼ (Wt)t�0 is a standard Wienerprocess defined on a probability space (�,F ,P), and

r4 0 is the interest rate. Recall that a European call

option (cf. Samuelson 1965, Black and Scholes 1973,

Merton 1973) is a financial contract between a seller/

hedger and a buyer/holder entitling the latter to sell the

underlying stock at a specified strike price K4 0 at a

specified maturity time T4 0. Standard hedging argu-

ments based on self-financing portfolios imply that the

arbitrage-free price of the option is given by

V ¼eE e�rTðXT � KÞþ ð3Þ

where the expectation ~E is taken with respect to the

(unique) equivalent martingale measure ~P (see, for exam-

ple, Karatzas and Shreve 1998 for a modern exposition).

After receiving the amount V from the buyer, the seller

can perfectly hedge his position at time T through trading

in the underlying stock and bond, and this enables him to

meet his obligation without any risk. On the other hand,

since the holder can also trade in the underlying stock and

bond, he can perfectly hedge his position in the opposite

direction and completely eliminate any risk too (upon

exercising at T). Thus, the rational performance is risk-

free, at least from this theoretical standpoint.2. There are many reasons, however, why this theoret-

ical risk-free standpoint does not quite translate into real

world markets. Without addressing any of these issues

more explicitly, in this section we will analyse the rational

performance from the standpoint of a true buyer. By ‘true

buyer’ we mean a buyer who has no ability or desire to

sell the option nor to hedge his own position. Thus, every

true buyer will exercise the option at time T in accordance

with the rational performance. For more details on the

motivation and interest for considering a true buyer in

this context, see Peskir and Samee (2011).3. With this in mind, we now return to the European

call holder and recall that he has the right to sell the stock

at the strike price K at the maturity time T. Thus his

payoff can be expressed as

e�rTðXTð�Þ � KÞþ ð4Þ

where XT¼XT(�) represents the stock/market

price at time T under the actual probability measure P.Recall also that the unique strong solution to (1) is

given by

Xt ¼ Xtð�Þ ¼ x exp �Wt þ ��2

2

� �t

� �ð5Þ

under P for t2 [0,T] where �2 IR is the actual drift. Note

that �}XT(�) is strictly increasing so that �} e�rT

(XT(�)�K)þ is (strictly) increasing on IR (when non-

zero). Moreover, it is well known that LawðXð�Þ j ~PÞ is thesame as Law(X(r) jP). Combining this with (3) above we

see that if �¼ r then the return is ‘fair’ for the buyer, in

the sense that

V ¼ E e�rTðXTð�Þ � KÞþ ð6Þ

where the left-hand side represents the value of his

investment and the right-hand side represents

the expected value of his payoff. On the other hand,

96 G. Peskir and F. Samee

if �4 r then the return is ‘favourable’ for the buyer, in

the sense that

V5E e�rTðXTð�Þ � KÞþ ð7Þ

and if �5 r then the return is ‘unfavourable’ for the

buyer, in the sense that

V4E e�rTðXTð�Þ � KÞþ ð8Þ

with the same interpretations as above. Recall that the

actual drift � is unknown at time t¼ 0 and also difficult to

estimate at later times t2 (0,T] unless T is unrealistically

large.4. The brief analysis above shows that whilst the actual

drift � of the underlying stock price is irrelevant in

determining the arbitrage-free price of the option, to a

(true) buyer it is crucial, and he will buy the option if he

believes that �4 r. If this turns out to be the case then on

average he will make a profit. Thus, after purchasing the

option, the call holder will be happy if the observed stock

price movements reaffirm his belief that �4 r.The British call option seeks to address the opposite

scenario: ‘What if the call holder observes stock price

movements which change his belief regarding the actual

drift and cause him to believe that �5 r instead?’ In this

contingency the British call holder is effectively able to

substitute this unfavourable drift with a contract drift and

minimise his losses. In this way he is endogenously

protected from any stock price drift smaller than the

contract drift. The value of the contract drift is therefore

selected to represent the buyer’s expected level of toler-

ance for the deviation of the actual drift from his original

belief (see figure 1). It will be shown below that the

practical implications of this protection feature are most

remarkable as not only can the British call holder exercise

at or below the strike price to a substantial reimbursement

of the original option price (covering the ability to sell in a

liquid option market completely endogenously) but also

when the stock price movements are favourable he will

generally receive high returns (see section 6 for further

details).5. Releasing now the true-buyer’s perspective and

considering the real world buyer instead, observe that a

call holder who believes (based upon his observations)that �5 r may choose/attempt to sell his contract.However, in a real financial market the price at whichhe is able to sell will be determined by the market (as thebid price of the bid–ask spread) and this may also involveadditional transaction costs and/or taxes. Moreover,unless the exchange is fully regulated, it may be increas-ingly difficult to sell the option when out-of-the-money(e.g. in most practical situations of over-the-countertrading such an option will expire worthless). The lattertherefore strongly correlates the buyer’s risk exposure tothe liquidity of the option market. We remark that theliquidity of the option market (alongside possible trans-action costs and tax considerations) can change duringthe term of the contract. This for example can be causedby extreme news events either specific (to the underlyingstock) or systemic in nature (such as the recent turbulentevents in the financial industry). The protection affordedto the British call option holder, on the other hand, isendogenous, i.e. it is always in place regardless of whetherthe option market is liquid or not.

3. The British call option: Definition and basic properties

We begin this section by presenting a formal definition ofthe British call option. This is then followed by a briefanalysis of the optimal stopping problem and the free-boundary problem characterising the arbitrage-free priceand the rational exercise strategy. These considerationsare continued in sections 4 and 5 below.

1. Consider the financial market consisting of a riskystock X and a riskless bond B whose prices evolve asequations (1) and (2) respectively, where �2 IR is theappreciation rate (drift), �4 0 is the volatility coefficient,W¼ (Wt)t�0 is a standard Wiener process defined on aprobability space (�,F ,P), and r4 0 is the interest rate.We assume that the stock does not pay dividends andthere are no transaction costs associated with its trade.Let a strike price K4 0 and a maturity time T4 0 begiven and fixed.

Definition 3.1: The British call option is a financialcontract between a seller/hedger and a buyer/holderentitling the latter to exercise at any (stopping) time �prior to T whereupon his payoff (deliverable immediately)is the ‘best prediction’ of the European payoff (XT�K)þ

given all the information up to time � under the hypothesisthat the true drift of the stock price equals �c (see figure 2).

rc

tolerablecritical favourable

observations

0

Figure 1. An indication of the economic meaning of thecontract drift.

•Buyer Seller•

V

E ( )( )+

-K-TXc

Figure 2. Definition of the British call option in terms of itsprice and payoff.

The British call option 97

The quantity �c is defined in the option contract and

we refer to it as the ‘contract drift’. Recalling our

discussion in section 2 above it is natural that the contract

drift satisfies the right-hand inequality in

05�c 5 r ð9Þ

since otherwise the British call holder could beat the

interest rate r by simply exercising immediately (a formal

argument confirming this economic reasoning will be

given shortly below). We will also show below that the

left-hand inequality in (9) must be satisfied as well since

otherwise it would be optimal to exercise at time T and

the British call option would reduce to the European call

option. Recall also from section 2 above that the value of

the contract drift is selected to represent the buyer’s

expected level of tolerance for the deviation of the true

drift � from his original belief (see figure 1). It will be

shown in section 6 below that this protection feature has

remarkable implications both in terms of liquidity and

return.2. Denoting by (F t)0�t�T the natural filtration gener-

ated by X (possibly augmented by null sets or in some

other way of interest) the payoff of the British call option

at a given stopping time � can be formally written as

E�c ðXT � KÞþj F ��

ð10Þ

where the conditional expectation is taken with respect to

a new probability measure P�c under which the stock

price X evolves as

dXt ¼ �cXt dtþ �Xt dWt ð11Þ

with X0¼ x in (0,1). Comparing equations (1) and (11)

we see that the effect of exercising the British call option is

to substitute the true (unknown) drift of the stock price

with the contract drift for the remaining term of the

contract.3. Stationary and independent increments of W gov-

erning X imply that

E�c ðXT � KÞþj F t

� �¼ G�c ðt,XtÞ ð12Þ

where the payoff function G�c can be expressed as

G�c ðt, xÞ ¼ EðxZ�c

T�t � KÞþ ð13Þ

and Z�c

T�t is given by

Z�c

T�t ¼ exp �WT�t þ �c ��2

2

� �ðT� tÞ

� �ð14Þ

for t2 [0,T] and x2 (0,1). Hence one finds that equation

(13) can be rewritten as follows:

G�c ðt,xÞ

¼ x e�cðT�tÞ�1

�ffiffiffiffiffiffiffiffiffiffiT� tp log

x

K

þ �cþ

�2

2

� �ðT� tÞ

� �� K�

1


x

K

þ �c�

�2

2

� �ðT� tÞ

� �� ð15Þ

for t2 [0,T) and x2 (0,1) where � is the standard

normal distribution function given by �ðxÞ ¼ ð1=ffiffiffiffiffiffi2ppÞ�R x

�1e�y

2=2 dy for x2 IR.It may be noted that the expression for G�c ðt, xÞ

multiplied by e��cðT�tÞ coincides with the Black–Scholes

formula for the arbitrage-free price of the European call

option (written for the remaining term of the contract)

where the interest rate equals the contract drift �c. Hence

the British call option can be formally viewed as an

American option on the undiscounted European call

option written on a stock paying dividends at rate

�¼ r��c4 0. Since the payoff (i.e. the European call

value) is undiscounted we see that a direct financial

interpretation in terms of compound options is not

possible (cf. subsection 3.3 of Peskir and Samee 2011

for further details).4. Standard hedging arguments based on self-financing

portfolios (with consumption) imply that the arbitrage-

free price of the British call option is given by

V ¼ sup0��T

eE e�r�E�c ðXT � KÞþj F ��

ð16Þ

where the supremum is taken over all stopping times � ofX with values in [0,T] and ~E is taken with respect to the

(unique) equivalent martingale measure ~P. Making use of

equation (12) above and the optional sampling theorem,

upon enabling the process X to start at any point x in

(0,1) at any time t2 [0,T], we see that the problem (16)

extends as follows:

Vðt, xÞ ¼ sup0��T�t

eEt,x e�r�G�c ðtþ �,Xtþ�Þ½ � ð17Þ

where the supremum is taken over all stopping times � ofX with values in [0,T� t] and ~Et,x is taken with respect to

the (unique) equivalent martingale measure ~Pt,x under

which Xt¼x. Since the supremum in equation (17) is

attained at the first entry time of X to the closed set where

V equals G�c , and LawðXð�Þ j ~PÞ is the same as

Law(X(r) jP), it follows from the well-known flow struc-

ture of the geometric Brownian motion X that

Vðt, xÞ ¼ sup0��T�t

E e�r�G�c ðtþ �, xX�Þ½ � ð18Þ

for t2 [0,T] and x2 (0,1) where the supremum is taken

as in equation (17) above and the process X¼X(r) under

P solves

dXt ¼ rXt dtþ �Xt dWt ð19Þ

with X0¼ 1. As it will be clear from the context which

initial point of X is being considered, as well as whether

the drift of X equals r or not, we will not reflect these facts

directly in the notation of X (by adding a superscript or

similar).5. We see from equation (13) that

x�G�c ðt, xÞ is convex ð20Þ

and strictly increasing on (0,1) with G�c ðt, 0Þ ¼ 0 and

G�c ðt,1Þ ¼ 1 for any t2 [0,T] given and fixed. One also

sees that G�c ðT, xÞ ¼ ðx� KÞþ for x2 (0,1) showing that


the British call payoff coincides with the European/

American call payoff at the time of maturity. Moreover, if�c� 0 then from equations (13) and (14) we see that

G�c ðt, xÞ5 e�rðT�tÞ EðxZrT�t � KÞþ ð21Þ

for t2 [0,T) and x2 (0,1), where the term on the right-

hand side can be recognised as the payoff obtained by

choosing �¼T� t in equation (18). From the strict

inequality (21), and the fact that the supremum in (21)is attained at the first entry time of X to the (closed) set

where V equals G�c , we therefore see that it is not optimal

to stop in (18) before the time of maturity when �c� 0

(i.e. the optimal stopping time �� in equation 18 equals

T� t). This establishes the claim about the left-hand

inequality following (9) above. Denoting by VE(t,x) the

arbitrage-free price of the European call option (given bythe term on the right-hand side of inequality (21) above) it

follows therefore using (13) and (14) that V(t, x)¼VE(t,x)

if �c� 0 and V(t,x)4VE(t, x) if �c4 0 for all t2 [0,T)

and x2 (0,1). In particular, this confirms that the British

call option is more expensive than the European call

option whenever (9) holds. While this fact is to beexpected (owing to the presence of the early exercise

feature) it may be noted that V and G�c tend to stay much

closer together than the two functions for the European

call option (a financial interpretation of this phenomenon

will be addressed in section 6 below). Finally, from (18)

and (20) we easily find that

x�Vðt, xÞ is convex ð22Þ

and (strictly) increasing on (0,1) with V(t, 0)¼ 0 and

V(t,1)¼1 for any t2 [0,T) given and fixed, and one

likewise sees that V(T, x)¼ (x�K)þ for x2 (0,1). In this

sense the value function of the British call option is

similar to the value function of the European call option

(a snapshot of the former function is shown in figure 3).The most important technical difference, however, is that

whilst the formal American call boundary bA is trivial

(non-existing) this is not the case for the British call

boundary b whenever (9) holds.6. To gain a deeper insight into the solution to the

optimal stopping problem (18), let us note that Ito’s

formula yields

e�rsG�c ðtþ s,XtþsÞ

¼ G�c ðt, xÞ þ

Z s

0

e�ruH�c ðtþ u,XtþuÞ duþMs ð23Þ

where the function H�c ¼ H�c ðt, xÞ is given by

H�c ¼ G�ct þ r x G�c

x þ�2

2x2G�c

xx � r G�c ð24Þ

and Ms ¼ �R s0 e�ruXu G

�cx ðtþ u,XtþuÞ dWu defines a con-

tinuous martingale for s2 [0,T� t] with t2 [0,T). By the

optional sampling theorem we therefore find

E e�r�G�c ðtþ �,xX�Þ½ �

¼ G�c ðt, xÞ þ E

Z �

0

e�ruH�c ðtþ u, xXuÞ du

� �ð25Þ

for all stopping times � of X solving equation (19) with

values in [0,T� t] with t2 [0,T) and x2 (0,1) given and

fixed. On the other hand, it is clear from equation (13)

that the payoff function G�c satisfies the Kolomogorov

backward equation (or the undiscounted Black–Scholes

equation where the interest rate equals the contract drift)

G�ct þ �c x G�c

x þ�2

2x2G�c

xx ¼ 0 ð26Þ

so that from equation (24) we see that

H�c ¼ ðr� �cÞ x G�cx � rG�c : ð27Þ

This representation shows in particular that if �c� r then

H�c 5 0 so that from equation (25) we see that it is always

optimal to exercise immediately as pointed out following

(9) above. Moreover, inserting the expression for G�c

from equation (15) into equation (27), it is easily verified

that

H�c ðt,xÞ ¼ rK�1


x

K

þ �c�

�2

2

� �ðT� tÞ

� �� c xe

�cðT�tÞ�

�1

�ffiffiffiffiffiffiffiffiffiffiT� tp

�log

x

K

þ �cþ

�2

2

� �ðT� tÞ

��ð28Þ

for t2 [0,T) and x2 (0,1).A direct examination of the function H�c in equation

(28) shows that there exists a continuous (smooth)

function h : [0,T]! IR such that

H�c ðt, hðtÞÞ ¼ 0 ð29Þ

for all t2 [0,T) with H�c ðt, xÞ5 0 for x4 h(t) and

H�c ðt, xÞ4 0 for x5 h(t) when t2 [0,T) is given and

fixed. In view of equation (25) this implies that no point

0 b(t)

x V(t,x)

x (x-K)+

x G (t,x)c

K

Figure 3. The value/payoff functions of the British/Europeancall options.


(t, x) in [0,T)� (0,1) with x5 h(t) is a stopping point(for this one can make use of the first exit time from

a sufficiently small time–space ball centred at the point).Likewise, it is also clear and can be verified that if

x4 h(t) and t5T is sufficiently close to T then it isoptimal to stop immediately (since the gain obtained from

being below h cannot offset the cost of getting thereowing to the lack of time). This shows that the optimal

stopping boundary b separating the continuation set fromthe stopping set satisfies b(T)¼ h(T) and this value equals

rK/�c as is easily seen from (28). Note that rK/�c4K

since �c satisfies (9) above.7. Standard Markovian arguments lead to the follow-

ing free-boundary problem (for the value functionV¼V(t, x) and the optimal stopping boundary b¼ b(t)

to be determined):

Vt þ rxVx þ ð�2=2Þx2Vxx � rV ¼ 0

for x 2 ð0, bðtÞÞ and t 2 ½0,TÞ ð30Þ

Vðt,xÞ ¼ G�c ðt,xÞ for x ¼ bðtÞ and t 2 ½0,T�

(instantaneous stopping) ð31Þ

Vxðt,xÞ ¼ G�cx ðt,xÞ for x ¼ bðtÞ and t 2 ½0,T�

ðsmooth fitÞ ð32Þ

VðT, xÞ ¼ ðK� xÞþ for x 2 ð0, bðTÞÞ

with bðTÞ ¼ rK=�c

ð33Þ

Vðt, 0Þ ¼ 0 for t 2 ½0,T� ð34Þ

where we also set Vðt, xÞ ¼ G�c ðt, xÞ for x4 b(t) and

t2 [0,T] (see, for example, Peskir and Shiryaev 2006).It can be shown that this free-boundary problem has a

unique solution V and b which coincide with the valuefunction (18) and the optimal stopping boundary, respec-

tively. This means that the continuation set is given byC ¼ fV4G�c g ¼ f ðt, xÞ 2 ½0,TÞ � ð0,1Þ j x5 bðtÞ g and

the stopping set is given by D ¼ fV ¼ G�cg ¼

f ðt,xÞ 2 ½0,T� � ð0,1Þ j x � bðtÞ g [ f ðT, xÞ j x 2 ð0, bðTÞÞ g

so that the optimal stopping time in equation (18) isgiven by

�b ¼ inf t 2 ½0,T� jXt � bðtÞ �

ð35Þ

This stopping time represents the rational exercise strat-

egy for the British call option and plays a key role infinancial analysis of the option.

Depending on the size of the contract drift �c satisfying

(9) we distinguish three different regimes for the positionand shape of the optimal stopping boundary b (see

figure 4). Firstly, when �c2 (0, r) is close to 0 then b is adecreasing function of time. Secondly, if �c2 (0, r) is close

to r then b is a skewed S-shaped function of time. Thirdly,there is an intermediate case where b can take either of the

two shapes depending on the size of T. Moreover, thesecond regime becomes dominant when T is large in the

sense that b(0) tends to 0 as T!1. These three regimes

are not disconnected and if we let �c run from 0 to r thenthe optimal stopping boundary b moves from the 1function to the 0 function on [0,T) gradually passingthrough the three shapes above and always satisfyingb(T)¼ rK/�c (exhibiting also a singular behaviour at T inthe sense that b0(T� )¼�1). We will see in section 6below that the three regimes have three different eco-nomic interpretations and their fuller understanding isimportant for a correct/desired choice of the contract drift�c in relation to the interest rate r and other parameters inthe British call option. Note that this structure differsfrom the three-regime structure in the British put option(Peskir and Samee 2011) where the optimal stoppingboundary bp can be skewed U-shaped so that bp(0) tendsto 1 as T!1.

Fuller details of the analysis above go beyond our aimsin this paper and for this reason will be omitted. It shouldbe noted however that one of the key elements whichmakes this analysis more complicated (in comparisonwith the American put option) is that b is not necessarily amonotone function of time. In the next section we willderive simpler equations which characterise V and buniquely and can be used for their calculation (section 6).

8. We conclude this section with a few remarks on thechoice of the volatility parameter in the British payoffmechanism. Unlike its classical European or Americancounterpart, it is seen from equation (15) that thevolatility parameter � appears explicitly in the Britishpayoff (10) and hence should be prescribed explicitly inthe contract specification. Note that this feature is alsopresent in the British put option (cf. Peskir and Samee2011) and its appearance can be seen as a directconsequence of ‘optimal prediction’. Considering thisfrom a practical perspective, a natural question (commonto all options with volatility-dependent payoffs) is ‘Whatvalue of the parameter � should be used in the contractspecification/payoff?’. Both counterparties to the option

T

c

rK

0

K

b

as c 0

cas r

(i)

(ii)

(iii)

Figure 4. A computer drawing showing how the rationalexercise boundary of the British call option changes as onevaries the contract drift.


trade must agree on the value of this parameter, or at leastagree on how it should be calculated from future marketobservables, at the initiation of the contract. However,whilst this question has many practical implications,it does not pose any conceptual difficulties under thecurrent modelling framework. Since the underlying pro-cess is assumed to be a geometric Brownian motion, the(constant) volatility of the stock price is effectively knownby all market participants, since one may take any of thestandard estimators for the volatility over an arbitrarilysmall time period prior to the initiation of the contract.This is in direct contrast to the situation with stock pricedrift �, whose value/form is inherently unknown, nor canbe reasonably estimated (without an impractical amountof data), at the initiation of the contract. In this sense, itseems natural to provide a true buyer with protectionfrom an ambiguous (Knightian) drift rather than a‘known’ volatility, at least in this canonical Black–Scholes setting. We remark however that as soon as onedeparts from the current modelling framework, and getscloser to a more practical perspective, it is clear that thespecification of the contract volatility may indeed becomevery important. In this wider framework one is naturallyled to consider the relationship between the realised/implied volatility and the ‘contract volatility’ using thesame/similar rationale as for the actual drift and the‘contract drift’ in the text above. Owing to the funda-mental difference between the drift and the volatility inthe canonical (Black–Scholes) setting, and the highlyapplied nature of such modelling issues, these features ofthe British payoff mechanism are left as the subject offuture development.

4. The arbitrage-free price and the rational

exercise boundary

In this section we derive a closed form expression for thearbitrage-free price V in terms of the rational exerciseboundary b (the early-exercise premium representation)and show that the rational exercise boundary b itself canbe characterised as the unique solution to a nonlinearintegral equation (theorem 1). We note that the formerapproach was originally applied in the pricing ofAmerican options in Kim (1990), Jacka (1991) and Carret al. (1992), and the latter characterisation was estab-lished in Peskir (2005b) (for more details see, for example,Peskir and Shiryaev 2006).

We will make use of the following functions in theorem1 below:

Fðt, xÞ ¼ G�c ðt, xÞ � e�rðT�tÞGrðt, xÞ ð36Þ

Jðt, x, v, zÞ ¼ �e�rðv�tÞZ 1z

H�c ðv, yÞ f ðv� t, x, yÞ dy ð37Þ

for t2 [0,T), x4 0, v2 (t,T) and z4 0, where thefunctions Gr and G�c are given in equations (13) and(15) above (upon identifying �c with r in the former case),the function H�c is given in equations (24) and (28) above,

and y} f(v� t, x, y) is the probability density function of

xZrv�t from equation (14) above (with �c replaced by r

and T� t replaced by v� t) given by

f ðv� t, x, yÞ ¼1

� yffiffiffiffiffiffiffiffiffiffiv� tp ’

1

�ffiffiffiffiffiffiffiffiffiffiv� tp

�� log

y

x

� r�

�2

2

� �ðv� tÞ

� ��ð38Þ

for y4 0 (with v� t and x as above) where ’ is the

standard normal density function given by ’ðxÞ ¼ð1=

ffiffiffiffiffiffi2ppÞ e�x

2=2 for x2 IR. It should be noted that

J(t, x, v, b(v))4 0 for all t2 [0,T), x4 0 and v2 (t,T)

since H�c ðv, yÞ5 0 for all y4 b(v) as b lies above h (recall

equation (29) above). Finally, it can be verified using

standard means that

Jðt, x,T� , zÞ

¼ �c x �1

�ffiffiffiffiffiffiffiffiffiffiffiT� tp log

x

z _ K

þ rþ

�2

2

� �ðT� tÞ

� �� r K e�rðT�tÞ�

1

�ffiffiffiffiffiffiffiffiffiffiffiT� tp

�� log

x

z _ K

þ r�

�2

2

� �ðT� tÞ

� ��ð39Þ

for t2 [0,T), x4 0 and z4 0. This expression is useful in

a computational treatment of equation (41) below. The

main result of this section may now be stated as follows.

Theorem 4.1: The arbitrage-free price of the British call

option admits the following early-exercise premium

representation

Vðt, xÞ ¼ e�rðT�tÞ Grðt, xÞ þ

Z T

t

Jðt, x, v, bðvÞÞ dv ð40Þ

for all (t,x)2 [0,T]� (0,1), where the first term is the

arbitrage-free price of the European call option and the

second term is the early-exercise premium.The rational exercise boundary of the British call option

can be characterised as the unique continuous solution

b: [0,T]! IRþ to the nonlinear integral equation

Fðt, bðtÞÞ ¼

Z T

t

Jðt, bðtÞ, v, bðvÞÞ dv ð41Þ

satisfying b(t)� h(t) for all t2 [0,T] where h is defined by

equation (29) above.

Proof: The proof given below is similar to the proof of

theorem 1 in Peskir and Samee (2011) and we present all

details for completeness (as well as to demonstrate a full

symmetry between the arguments). Alternatively, one

could also exploit the British put–call symmetry relations

derived in section 5 below and deduce either of the two

theorems from the existence of the other. In the present

proof we first derive equation (40) and show that the

rational exercise boundary solves equation (41). We then

show that equation (41) cannot have other (continuous)

solutions.


1. Let V: [0,T]� (0,1)! IR and b: [0,T]! IRþdenote the unique solution to the free-boundary problem

(30)–(34) (where V extends as G�c above b), set

Cb¼ {(t, x)2 [0,T)� (0,1) jx5 b(t)} and Db¼ {(t, x)2

[0,T)� (0,1) jx4 b(t)}, and let ILXVðt,xÞ ¼ rxVxðt,xÞþ

ð�2=2Þx2Vxxðt,xÞ for (t, x)2Cb[Db. Then V and b are

continuous functions satisfying the following conditions:

(i) V is C1,2 on Cb[Db;(ii) b is of bounded variation;(iii) P(Xt¼ c)¼ 0 for all t2 [0,T] and c4 0;(iv) Vtþ ILXV� rV is locally bounded on Cb[Db

(recall that V solves equation (30) and coincides

with G�c on Db);(v) x}V(t,x) is convex on (0,1) for every t2 [0,T]

(recall (22) above); and(vi) t�Vxðt, bðtÞ�Þ ¼ G�c

x ðt, bðtÞÞ is continuous on

[0,T] (recall that V satisfies the smooth-fit condi-

tion (32) at b).

From these conditions we see that the local time–space

formula (Peskir 2005a) is applicable to (s, y)} e�rs

V(tþ s, xy) with t2 [0,T) and x4 0 given and fixed.

This yields

e�rsVðtþ s,xXsÞ

¼Vðt,xÞþ

Z s

0

e�rv Vtþ ILXV� rVð Þðtþ v,xXvÞ

� IðxXv 6¼ bðtþ vÞÞ dvþMbs þ

1

2

Z s

0

e�rvðVxðtþ v,xXvþÞ

�Vxðtþ v,xXv�ÞÞ IðxXv ¼ bðtþ vÞÞ d‘bvðXxÞ ð42Þ

where Mbs ¼�

R s0 e�rvxXvVxðtþv,xXvÞIðxXv 6¼bðtþvÞÞdBv

is a continuous local martingale for s2 [0,T� t] and

‘bðXxÞ¼ð‘bvðXxÞÞ0�v�s is the local time of Xx

¼ (xXv)0� v�s

on the curve b for s2 [0,T� t]. Moreover, since V satisfies

equation (30) on Cb and equals G�c on Db, and the

smooth-fit condition (32) holds at b, we see that equation

(42) simplifies to

e�rsVðtþ s, xXsÞ ¼ Vðt,xÞ þ

Z s

0

e�rvH�c ðtþ v, xXvÞ

� IðxXv 4 bðtþ vÞÞ dvþMbs ð43Þ

for s2 [0,T� t] and (t, x)2 [0,T)� (0,1).2. We show that Mb ¼ ðMb

s Þ0�s�T�t is a martingale for

t2 [0,T). For this, note that from equations (15), (27) and

(28) (or calculating directly) we find that

G�cx ðt,xÞ ¼ e�cðT�tÞ�

1

�ffiffiffiffiffiffiffiffiffiffiffiT� tp

�

� logx

K

þ �c þ

�2

2

� �ðT� tÞ

� ��ð44Þ

for t2 [0,T) and x4 0. Moreover, by (22) and (32) we see

that

0 � Vxðt, xÞ � G�cx ðt, bðtÞÞ ð45Þ

for all t2 [0,T) and all x2 (0, b(t)). Combining (44) and(45) we conclude that

0 � Vxðt, xÞ � e�cðT�tÞ ð46Þ

for all t2 [0,T) and all x4 0 (recall that V equals G�c

above b). Hence we find that

EhMb,MbiT�t

¼�2x2E

Z T�t

0

e�2rvX2v V

2xðtþv,xXvÞIðxXv 6¼bðtþvÞÞdv

� ��2x2e�cðT�tÞ

Z T�t

0

EX2v dv¼�

2x2e�cðT�tÞ

Z T�t

0

ervdv

¼�2x2

re�cðT�tÞ erðT�tÞ�1

� �51 ð47Þ

from where it follows that Mb is a martingale as claimed.3. Replacing s by T� t in equation (43), using the fact

that VðT, xÞ ¼ G�c ðT, xÞ ¼ ðx� KÞþ for x4 0, taking Eon both sides and applying the optional samplingtheorem, we get

e�rðT�tÞEðxXT�t�KÞþ

¼Vðt,xÞþ

Z T�t

0

e�rvE H�c ðtþ v,xXvÞ IðxXv4bðtþ vÞÞ½ �dv

¼Vðt,xÞ�

Z T

t

Jðt,x,v,bðvÞÞ dv ð48Þ

for all (t, x)2 [0,T)� (0,1). Recognising the left-handside of equation (48) as e�r(T�t) Gr(t, x) we see that thisyields the representation (40). Moreover, sinceVðt, bðtÞÞ ¼ G�c ðt, bðtÞÞ for all t2 [0,T] we see from equa-tion (40) that b solves equation (41). This establishes theexistence of the solution to equation (41). We now turn toits uniqueness.

4. We show that the rational exercise boundary is theunique solution to equation (41) in the class of continuousfunctions t} b(t) on [0,T] satisfying b(t)� h(t) for allt2 [0,T]. For this, take any continuous functionc: [0,T]! IR which solves equation (41) and satisfiesc(t)� h(t) for all t2 [0,T]. Motivated by the representation(48) above define the function Uc : [0,T]� (0,1)! IR bysetting

Ucðt, xÞ ¼ e�rðT�tÞE G�c ðT, xXT�tÞ½ �

�

Z T�t

0

e�rvE H�c ðtþ v, xXvÞ½

� IðxXv 4 cðtþ vÞÞ� dv ð49Þ

for (t, x)2 [0,T]� (0,1). Observe that c solving equation(41) means exactly that Ucðt, cðtÞÞ ¼ G�c ðt, cðtÞÞ for allt2 [0,T] (recall that G�c ðT, xÞ ¼ ðx� KÞþ for all x4 0).

(i) We show that Ucðt, xÞ ¼ G�c ðt,xÞ for all(t, x)2 [0,T]� (0,1) such that x� c(t). For this, takeany such (t, x) and note that the Markov property of Ximplies that

e�rsUcðtþ s, xXsÞ

�

Z s

0

e�rvH�c ðtþ v, xXvÞ IðxXv 4 cðtþ vÞÞ dv ð50Þ


is a continuous martingale under P for s2 [0,T� t].Consider the stopping time

�c ¼ inf f s 2 ½0,T� t� j xXs � cðtþ sÞ g ð51Þ

under P. Since Ucðt, cðtÞÞ ¼ G�c ðt, cðtÞÞ for all t2 [0,T] andUcðT, xÞ ¼ G�c ðT, xÞ for all x4 0 we see thatUcðtþ �c, xX�c Þ ¼ G�c ðtþ �c, xX�c Þ. Replacing s by �c informula (50), taking E on both sides and applying theoptional sampling theorem, we find that

Ucðt,xÞ ¼ E e�r�c Ucðtþ�c,xX�c Þ� ��E

Z �c

0

e�rvH�c ðtþ v,xXvÞ IðxXv4cðtþ vÞÞ dv

� �¼ E e�r�c G�c ðtþ�c,xX�c Þ

� ��E

Z �c

0

e�rvH�c ðtþ v,xXvÞ dv

� �¼G�c ðt,xÞ ð52Þ

where in the last equality we use equation (25). This showsthat Uc equals G�c above c as claimed.

(ii) We show that Uc(t, x)�V(t, x) for all (t, x)2[0,T]� (0,1). For this, take any such (t, x) and considerthe stopping time

�c ¼ inf f s 2 ½0,T� t� j xXs � cðtþ sÞ g ð53Þ

under P. We claim that Ucðtþ �c, xX�c Þ ¼G�c ðtþ �c, xX�c Þ. Indeed, if x� c(t) then �c¼ 0 so thatUcðt, xÞ ¼ G�c ðt,xÞ by (i) above. On the other hand, ifx5 c(t) then the claim follows since Ucðt, cðtÞÞ ¼G�c ðt, cðtÞÞ for all t2 [0,T] and UcðT, xÞ ¼ G�c ðT,xÞfor all x4 0. Replacing s by �c in (50), taking E onboth sides and applying the optional sampling theorem,we find that

Ucðt,xÞ

¼ E e�r�c Ucðtþ �c,xX�c Þ� ��E

Z �c

0


� �¼ E e�r�c G�c ðtþ �c,xX�c Þ

� ��Vðt,xÞ ð54Þ

where in the second equality we used the definition of �c.This shows that Uc

�V as claimed.(iii) We show that c(t)� b(t) for all t2 [0,T]. For this,

suppose that there exists t2 [0,T) such that c(t)4 b(t).Take any x� c(t) and consider the stopping time

�b ¼ inf f s 2 ½0,T� t� jxXs � bðtþ sÞ g ð55Þ

under P. Replacing s with �b in (43) and (50), taking E onboth sides of these identities and applying the optionalsampling theorem, we find

E e�r�b Vðtþ �b, xX�b Þ� �¼ Vðt, xÞ þ E

Z �b

0

e�rvH�c ðtþ v, xXvÞ dv

� �ð56Þ

E e�r�b Ucðtþ�b,xX�b Þ� �

¼Ucðt,xÞ

þE

Z �b

0

e�rvH�c ðtþ v,xXvÞ

�� IðxXv4cðtþ vÞÞ dv

i: ð57Þ

Since x� c(t) we see by (i) above that Ucðt, xÞ ¼

G�c ðt, xÞ ¼ Vðt, xÞ where the last equality follows since x

lies above b(t). Moreover, by (ii) above we know that

Ucðtþ �b, xX�b Þ � Vðtþ �b, xX�bÞ so that equations (56)

and (57) imply that

E

Z �b

0


�� IðxXv � cðtþ vÞÞ dv

�� 0: ð58Þ

The fact that c(t)4 b(t) and the continuity of the

functions c and b imply that there exists "4 0 sufficiently

small such that c(tþ v)4 b(tþ v) for all v2 [0, "].Consequently the P-probability of Xx

¼ (xþXv)0�v�"spending a strictly positive amount of time (w.r.t. the

Lebesgue measure) in this set before hitting b is strictly

positive. Combined with the fact that b lies above h this

forces the expectation in (58) to be strictly negative and

provides a contradiction. Hence c� b as claimed.(iv) We show that b(t)¼ c(t) for all t2 [0,T]. For this,

suppose that there exists t2 [0,T) such that c(t)5 b(t).

Take any x2 (c(t), b(t)) and consider the stopping time

�b ¼ inf f s 2 ½0,T� t� jxXs � bðtþ sÞ g ð59Þ

under P. Replacing s with �b in (43) and (50), taking E on

both sides of the resulting identities and applying the

optional sampling theorem, we find

E e�r�b Vðtþ �b, xX�bÞ� �

¼ Vðt, xÞ ð60Þ

E e�r�b Ucðtþ �b,xX�bÞ� �

¼Ucðt,xÞ

þE

Z �b

0


� �: ð61Þ

Since c� b by (iii) above and Uc equals G�c above c by (i)

above, we see that Ucðtþ �b, xX�b Þ ¼ G�c ðtþ �b, xX�bÞ ¼Vðtþ �b, xX�b Þ, where the last equality follows since V

equals G�c above b (recall also that UcðT, xÞ ¼

G�c ðT, xÞ ¼ VðT, xÞ for all x4 0). Moreover, by (ii) we

know that Uc�V so that equations (60) and (61) imply

that

E

Z �b

0


�IðxXv 4 cðtþ vÞÞ dv

�� 0: ð62Þ

But then as in (iii) above the continuity of the functions c

and b combined with the fact that c lies above h forces the

expectation in (62) to be strictly negative and provides a

contradiction. Thus c¼ b as claimed and the proof is

complete. œ


5. The British put–call symmetry

In this section we derive the ‘British put–call symmetry’

relations which express the arbitrage-free price and the

rational exercise boundary of the British call option in

terms of the arbitrage-free price and the rational exercise

boundary of the British put option where the roles of the

contract drift and the interest rate have been swapped

(theorem 2). We note that similar symmetry relations are

known to be valid for the American put and call options

written on stocks paying dividends (see Carr and Chesney

1996) where the roles of the dividend yield and the interest

rate have been swapped instead. The idea of examining

these relations in the British option setting is due to

Kristoffer Glover.1. In the setting of section 3 let us put

G�cc ðt, x;KÞ ¼ EðxZ�c

T�t � KÞþ ð63Þ

G�cp ðt, x;KÞ ¼ EðK� xZ�c

T�tÞþ

ð64Þ

to denote the payoff functions of the British call and put

options respectively, where the contract drift �c4 0 and

the strike price K4 0 are given and fixed, (t, x)2 [0,T]�

(0,1), and Z�c

T�t is given in equation (14) above. Let us

also put

V�cc ðt, x;K, rÞ ¼ sup

0��T�tE e�r�G�c

c ðtþ �, xXr�;KÞ

� �ð65Þ

V�cp ðt, x;K, rÞ ¼ sup

0��T�tE e�r�G�c

p ðtþ �, xXr�;KÞ

h ið66Þ

to denote the arbitrage-free prices of the British call and

put options respectively, where the interest rate r4 0 is

given and fixed, (t, x)2 [0,T]� (0,1), and the process Xr

solves equation (19) above with Xr0 ¼ 1. Recall that the

supremums in (65) and (66) are taken over all stopping

times � of Xr with values in [0,T� t]. The following

lemma lists basic properties of the payoff functions (63)

and (64) that will be useful in the proof of theorem 2

below.

Lemma 5.1: The following relations are valid:

G�ca ðt, x;KÞ ¼ xG�c

a t, 1;K=xð Þ ð67Þ

G�ca ðt, x;KÞ ¼ K G�c

a t, x=K; 1ð Þ ð68Þ

G�ca ðt, x;KÞ ¼ G0

a t, x e�cðT�tÞ;K� �

ð69Þ

G0cðt, x; 1Þ ¼ xG0

p t, 1=x; 1ð Þ ð70Þ

G0pðt, x; 1Þ ¼ xG0

c t, 1=x; 1ð Þ ð71Þ

for all values of the parameters, where the subscript ‘a’ in

(67)–(69) stands for either ‘c’ or ‘p’.

Proof: Since (67)–(69) are evident from the definitions

(63) and (64) we focus on relation (70). For this, note that

by the Girsanov theorem we have

G0cðt, x; 1Þ ¼ EðxZ0

T�t � 1Þþ ¼ xE Z0T�t 1�

1

xZ0T�t

� �þ� �

¼ xE

"exp �BT�t �

�2

2ðT� tÞ

� �

� 1�1

xexp ��BT�t þ

�2

2ðT� tÞ

� �� þ#

¼ xeE 1� 1

xexp

�� ½BT�t � �ðT� tÞ�

� �2ðT� tÞ þ�2

2ðT� tÞ

�!þ

¼ xeE 1�1

xexp ��eBT�t �

�2

2ðT� tÞ

� �� þ

¼ xE 1�1

xZ0

T�t

� �þ¼ xG0

p t,1

x; 1

� �ð72Þ

where ~E is taken with respect to ~P given by

d ~P ¼ exp �BT�t � ð�2=2ÞðT� tÞ

� �dP and ~Bs ¼ Bs � �s is

a standard Brownian motion under ~P for s2 [0,T� t]

(recall also that � ~BT�t¼law ~BT�t under ~P). This establishes

relation (70) from where relation (71) follows as well. œ

The main result of this section may now be stated as

follows.

Theorem 5.2: The arbitrage-free price of the British call

option with the contract drift �c and the interest rate r

satisfying (9) can be expressed in terms of the arbitrage-free

price of the British put option with the contract drift r and

the interest rate �c as

V�cc ðt,x;K, rÞ ¼

x

Ke�cðT�tÞ

� Vrp t,

K2

xe�ðrþ�cÞðT�tÞ;K,�c

� �ð73Þ

for all (t, x)2 [0,T]� (0,1).The rational exercise boundary of the British call option

with contract drift �c and interest rate r satisfying (9) can

be expressed in terms of the rational exercise boundary of

the British put option with contract drift r and interest rate

�c as

b�c,rc ðtÞ ¼ e�ðrþ�cÞðT�tÞ

K2

br,�cp ðtÞ

ð74Þ

for all t2 [0,T].

Proof: We first derive the identity (73). For this, fix any

(t, x)2 [0,T]� (0,1) and take any stopping time � of Xr

with values in [0,T� t]. Using the facts from lemma 1


indicated below and applying the Girsanov theorem we

find that

E e�r�G�cc tþ �,xXr

�;K� ��

¼ð68Þ

KE e�r�G�cc tþ �,

x

KXr�;1

h i¼ð69Þ

KE e�r�G0c tþ �,

x

Ke�cðT�t��ÞXr

�;1 h i

¼ð70Þ

KE e�r�x

Ke�cðT�t��ÞXr

�G0p tþ �,

K

xe��cðT�t��Þ

1

Xr�

;1

� �� ¼ xe�cðT�tÞE

�e�B��

�2

2 � e��c�

�G0p tþ �,

K

xe��cðT�tÞe��B�þ �c�rþ

�2

2

� ��;1

� ��¼ xe�cðT�tÞeE�e��c�

�G0p tþ �,

K

xe��cðT�tÞe��ðB��Þ��

2�þ �c�rþ�2

2

� ��;1


�G0p tþ �,

K

xe��cðT�tÞe��

eB�þ �c�r��2

2

� ��;1


�G0p

�tþ �,

K

xe�ðrþ�cÞðT�tÞerðT�t��Þe��

eB�þ �c��2

2

� ��;1

��¼ð69Þ

xe�cðT�tÞeE e��c�½

�Grp tþ �,

K

xe�ðrþ�cÞðT�tÞ eX�c

� ;K

K

� ��¼ð68Þ x

Ke�cðT�tÞeE e��c�½

�Grp tþ �,

K2

xe�ðrþ�cÞðT�tÞ eX�c

� ;K

� ��ð75Þ

where ~E is taken with respect to ~P given by d ~P ¼expð�B� �

�2

2 �Þ dP and ~Bt ¼ Bt � �t is a standard

Brownian motion under ~P for t2 [0,T] (recall also that

� ~B ¼law ~B under ~P). Taking the supremum over all stopping

times � of Xr with values in [0,T� t] on both sides of

equation (75) we obtain the identity (73) as claimed.To prove equation (74) note that by setting � 0 in

equation (75) we find that

G�cc ðt, x;KÞ ¼

x

Ke�cðT�tÞ Gr

p t,K2

xe�ðrþ�cÞðT�tÞ;K

� �ð76Þ

for all (t, x)2 [0,T]� (0,1). A direct comparison of

identity (73) and equation (76) shows that

V�cc ðt, x;K, rÞ4G�c

c ðt, x;KÞ ð77Þ

if and only if the following inequality is satisfied:

Vrp t,

K2

xe�ðrþ�cÞðT�tÞ;K,�c

� �4Gr

p t,K2

xe�ðrþ�cÞðT�tÞ;K

� �ð78Þ

for all (t, x)2 [0,T]� (0,1). As the two relations (77) and(78) characterise the continuation sets for the optimalstopping problems (65) and (66) respectively, it followsthat equation (74) holds and the proof is complete. œ

Remark 1: Note that the previous arguments extend tothe case when the strike price in equations (63) and (65) isnot necessarily equal to the strike price in equations (66)and (66). More precisely, replacing the strike price K inequations (63) and (65) by Kc4 0 and the strike price K inequations (64) and (66) by Kp4 0, and noting that thefinal equality in (75) remains valid if we replace K/K byKp/Kp, we obtain the following extensions of (73) and (74)respectively:

V�cc ðt,x;Kc, rÞ ¼

x

Kpe�cðT�tÞ

�Vrp t,

KcKp

xe�ðrþ�cÞðT�tÞ;Kp,�c

� �ð79Þ

b�c,rc ðtÞ ¼ e�ðrþ�cÞðT�tÞ

KcKp

br,�cp ðtÞ

ð80Þ

for all (t,x)2 [0,T]� (0,1). This can be rewritten in amore symmetric form as follows:

V�cc t, xKp e

��cðT�tÞ; yKc, r� �¼ Vr

p t, yKc e�rðT�tÞ;xKp,�c

� �ð81Þ

b�c,rc ðtÞb

r,�cp ðtÞ ¼ e�ðrþ�cÞðT�tÞ KcKp ð82Þ

for all t2 [0,T] and all x, y2 (0,1).

6. Financial analysis of the British call option

In the present section we firstly discuss the rationalexercise strategy of the British call option, and then anumerical example is presented to highlight the practicalfeatures of the option. We draw comparisons with theEuropean call option in particular because the latteroption is widely traded and well understood (recallingthat the American call option written on a stock payingno dividends reduces to the European call option since itis rational to exercise at maturity). In the financialanalysis of the option returns presented below we mainlyaddress the question as to what the return would be if thestock price enters the given region at a given time (i.e. wedo not discuss the probability of the latter event nor dowe account for any risk associated with its occurrence).Such a ‘skeleton analysis’ is both natural and practicalsince it places the question of probabilities and risk underthe subjective assessment of the option holder (irrespec-tive of whether the stock price model is correct or not).

1. In section 3 above we saw that the rational exercisestrategy of the British call option (the optimal stoppingtime (35) in the problem (18) above) changes as one variesthe contract drift �c. This is illustrated in figure 4 above.To explain the economic meaning of the three regimesappearing on this figure, let us first recall that it is naturalto set �c5 r. Indeed, if one sets �c� r then it is always


optimal to stop at once in the problem (18). In this casethe buyer is overprotected. By exercising immediately hewill beat the interest rate r and moreover he will avoid anydiscounting of his payoff. We also noted above that therational exercise boundary b in figure 4 satisfiesb(T)¼ rK/�c. In particular, when �c¼ r then b(T)¼Kand b extends backwards in time to zero. The interest rater represents a borderline case and any �c4 0 strictlysmaller than r will inevitably lead to a non-trivial rationalexercise boundary (when the initial stock price is suffi-ciently low). It is clear however that not all of thesesituations will be of economic interest and in practice �c

should be set further away from r in order to avoidoverprotection (note that most generally overprotectionrefers to the case where the initial stock price lies aboveb(0)). On the other hand, when �c# 0 then b(T) increasesup to infinity and b disappears in the limit, so that it isnever optimal to stop before the maturity time T and theBritish call option reduces to the European call option.In the latter case a zero contract drift represents aninfinite tolerance of unfavourable drifts and the Britishcall holder will never exercise the option before the time ofmaturity. This brief analysis shows that the contract drift�c should not be too close to r (since in this case the buyeris overprotected) and should not be too close to zero(since in this case the British call option effectively reducesto the European call option). We remark that the latterpossibility is not fully excluded, however, especially ifsharp increases in the stock price are possible or likely.

2. Further to our comments above, consider theposition and shape of the rational exercise boundaries infigure 4, and assume that the initial stock price equals K.In this case the boundary (ii) represents overprotection.Indeed, although not larger than r, the contract drift �c isfavourable enough so that the additional incentive ofavoiding discounting makes it rational to exercise imme-diately. On the other hand, it may be noted that theposition of the boundaries (i) and (iii) suggests that thebuyer should exercise the British call option rationallywhen observed price movements are favourable. As thiseffect is analogous to the same effect in the British putoption we refer to Peskir and Samee (2011) for furtherdetails. In addition we remark that the position of theboundaries in figure 4 is largely determined by thepresence of discounting in the problem. As the stockprice increases, the payoff function (13) increases as well(for fixed time), and the effect of discounting impactsmore heavily on any decision to continue. The British callholder thus becomes more inclined to exercise simply tobeat the discounting effect and the higher (more favour-able) the contract drift the more so this is true.

3. Figure 5 shows the rational exercise boundaries ofthe British call option for a fixed parameter set (chosen topresent the practical features of the option in a fair andrepresentative way). The values of the contract drift �c

have been selected to produce boundaries of type (i) and(iii) in figure 4. As already indicated above, these valuesalso depend strongly upon the volatility coefficient �.When � is large then the British call holder will have agreater tolerance for unfavourable drifts since the high

volatility is more likely to drown out the effect of the driftand as such the buyer can still record favourable returns.In this case the contract drift �c must be set further awayfrom the interest rate r in order to avoid overprotection.This is also seen from the fact that the rational exerciseboundary b becomes more S-skewed as the volatilitycoefficient � increases so that b(0) can become small.Assuming again that the initial stock price equals K, theprice of the European call option is 0.2032. The price ofthe British call option is 0.2034 if �c¼ 0.05 and 0.2052 if�c¼ 0.07. Note that the closer the contract drift gets to r,the stronger is the protection feature provided, and themore expensive the British call option becomes.Moreover, in terms of price sizes it can be seen that thisexample is not isolated since the price of the British calloption stays very close to the price of the European calloption unless the contract drift �c is unrealistically closeto the interest rate r (so that the buyer is overprotectedand the situation is uninteresting economically). Recallalso that when �c# 0 then in the limit it is not rational toexercise before the time of maturity and the price of theBritish call option reduces to the price of the Europeancall option. The fact that the price of the British calloption is very comparable to the price of the Europeancall option (in most of the situations that are of interestfor trading) is of considerable practical value (given theadditional benefits to be discussed shortly below). It alsoimplies (as a rule of thumb) that the deltas for bothoptions (hedge ratios) stay very close to each other as wellas that many of the sensitivities of the British call option(option Greeks) will be very similar to those of theEuropean call option. Further details of such an analysiscan be readily generated using the formulaederived above.

4. Table 1 shows the power of the protection feature inpractice. For example, if the stock price is at K halfway tomaturity (clearly representative of unfavourable pricemovements) then the British call holder can exerciseimmediately to a payoff which represents a

Figure 5. A computer drawing showing the rational exerciseboundaries of the British call option with K¼ 10, T¼ 1, r¼ 0.1,�¼ 0.4 when the contract drift �c equals 0.05 and 0.07.


reimbursement of 62–65% of his original investment.Compare this with a ‘formal American call’ holder who inthis contingency is out-of-the-money and would receivezero payoff upon exercise. We also see that the size of thereimbursement received by the British call holder dependsupon the contract drift. The closer the contract drift is tor, the more protection the British call holder is afforded,and thus the greater his reimbursement will be. We note inaddition that setting V¼VK and G�c ¼ G�c

K to indicatedependence on the strike price K, we have VK(t,x)¼KV1(t, x/K) and G�c

K ðt,xÞ ¼ KG�c

1 ðt, x=KÞ, so that thereturn does not depend on the size of the strike price(when properly scaled). The same fact is also valid for thereturns of the European call option considered in tables 2and 3.

5. Tables 2 and 3 show the returns that the Britishcall holder can extract when the price movements arefavourable or unfavourable respectively. We focus on theBritish call option with �c¼ 0.07 since in this case the

rational exercise boundary is closer to the strike price Kand this makes the comparison more interesting econom-ically. Tables 2 and 3 also show the returns observed uponselling the European call option (at the arbitrage-freeprice) in the same contingency. This is motivated by thefact that in practice the option holder may also choose tosell his option at any time during the term of the contract,and in this case one may view his ‘payoff’ as the price hereceives upon selling. In particular, apart from being ableto ‘cash in’ immediately upon favourable stock pricemovements (table 2), the European call holder is to someextent protected from unfavourable price movements byhis ability to sell the contract (table 3). Upon inspection oftables 2 and 3 we see that exercising the British call optionproduces remarkably similar returns to selling theEuropean call option (at the arbitrage-free price) irre-spective of whether stock price movements are favourableor unfavourable. However, in a real financial market theoption holder’s ability and/or desire to sell his contract

Table 2. Returns (%) observed upon exercising the British call option (with �c¼ 0.07) above the strike price K compared withreturns received upon selling the European call option in the same contingency.

Time (months)

0 2 4 6 8 10 12

Exercise at 20 (British call) 561 548 535 522 510 499 487Selling at 20 (European call) 541 533 525 516 508 500 492Exercise at 18 (British call) 459 446 434 422 411 400 390Selling at 18 (European call) 445 436 427 419 410 402 394Exercise at 16 (British call) 359 347 335 323 312 302 292Selling at 16 (European call) 350 341 331 322 312 304 295Exercise at b (British call) 238 257 274 287 292 278 209Selling at b (European call) 235 254 272 287 293 280 211Exercise at 12 (British call) 175 163 151 138 124 109 98Selling at 12 (European call) 174 163 152 140 126 111 98Exercise at 11 (British call) 135 124 112 99 84 68 49Selling at 11 (European call) 135 125 113 101 87 70 49

Notes: The returns are expressed as a percentage of the original option price paid by the buyer (rounded to the nearest integer), i.e.

Rðt,xÞ=100 ¼ G�c ðt,xÞ=Vð0,KÞ and RE(t, x)/100¼VE(t, x)/VE(0,K), respectively. The parameter set is the same as in figure 5 (K¼ 10, T¼ 1, r¼ 0.1,

�¼ 0.4) and the initial stock price equals K.

Table 1. Returns (%) observed upon exercising the British call option at and below the strike price K.

Time (months)

0 2 4 6 8 10 12

Exercise at K with �c¼ 0.07 99 88 77 65 51 35 0Exercise at K with �c¼ 0.05 93 84 74 62 50 34 0Exercise at 9 with �c¼ 0.07 68 58 49 38 27 13 0Exercise at 9 with �c¼ 0.05 63 55 46 36 26 13 0Exercise at 8 with �c¼ 0.07 42 35 27 19 11 03 0Exercise at 8 with �c¼ 0.05 39 32 25 18 10 03 0Exercise at 7 with �c¼ 0.07 23 18 12 07 03 0.4 0Exercise at 7 with �c¼ 0.05 21 16 12 07 03 0.3 0Exercise at 6 with �c¼ 0.07 11 07 04 02 01 0.0 0Exercise at 6 with �c¼ 0.05 10 07 04 02 01 0.0 0Exercise at 5 with �c¼ 0.07 04 02 01 0.3 0.0 0.0 0Exercise at 5 with �c¼ 0.05 03 02 01 0.3 0.0 0.0 0


Rðt,xÞ=100 ¼ G�c ðt,xÞ=Vð0,KÞ. The parameter set is the same as in figure 5 (K¼ 10, T¼ 1, r¼ 0.1, �¼ 0.4) and the initial stock price equals K.


may depend upon a number of exogenous factors. These

include his ability to access the option market, the

transaction costs and/or taxes involved in selling theoption (i.e. friction costs), and in particular the liquidity

of the option market itself (which in turn determines the

market/liquidation price of the option). Recall also from

section 2 that these factors can change during the term of

the contract and that these changes can be difficult to

predict. We want to make it clear however that the Britishcall option (as considered in this paper) does not include

any of the exogenous factors explicitly in the pricing

model (nor does it aim to model their changes). Rather,

and crucially, the protection feature of the British call

option is intrinsic to it, that is, it is completely endoge-

nous. It is inherent in the payoff function itself (obtained

as a consequence of optimal prediction), and as such it is

independent of any exogenous factors. From this point of

view the British call option is a particularly attractive

financial instrument for the buyer who is unable to access

the market freely to sell his contract, when the friction

costs involved in doing so are significant, or when the

market for the contract is not perfectly liquid.6. Finally in table 4 we highlight a remarkable and

peculiar aspect of the British call option (also seen in the

British put option, see Peskir and Samee 2011)) which

was touched upon in section 3 above. We observed there

Table 4. Returns (%) observed upon (i) exercising and (ii) selling the British call option (with �c¼ 0.07) at and below the rationalexercise boundary.

Time (months)

0 2 4 6 8 10 12

Exercise at b 238 257 274 287 292 278 209Selling at b 238 257 274 287 292 278 209Exercise at 12 175 163 151 138 124 109 98Selling at 12 175 164 152 139 125 110 98Exercise at 11 135 124 112 99 84 68 49Selling at 11 136 125 113 100 86 69 49Exercise at K 99 88 77 65 51 35 0Selling at K 100 90 78 66 53 36 0Exercise at 9 68 58 49 38 27 13 0Selling at 9 69 60 50 39 28 14 0Exercise at 8 42 35 27 19 11 03 0Selling at 8 44 36 28 20 11 04 0Exercise at 7 23 18 12 07 03 0.4 0Selling at 7 24 19 13 08 03 0.5 0Exercise at 6 11 07 04 02 01 0.0 0Selling at 6 11 08 05 02 01 0.0 0Exercise at 5 04 02 01 0.3 0.0 0.0 0Selling at 5 04 02 01 0.4 0.1 0.0 0


Reðt,xÞ=100 ¼ G�c ðt,xÞ=Vð0,KÞ and Rs(t,x)/100¼V(t,x)/V(0,K). The parameter set is the same as in figure 5 (K¼ 10, T¼ 1, r¼ 0.1, �¼ 0.4) and the

initial stock price equals K.

Table 3. Returns (%) observed upon exercising the British call option (with �c¼ 0.07) at and below the strike price K comparedwith returns received upon selling the European call option in the same contingency.

Time (months)

0 2 4 6 8 10 12

Exercise at K (British call) 99 88 77 65 51 35 0Selling at K (European call) 100 90 79 67 53 36 0Exercise at 9 (British call) 68 58 49 38 27 13 0Selling at 9 (European call) 69 60 50 40 28 14 0Exercise at 8 (British call) 42 35 27 19 11 03 0Selling at 8 (European call) 44 36 28 20 12 04 0Exercise at 7 (British call) 23 18 12 07 03 0.4 0Selling at 7 (European call) 24 19 13 08 03 0.5 0Exercise at 6 (British call) 11 07 04 02 01 0.0 0Selling at 6 (European call) 11 08 05 02 01 0.0 0Exercise at 5 (British call) 04 02 01 0.3 0.0 0.0 0Selling at 5 (European call) 04 02 01 0.4 0.1 0.0 0


Rðt,xÞ=100 ¼ G�c ðt,xÞ=Vð0,KÞ and RE(t,x)/100¼VE(t, x)/VE(0,K), respectively. The parameter set is the same as in figure 5 (K¼ 10, T¼ 1, r¼ 0.1,

�¼ 0.4) and the initial stock price equals K.


(see figure 3) that the value function and the payoff

function of the British call option stay close together in

the continuation set. The extent to which this is true is

made apparent by table 4. For this particular choice of

contract drift we see that the value function stays above

the payoff function (both expressed as percentage returns)

within a margin of 2%. The tightness of this relationship

will be affected by the choice of the contract drift (indeed

the closer the contract drift is to r the stronger the

protection and the tighter the relationship will be). From

table 4 we see that (i) exercising in the continuation set

produces a remarkably comparable return to selling the

contract in a liquid option market; and (ii) even if the

option market is perfectly liquid it may still be more

profitable to exercise rather than sell (when the friction

costs exceed the margin of 2% for instance).

Acknowledgements

We are grateful to Credit Suisse London for hospitality

and support. Special thanks go to David Shorthouse,

Laurent Veilex and Stephen Zoio for stimulating and

informative discussions.

References

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Du Toit, J. and Peskir, G., The trap of complacency inpredicting the maximum. Ann. Probab., 2007, 35, 340–365.

Jacka, S.D., Optimal stopping and the American put. Math.Finance, 1991, 1, 1–14.

Karatzas, I. and Shreve, S.E.,Methods of Mathematical Finance,1998 (Springer: New York).

Kim, I.J., The analytic valuation of American options. Rev.Financ. Stud., 1990, 3, 547–572.

Merton, R.C., Theory of rational pricing. Bell J. Econ. MgmtSci., 1973, 4, 141–183.

Peskir, G., A change-of-variable formula with local time oncurves. J. Theoret. Probab., 2005a, 18, 499–535.

Peskir, G., On the American option problem. Math. Finance,2005b, 15, 169–181.

Peskir, G. and Samee, F., The British put option. Appl. Math.Finance, 2011, 18, 537–563.

Peskir, G. and Shiryaev, A.N., Optimal Stopping and Free-Boundary Problems, Lectures in Mathematics, ETH, Zurich,2006 (Birkhauser: Basel, Switzerland).

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The British call option