3 Passport Option

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The Passport Option

Abstract

A passport option1 grants its holder the right to engage in a short/long trading strategy of

his own choice, while obligating the option writer to cover any net losses on the strategy.

Using results from stochastic control theory, we identify the optimal trading strategy and

derive a non-linear PDE satisfied by the asset-deflated option price. A closed-form

pricing formula is derived in a special case. For the general case, the paper illustrates how

a Crank-Nicholson finite difference scheme can be used to price the option. The

numerical scheme is applicable to a number of variations on the basic option contract.


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1. The Passport Option

The passport option is a new contingent claim that, in effect, insures against cumulative

losses resulting from repeated short-term trading activities. Recently introduced to the

marketplace, the option should provide a useful new tool for risk managers and investors

alike. To formally define the passport option consider a time horizon [0,T] and H+1

discrete dates t t t t t H H0 1 2 1, , ,..., , , where t0 0= and t TH = . At each ti , i H= 0 1 1, ,2,...,an investor takes a position u ti( ) [ , ] 11 in an asset S; the position is held over theinterval ( , ]t ti i+1 . The choice of u ti( ) is determined by the investor at time ti . The

accumulated gain of the asset holdings over [ , ]0 T is

u t S t S t ii

H

i i( )[ ( ) ( )]=

+ 0

1

1 . (1)

Under the terms of the European passport option contract, the option holder has the

right, but not the obligation, to receive this gain at the option maturity T. The payout of

the option at Tis thus

u t S t S t ii

H

i i( )[ ( ) ( )]=

+

+

0

1

1 (2)

where we have used the notation z MAX z+ = ( ,0) .

The most salient feature of the passport option is obviously the H rights to alter the

position in the underlying asset. While in practice the choice of the holding strategy u

might be "irrational" and influenced by personal beliefs or external hedging needs 2, the

seller of the option clearly must assume that the buyer will attempt to maximize his

financial gain on the contract. Unless the buyer of the option contractually agrees to a

specific strategy, the passport option will consequently always command a price that

corresponds to the strategy u for which the option value is maximized. Determination of

this strategy is, as we shall see, generally a non-trivial problem.

The rest of this paper is organized as follows. In Section 2, we provide notation and state

our process assumptions. Section 3 analyzes the pricing of European passport options for

the case where changes in holding strategy takes place in continuous time. In Section 4,

we address the numerical implementation of the results derived in Section 3. Section 5

contains various numerical results, and Section 6 discusses extensions of the numerical


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method to more complicated options, including those with time-discrete switching rights.

Finally, Section 7 contains our conclusions and some directions for further research.

2. A Continuous-Time Framework

Let us consider a standard Black-Scholes economy with two assets: a money-market

accountB and a dividend-paying stock3S. Assuming a constant risk-free interest rate r,B

evolves deterministically

B t rt ( ) exp( )= .

We assume that S pays a constant dividend yield of and follows a one-factor geometric

Brownian motion with constant volatility . According to Harrison and Pliska (1981),

the absence of arbitrage dictates the existence of a probability measureQ

(the "riskneutral" measure) under which the process for S can be written as the stochastic

differential equation (SDE)

dS t S t r dt dW t ( ) / ( ) ( ) ( )= + ; (3)

here, Wis a standard Brownian motion under Q.

To characterize the short/long strategy of the passport option holder, we now introduce an

adapted process { ( )}u t t T0 , where u t( ) [ , ] 11 . With this definition of u, we canintroduce the gain process4

w t u s dS st

( ) ( ) ( )= 0 (4a)or

dw t u t dS t w( ) ( ) ( ), ( )= =0 0 . (4b)

Equation (4a-b) is a continuous-time version of the discrete gain process (1). For

sufficiently small time increments t ti i+ 1 in (2), (4a-b) is a reasonable approximationalthough the value of the passport option will be biased high5 (see Section 6.2).

For a given holding strategy u, the payout at expiration Tof the European passport option

is defined as


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V T w T u( ) ( )=

+ .

As V Bu/ is a martingale under Q, the time toption value is

V t e E w T u r T t tQ( ) [ ( ) ]( )= + (5)

where EtQ[] denotes time texpectation under the probability measure Q.

We stress that the pricing equation (5) holds for anyfixedstrategy u. As discussed earlier,

allowing the option buyer to chose u freely in [-1,1] implies that the fair value Vof the

European passport option is the maximum

V t V t e E w T u u ur T t

t

Q( ) sup ( ) sup [ ( ) ][ , ] [ , ]( )= =

+1 1 1 1 (6)

where, from (4b),

dw t u t S t r dt u t S t dW t ( ) ( ) ( )( ) ( ) ( ) ( )= + . (7)

Equations (6) and (7) together form a stochastic control problem with w T( )+ being the

terminal bequestfunction and u the controls.

3. Solution

As the process (7) for w depends on S, we first attempt to introduce a shift of probability

measure that will remove S from the SDE for w . Specifically, we use the Girsanov

Theorem (Karatzas and Shreve (1994), Section 3.5) to introduce a new probability

measure Q under which

dW t dW t dt ( ) ( )= (8)

is a Brownian motion. We have the following proposition:

Proposition 1


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Define x t w t S t ( ) ( ) / ( )= , v t V t S t u u( ) ( ) / ( )= , and v t V t S t ( ) ( ) / ( )= . (5) and (6) can then

be written

v t e E x T u

T t

t

Q( ) [ ( ) ]( ) = + (9)

v t v t u u( ) sup ( )[ , ]= 1 1 (10)

where

dx t u t x t r dt u t x t dW t ( ) [ ( ) ( )]( ) [ ( ) ( )] ( )= + . (11)

Proof:First, notice that as V t B t u( ) / ( ) is a martingale under Q, we can use the Martingale

Representation Theorem (Duffie (1996), p. 287) to write

dV t V t rdt t dW t u u( ) / ( ) ( ) ( )= +

for some volatility function ( )t . From Itos lemma and (8)

dv t v t t dt t dW t

dt t dW t

u u( ) / ( ) ( ( ) ) ( ( ) ) ( )

( ( ) ) ( )

= + +

= +

2

whereby we conclude that v t eut( ) is a Q-martingale. That is, we have

v t e E v T e E x T u

T t

t

Q

u

T t

t

Q( ) [ ( )] [ ( ) ]( ) ( ) = = +

which is (9). (10) follows directly from (6). (11) follows from (7) and Itos lemma.

With Proposition 1, we have reduced the stochastic control problem (5), (6), and (7) to a

one-variable (x) setting. To proceed, we first need the following lemma:

Lemma 1


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In the absence of arbitrage, the S-deflated option value v is an increasing function ofx:

x x v t x v t x A B A B ( , ) ( , ) (12)

Moreover, v is a convex function ofx:

x x x v t x v t x v t x B A C A C B= + + ( ) ( , ) ( ) ( , ) ( , ),1 1 (13)

for [ , ]0 1 .

Proof:

To prove (12), consider two investors A and B, each having purchased identical T-

maturity passport options. The two investors have chosen different strategiesu

over [0,t],

resulting in different gains at time t. We assume thatA did at least as well asB, i.e.

w t w t A B( ) ( ) . (14)

Now assume thatBs optimal strategy going forward is u s t s T B* ( ), . From (5), we can

write the fair value VB ofBs option as

( )[ ]V t V t e E w t T w t B u B r T t tQ B BB( ) ( ) ( , ) ( )*( )= = +

+

where w t T u s dS sB Bt

T

( , ) ( ) ( )* .

AsBs optimal strategy may or may not be the same as As optimal strategy, we have the

following inequality for the fair value ofAs option

( )[ ]V t V t V t e E w t T w t V t A uA u A r T t tQ B A BA B( ) ( ) ( ) ( , ) ( ) ( )* *( )= = +

+

(15)

The last inequality follows from (14). Dividing (14) and (15) by S t( ) results in (12).

To show convexity, we introduce a third investor Cand replace (14) with the condition


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where

v

x 0 and

2

20

v

x . v satisfies the partial differential equation (PDE)

v

t

u x rv

x

u xv

x

v+ + =( )( ) ( )* *1

2

2 22

2(19a)

or, equivalently,

v

tx r

v

xx

v

xr

v

xx

v

xv + + + =( ) ( ) ( )

1

21 2 2

2

2

22

2(19b)

with boundary condition

v T x T ( ) ( )= + .

Proof:

By Theorem 11.3 in ksendahl (1995), the form of the SDE (11) forx under Q' allows us

to restrict the search for u* to functions depending only on time and the current level ofx,

i.e.

( )u t u t x t * *

( ) , ( )= .

Similarly, ( )v t v t x t ( ) , ( )= . Applying the Hamilton-Jacobi-Bellman (HJB) equation (see

e.g. ksendahl (1995), Theorem 11.1) on (9) and (10), we get

sup ( )( ) ( )[ , ]hv

th x r

v

xh x

v

xv + +

=1 1

2 2

2

2

1

20

. (20)

The supremum in (20) is obtained at ( )h u t x t =*

, ( ) .

Setting

C rv

x1 ( )

, Cv

x2

22

2

1

2

, Cv

tx r

v

xx

v

xv3

2 22

2

1

2 +

( )


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the maximum in (20) can be written

sup { ( ) }[ , ]h C h C xC h C + +1 1 3 1 22

22 . (21)

From the convexity properties outlined in Lemma 1, we know that 2 2 0v x/ . Asconsequently C2 0 , the maximum in (21) will always be obtained for either h = 1 or

h = 1. The obvious result is

u t x sign C xC sign r v

xx

v

x

* ( , ) ( ) ( )= =

1 2

2

2

22

.

The result v x/ 0 follows directly from Lemma 1. (19b) follows from (19a) byinsertion of (18) and using ( )u*

21= .

Not surprisingly, the optimal control is always either 1 or -1 (i.e. of the so-called "bang-

bang" type); allowing the option holder to take "fractional" positions (i.e. <


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The valuation PDE becomes

v

t

xv

x

v+ + =1

2

1 2 22

2( | | ) , v T x T ( ) ( )= + . (23)

When r= , S is a martingale under Q (see (3)). This fact might appear to make theresult (22) problematic: ifS is a martingale (under Q) without any predictable trend, why

is it possible to extract more value by following (22) than by using a naive strategy, say

u t x t ( , ( )) = 1? To answer this question, set r= and consider the case where the optionholder must make his last change of strategy at some predetermined intermediate time

s T( , )0 . Let us denote the fair value of this option Vs . At s, the option holder hasaccumulated a gain of w s( ) and is faced with the decision of setting either u s x s( , ( )) = 1or u s x s( , ( )) = 1. Depending on whether a short or a long position is chosen, theterminal payout is

u s x s( , ( )) = 1: ( )V T S T S s w ss++

= 1 ( ) ( ) [ ( ) ( )] (24a)

u s x s( , ( )) = 1: ( )V T S s w s S T s+

= + 1 ( ) [ ( ) ( )] ( ) . (24b)

(24a) is the payout of a regular call option with strike S s w s( ) ( ) , while (24b) is thepayout of a put option with strike S s w s( ) ( )+ . Applying the Black-Scholes formula(Black and Scholes (1973), Merton (1973)) with r= , we get

( )V s e

N d S s w s N d T s S s w s

w s S s w s

s T s c c

+ =

>

1 ( )

( ) [ ( ) ( )] , ( ) ( )

( ), ( ) ( )

( )

(25a)

V s eS s w s N d S s N d T s S s w s

S s w s

s T s p p

=

+ >

1 0( )

[ ( ) ( )] ( ) ( ) ( ), ( ) ( )

, ( ) ( )

( ) (25

b)

where


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dS s S s w s T s

T sc =

+

ln( ( ) / [ ( ) ( )]) ( )122

dS s S s w s T s

T sp =

+ +

ln( ( ) / [ ( ) ( )]) ( )122

andN() is the standard cumulative normal distribution.

Obviously, the option value at time s is

V s MAX V s V ss s s( ) ( ( ), ( ))= + 1 1 . (26)

It is easy to verify that the asymmetry of the Black-Scholes put and call formulas imply

that

w s V s V s( ) ( ) ( )> > +0 1 1 (27a)w s V s V s( ) ( ) ( )= = +0 1 1 (27b)w s V s V s( ) ( ) ( )<


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x i t ji x j t = = ,

for 0 i M, 0 j N and t TN = . x0 and xM represent and , respectively;

reasonable choices for these numbers are, say, me T t4 . Using the short notation

v v t xi j j i, ( , )= and u t x u j i i j

*,

*( , ) = , a mixed implicit-explicit finite differencediscretization of (19a) is given by

2 21

2 2

2 21

2 1

( ) ( )( )

( ) ( )

( ) ( )( )

,*

,*

,

,*

,

,*

,*

,

u x r u x v

u x v

u x r u x v

i j i x i j i i j

i j i t i j

i j i x i j i i j

+ + +

+ =

+

+

+

+

+

+ +

( ) ( ) ( )( )

( ) ( ) ( ( ) )

( ) ( ) ( )( )

,*

,*

,

,*

,

,*

,*

,

1

2 1 1 1

1

2 21 1

2 21

2 21 1

u x r u x v

u x v

u x r u x v

i j i x i j i i j

i j i t i j

i j i x i j i i j

(30)

where 0 1 and x t2 / . The boundary condition on this system of equations is

v xi N i, =+ . (31)

In (30), the parameter determines the time at which the partial derivatives w.r.t. x areevaluated. If = 1, thex-derivatives are evaluated at time tj and the differencing schemegives rise to the fully implicit finite difference method. If = 0, the x-derivatives areevaluated one time-step ahead, at tj+1 , and the differencing scheme is known as the

explicit finite difference method, also sometimes referred to as a trinomial tree. Finally,

when = 12 , the x-derivatives are evaluated at 12 1( )t tj j+ + and the scheme becomes theCrank-Nicholson method. Values of different from 0, 1/2, and 1 are possible, but littleused in practice.

For the PDE (19a), the dependence of the coefficients on x makes the explicit scheme

prone to instability8. We generally prefer the Crank-Nicholson scheme ( = 12 ), which hasthe highest order of convergence in t (see Smith (1985) for further discussion).

The discretization scheme (30) must be complemented by an algorithm to find the

optimal strategy ui j,* . Many such schemes are possible; here we present an algorithm


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based directly on the continuous-time result (18). In Section 6.2 we will discuss another

approach particularly suitable for discrete passport options.

Step 1:

Make a guess for the values of ui j,* . For example, we can set u ui j i j,

*,

*= +1 . The starting

value of ui N,* 1 at the second-to-last time-step tN1 can be computed analytically using the

Black-Scholes formula (as in (26)).

Step 2:

Solve the tri-diagonal system (30) for vi j, in the usual way (see e.g. Press et al (1992), p.

50-51). In this step, we must also impose appropriate boundary conditions at x0 and xM.

For example, we can set v j0 0, = and v v v v r M j M j M j M j t , , , ,( ) / ( )= + + + +1 1 1 1 1 , the latterbeing a discretization of the asymptotic behavior v x e r T t/ ( ) for largex.

Step 3:

Update the values ofui j,* using the following discretization of (18):

u sign

rv v v v

xv v v v v vi j

i j i j i j i j

x

i

i j i j i j i j i j i j

x

,

*

, , , ,

, , , , , ,

( )( ) ( )( )

( ) ( )( )=

+

+ + +

+ + + +

+ + + + +

1 1 1 1 1 1

2 1 1 1 1 1 1 1

2

1

2

2 1 2

.

(32)

Return to Step 2 until the values of ui j,* calculated in (32) remain unchanged from the

previous iteration.

As the switching boundary u* normally changes very little from one time-step to the next,

typically only one or two iterations are needed in Step 3. We note that the usage of (32) in

Step 3 requires v to be convex. In cases where this condition cannot be guaranteed (e.g.

for options with more complicated payouts than (31)) Step 3 must be modified. This issue

will be discussed in Section 6.3.

5. Numerical Examples

As a specific example, we now introduce a stock S with a time tvalue of $100 and a

volatility of 30%. We first assume that interest rates and dividends are 0, i.e. r= = 0 ,and consider the pricing of a European passport option with one year left to maturity.


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Depending on the value of the current gain, w t( ) , (29) gives the following theoretical

option values V(t)9:

Theoretical Value of European Passport OptionS t r T t ( ) $100; ; . ;= = = = = 0 0 3 1

w(t) V(t) V tt1 ( ) V tt

1( ) Premium

$100 $100.1566 $100.0000 $100.1493 $0.0073

$50 $51.6456 $50.0746 $51.4859 $0.1597

$20 $25.9063 $23.5344 $25.4406 $0.4658

$10 $18.8846 $17.0129 $18.1410 $0.7436

$0 $13.1381 $11.9235 $11.9235 $1.2145

-$10 $8.8808 $8.1410 $7.0129 $0.7398-$20 $5.8876 $5.4406 $3.5344 $0.4470

-$50 $1.5893 $1.4859 $0.0746 $0.1034

-$100 $0.1566 $0.1493 $0.0000 $0.0073

Table 1

In the table above, we have also listed the values of the at-the-money call (V tt1 ( )) and the

at-the-money put (V tt

1( ) ) defined in (24a-b). The "premium" column in the table isdefined as ( )V t MAX V t V t t t( ) ( ), ( ) 1 1 and, as mentioned earlier, can be interpreted as thevalue of the future rights to switch the position in S. As expected, this value is highest

around w t( ) = 0 ; i.e. when there is the most uncertainty about the future sign ofw(t) (andthus, from (23), the sign of the asset position u(t)).

In Figure 1 below we have graphed V, V tt1 ( ) , and V tt

1( ) against w. The peak of the

premium ( )V t MAX V t V t t t( ) ( ), ( ) 1 1 atx = 0 is evident.


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Option Value vs. Current Gain

0

5

10

15

20

25

30

-20 -15 -10 -5 0 5 10 15 20

Gain w (t ) ($)

Value($)

Passport Option

Call

Put

S t r T t ( ) $100; ; . ;= = = = = 0 03 1

Figure 1

We now turn to the application of the finite difference scheme outlined in Section 4.

Using a Crank-Nicholson scheme ( = 12 ), the same process parameters as above, and

w t( ) = 0 , we get the following prices as a function of the grid size:

Crank-Nicholson Numerical Value of European Passport Option

S t r T t w t ( ) $100; ; . ; ; ( ) $0= = = = = = 0 03 1Theoretical Value = $13.1381

Time-steps (N)

x-steps (M) 25 50 100

50 $13.0834 $13.0830 $13.0829

100 $13.1240 $13.1236 $13.1235

200 $13.1349 $13.1345 $13.1343

400 $13.1392 $13.1373 $13.1372

Table 2


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In generating Table 2, we have used odd-even averaging (i.e. the prices reported in the

table are the averages of the prices computed with Mand M+1 x-steps). The CPU time

whenM=400 andN=100 was 0.85 seconds (166 MHz Pentium).

We see from the table above that i) the numerical scheme converges to the theoretical

value, and ii) the number ofx-steps is clearly more important than the number of time-

steps. In Table 3 below, we have fixed the number of time-steps to N=50 and listed

option prices for various values ofw (with odd-even averaging). Again, it is clear that the

numerical scheme converges to the correct values.


S t r T t ( ) $100; ; . ;= = = = = 0 0 3 1N= 50

Gain w(t)

x-steps (M) -$20 -$10 $0 $10 $20

50 $5.9074 $8.9222 $13.0830 $18.9222 $25.9074

100 $5.8900 $8.8745 $13.1236 $18.8745 $25.8900

200 $5.8884 $8.8794 $13.1345 $18.8794 $25.8884

400 $5.8879 $8.8807 $13.1373 $18.8807 $25.8879

Theoretical $5.8876 $8.8808 $13.1381 $18.8846 $25.9063

Table 3

Now, let us turn to an example where r . Specifically let us set10 r = 5% and =

4.5%, and consider a 2-year passport option. With N = 100 and M = 800, the Crank-

Nicholson scheme gives the following results (with odd-even averaging):


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S t r T t ( ) $100; . . ;= = = = =5%; 4 5%; 0 3 2 N= 100;M= 800

w(t) V(t) V tt1 ( ) V tt1( ) Premium

$20 $28.2277 $25.2453 $26.7334 $1.4943

$10 $22.3741 $20.0208 $20.4249 $1.9492

$0 $17.4323 $15.7362 $14.8268 $1.6961

-$10 $13.5100 $12.2859 $10.0631 $1.2241

-$20 $10.4261 $9.5460 $6.2392 $0.8801

Table 4

In the figure below, we have graphed the optimal holding strategy u* as observed in the

finite difference lattice; notice the perhaps surprising presence of three distinctive

regions.

Optimal Holding Strategy vs. Time to Expiry and Current Gain

-50

-25

0

25

50

75

100

125

150

0.0 0.5 1.0 1.5 2.0

Time to Expiry T-t (Years)

Gainw

(t)($)

S t r( ) $100; . .= = = =5%; 45%; 03

u* = 1

u* = 1

u* = 1

Figure 2

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