Rubinstein 1

7/30/2019 Rubinstein 1

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The Quintessential Option Pricing Formula

Max Skipper & Peter Buchen

School of Mathematics and StatisticsUniversity of Sydney, Australia

(Version 3: September 2003)

Abstract

It might be argued that nothing more can be said about pricing options underthe Black-Scholes paradigm. We express the opposite view by presenting inthis paper a new formula that unies much of the existing literature on pric-ing exotic options within the Black-Scholes framework. The formula gives thearbitrage-free price of an M -binary (a generalised multi-asset, multi-periodbinary option), which is a fundamental building block for more complex ex-otic options. To demonstrate the utility of the formula, we apply it to pricingseveral well known exotics and also to a new option: a discretely monitored

call barrier option on the maximum of several assets.

Keywords : Exotic options, binaries, digitals, static replication.

1 Introduction

Any option or derivative that is not a plain vanilla call or put is generallyreferred to as an exotic option . One class of single asset exotics are those with

path-dependent payoffs. Examples include: Asian options, barrier options,lookback options, multi-period digitals, compound options, chooser optionsand many others. Multi-asset exotics, sometimes called rainbow options havealso become popular in the last couple of decades. Examples of these include:exchange options, basket options, min/max and best/worst options. Closed

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form price solutions have been found for most of the above 1 in the Black-

Scholes framework. While the following is by no means an exhaustive list, itnevertheless shows the wide range of published option price solutions withinthe Black-Scholes framework. Merton (1973), Reiner and Rubinstein (1991),Rich (1994),Heynen and Kat (1994), Buchen (2001) analyse barrier options,Geske (1979) and Rubinstein (1992) provide pricing formulae for compoundoptions; Margrabe (1978) and Carr (1988) price exchange options; Stulz(1982), Johnson (1987), Rich and Chance (1993) price options on the max-imum and minimum of several assets and their variants; Kemna and Vorst

(1990) price Geometric Asian options; Longstaff (1990) prices extendable op-tions; Conze and Viswanathan (1991) analyse lookback options; Rubinsteinand Reiner (1991) and Heynen and Kat (1996) price single asset and two-asset binaries; Rubinstein (1991) introduced and analysed chooser options;Zhang (1995) prices correlated digital options; while Buchen (2003) analysesthe class of dual-expiry exotics.

Pricing for non Black-Scholes dynamics (including stochastic volatility) willgenerally require numerical schemes such as Monte Carlo simulation ( e.g.Boyle 1977). Since most exotic options these days do require a stochas-tic volatility model, one might ask the point of continuing to price in theBlack-Scholes world. Apart from its intrinsic and academic value, one prac-tical application of the formula presented in this paper is that it can provideMonte Carlo schemes with good control variates, used for variance reduction.

It might seem implausible that many of the above options can be priced bya single universal formula. We present in this paper precisely such a formulawithin the Black-Scholes framework. In particular we derive the arbirage-freeprice of a generalised multi-asset, multi-period exotic binary option 2. We

1 Arithmetic Asian and basket options are the exception.2 Binary options are also called digital options.

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shall henceforth refer to these fundamental derivatives as M

binaries (see

Eq(16) for a formal denition).

We demonstrate that M binaries are building blocks for a wide class of ex-otic options. In this regard our approach is similar to that of Ingersoll (2000)who showed how digital options can be used to price more complex optionsincluding two-asset exotics. While Ingersoll derived several expressions fordigitals under varying exercise conditions, we adopt a very different approachin this paper. We are formally concerned with pricing only a single option:

the M -binary , which includes cash digitals, asset digitals and all possiblepower (or turbo) digitals. Many exotic options can be expressed as sim-ple static portfolios of these digitals. It follows from the principle of staticreplication that the arbitrage-free price of these exotics must then be thecorresponding value of their replicating portfolios.

While the formula we derive can price multi-variate, discretely monitoredbarrier and lookback options, certain enhancements to the theory presentedhere must be made to price continuously monitored versions of these options.Since these enhancements are not trivial we do not consider them in this pa-per. However preliminary results can be found in Skipper (2003).

The remainder of the paper is organised as follows. In Section 2 we denethe multi-asset, multi-period framework and because the formulation is muchmore involved than usual, we give considerable attention to establishing adescriptive and unifying notation. We develop the multi-asset, multi-periodprice dynamics in Section 3 and state the Main Theorem and universal for-mula in Section 4. In Section 5 we illustrate the use of the formula throughseveral examples including: asset and bond type binaries, compound options,strike reset options, geometric mean Asian options, quality options 3 includ-

3 We coin the term quality options to include any option whose payoff depends on the

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ing best and worst options, options on the maximum and minimum of several

assets, discretely monitored lookback and barrier options. Section 6 offersa short conclusion and the Appendix gives details of the proofs of the twoTheorems stated in the main text.

2 Set-up, Denitions and Notation

1. Let

I n = The index set

{1, 2, . . . , n

}V n = Any ndimensional column vectorAmn = Any m n matrixDn = Any n n diagonal matrix

Cn , C

n = Any n n covariance/correlation matrixGn (R) = Any zero-mean Gaussian n vector

with positive denite correlation matrix RC

n .

2. The asset parameter set A is the set of (constant) parameters:

A = [r, x i , q i , i , ij ]; (i, j I N ) (1)that underlie the N asset price dynamics. In the above: r is the risk freeinterest rate, x i , q i , i are the present value, dividend yield and volatility of asset i and ij is the correlation coefficient of the instantaneous log-returnsof asset i with asset j .

3. The tenor set T for an M -binary is the set of times:

T = [t, T 1, T 2, , T M , T ] (2)maximum or minimum price of a given set of assets.

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all monitoring times only that each asset involves at least one of these times.

We assemble all the relevant components X ik into a payoff vector X . The ar-rangement of these components is arbitrary, but each component correspondsto a unique pair ( i, k ) I N I M . It should be clear that for each indexi I N , k may assume any subset of values in I M ; similarly for each indexk I M , i may take a corresponding subset of values in I N . Because it ismore informative, rather than re-label the components with a new index, weretain both indices ( i, k ) and call such vectors: dual-index vectors. Clearly,

dual-index vectors (and matrices) become single-index when there is only oneunderlying asset, or when there is only one future monitoring time. We referto the dimension n of X as the payoff dimension . In general it differs fromboth N (the number of assets) and M (the number of time periods) and maytake any value in the range N n NM .

Example 1To illustrate the type of payoff structures we shall be pricing, consider aderivative whose expiry T payoff is V

T = f [X

1(T

1), X

1(T

2), X

2(T

1), X

3(T

2)]

for some function f , with T 1 < T 2 T . This derivative is multi-asset,multi-period in the sense we have dened above with N = 3 assets, M =2 periods and payoff dimension n = 4. One representation of the dual-index payoff vector (any ordering of the components is permissible) is X =(X 11 , X 12 , X 21, X 32) . The payoff function for this example would then bewritten as V T (X ) = f (X ).

It should be clear that multi-asset, multi-period binaries are very complex

derivatives requiring many parameters to dene them in full. We thereforeadopt a notation that is designed to both simplify mathematical formulaeand aid understanding. It is evident from the preceding discussion, that thisnotation is vector and matrix oriented. As described below, we also employsome deliberate abuse of standard matrix rules to enhance the clarity of the

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exposition.

If x V n is a vector (we use the notation x to represent its trans-pose) then we shall freely use expressions like ex , log x , x 2, x , . . .to denote similar vectors obtained by component-wise function eval-uation. Thus log x for example, denotes the vector with components(log x1, log x2, . . . , log xn ) .

We often ignore the difference between a vector (or matrix) and itscomponents. Hence we shall feel free to write x = x i or X = X ik for

single and dual-index vectors and R = ij or R = ij kl for matrices.The second matrix here is dened in Eq(11) and is a dual-index matrixwith components drawn from the asset-time index pairs ( i, k ) and ( j, l ).

Let x V n be a column vector of dimension n and let AAmn be anm n matrix. Then dene the mdimensional vector x AV m by 5

x A = exp( A log x ). (5)

The matrix A is the exercise condition matrix referred to earlier. A spe-cial case of Eq(5) is the choice A = where V n is an ndimensionalcolumn vector. The result is the scalar quantity

x = e log x = x 11 x 22 . . . x

nn . (6)

The parameter is the payoff index vector referred to in Eq(4).

We let 1 and 0 denote column vectors with every component equal to 1and 0 respectively. Also 1 j denotes the column vector with j

component

equal to 1 and all others equal to zero, and I n denotes the ndimensionalidentity matrix.

5 We also allow A to be n m , in which case x A = exp( A log x ) for compatibility.

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We dene the m

dimensional indicator function by

I m (x > a ) =m

i=1

I (xi >a i ). (7)

Since I (x a) it is a simple matter to change less-thanindicators into greater-than indicators as follows. Let S Dm bea diagonal matrix with all diagonal components equal to 1. Thenthe indicator I m (S x > S a ) has icomponent I (x i > a i ) if S ii = 1,and has icomponent I (x i < a i ) if S ii = 1. We shall say that anoption has exercise dimension m, if its expiry payoff depends on somemdimensional indicator function. In practice, for any M -binary werequire 1 m n where n is the payoff dimension. Matrix S is theexercise indicator matrix of Eq(4).

Option prices in the multi-variate Black-Scholes framework often in-volve the multi-variate normal distribution function. This is denotedhere by N m (d ; C ), where d V m is an mdimensional column vec-tor and C Cm a positive denite mdimensional correlation matrix.Thus if Z

G(C ) is an mdimensional Gaussian random vector withzero mean, unit variances and correlation matrix C , i.e. Z N (0, C ),then

N m (d ; C )def = E {I m (Z < d )}. (8)

Special cases of Eq(8) are the uni-variate ( m = 1) and bi-variate(m = 2) normal distribution functions. These will be written in theirmore usual notations N (d) and N (d1, d2; ) respectively, where is acorrelation coefficient.

Of course a correlation matrix is just a normalised covariance ma-trix. In particular, if C is a positive denite covariance matrix withD 2 = diag( C ) containing the variances on the main diagonal, then thecorresponding correlation matrix is C = D 1CD 1.

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3 Multi-Variate Asset Dynamics

We adopt the standard Black-Scholes framework extended to multiple assetswith parameters dened by Eq(1). Thus the asset price process X i (s) fori I N and s(t, T ] dened in the previous Section, satises the risk-neutralstochastic differential equation (SDE) for all s > t

dX i (s) = X i (s)[(r q i )ds + idW i (s)]; X i (t) = xi (9)where W i (s) are correlated Brownian motions with E {dW i (s)dW j (s)} =ij ds. Using Itos Lemma and the stationarity of W i (s), this SDE has asolution that can be expressed as

log X i (s)d= log xi + ( r q i 12 2i )(s t) + i W i (s t) (10)

It follows from this representation and the property E {W i ( k )W i ( l)} =min( k , l) that log X i (T k ) is Gaussian with correlation coefficients deter-mined by

R = corr {log X ik , log X jl }= ij kl ; kl =min( k , l)

k l (11)where k = T k t.

Further, since the W i ( k ) are Gaussian with zero mean and variance equal to k , we express Eq(10) at the monitoring times s = T k in the equivalent form

log X i (T k )d= log xi + ( r q i 122i ) k + i k Z i ( k ). (12)

Z i ( k ) Gn (R) are Gaussian random variables with (dual-index) correla-tion matrix R =

ij

kl. The

ijare correlation coefficients of instantaneous

log-returns across different assets at the same time; the kl are correlationcoefficients across different times for the same asset. In matrix form, Eq(12)has the representation

log X d= log x + + Z (13)

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where x is the present value of X (evaluated at k = 0 for all k

I M ),

= ( r q i 12 2i ) k V n (14) = diag( i k ) Dn (15)

and Z Gn (R) is a Gaussian vector with correlation matrix R dened byEq(11).

4 The Main Result

In this Section we give a precise denition of the generalised multi-asset,multi-period M -binary in terms of its expiry payoff V T (X ) and state itsarbitrage-free present value V (x , t ) assuming the multi-variate Black-Scholesrepresentation of the payoff vector X as stated in Eqs(12)-(15). This is themain result of the paper. The proof, which depends on three well knownLemmas is relegated to the Appendix and the notation used is that of Sec-tion 2 and Section 3.

Denition (M -binary )An M -binary with payoff parameter set P , dimension set D , tenor set T andpayoff vector X V n is a multi-asset, multi-period binary option with expiryT payoff function

V T (X ) = X I m (S X A >S a ) (16)

Theorem 1 (Main Theorem)The arbitrage-free present value of an M -binary as dened above, under log-

normal asset price dynamics with asset parameter set A is given by

V (x , t ) = x N m (S d ; SCS ) (17)

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where

= exp r + + 12 ; = T t (18)d = D 1 log x A / a + A( + ) V m (19)C = D 1(AA )D 1 C

m (20)

and

= R Cn (21)D 2 = diag( AA )

Dm (22)

Proof : See the Appendix.

The payoff function in Eq(16) is very general. It includes cash digitals (when = 0 giving X = 1), asset digitals (when = 1 jk resulting in X = X jk )and general power (or turbo) payoffs otherwise. These payoffs are contingenton a wide selection of possible exercise scenarios determined principally bythe exercise condition matrix A. The importance of the matrix A is illus-trated in the example below and the many applications that follow. Theterm log x A / a in Eq(19) is simply a shorthand notation for the vector(A log x log a ).

Example 2As an example of an M -binary consider the derivative with expiry payoff

V T (X ) = X 1X 2 I ( X 1X 3 >X 2)I (X 4


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It is also worth noting that calculations are considerably reduced if A = I nis an ndimensional identity matrix. This leads to the direct equivalencesD = and C = R. In such cases the exercise conditions are simple: i.e.exercise is determined by whether asset prices at certain monitoring timesare above or below a corresponding exercise price. We shall meet severalexamples (as in Example 2 above) where A is not an identity matrix and thecorresponding exercise condition is non-simple.

5 Applications

In order to apply Theorem 1 to price a multi-asset, multi-period exotic optionit is necessary to carry out some preparatory steps, as described below.

1. Static ReplicationThe rst step is to assign the parameter set A for the given asset dynam-ics and construct the payoff vector X . In all the ensuing applicationswe shall assume that A has a structure of the form of Eq(1) and forsimplicity we set all dividend yields q i = 0. Then represent the optionpayoff at expiry T as a portfolio (or linear combination) of M binarieseach of the algebraic form of Eq(16).

2. Specify Binary InputsFor each such M -binary obtain the dimension, tenor and payoff param-eter sets: D , T and P dened in Section 2. Many of these quantities willbe dual-index vectors and matrices based on the structure of the payoff vector X = X ik . Recall that it doesnt matter how the componentsare assembled into X , as long as all similar quantities are assembledthe same way.

The asset parameter set A determines the additional quantities [ , , ]dened respectively by Eqs(14), (15) and (21). Similarly, the tenor set

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T allows calculation of the time dependent correlation coefficients kldened by Eq(11).

3. Compute the OutputsThe third step is to compute the output parameters [ , D, d , C ] fromEqs (18), (22), (19) and (20) and obtain the price V (x , t ) from Eq(17).Finally, the exotic option price will be equal to the combined price of the replicating portfolio of these M binaries .

We basically adopt this three step procedure in all the applications that

follow.

5.1 Asset and Bond Binaries

Consider rst a class of generalised asset and bond type binary options 6 withsimple exercise conditions. These have often be used as building blocks formore complex exotics. Assume the tenor and dimension structures of Eqs(2)and (3) with T = T M .

On the expiry date T , the bond binary pays one unit of cash provided theprices X = X ik = X i (T k ), are above (or below) corresponding exercise pricesa = a ik . The payoff and exercise dimensions are equal and assumed to havevalue n, not necessarily equal to NM . For these binaries, two of the fourpayoff parameters are xed: = 0 and A = I n . We therefore use the symbolB(x , t ; S, a ) to refer to this bond binary.

The asset binary, similarly referred to by the symbol A j (x , t ; S, a ), is identical

to the above except that it pays one unit of asset- j at the expiry date T . Itspayoff differs only in the value of the parameter = 1 jM .

6 These are also known as asset-or-nothing and cash-or-nothing options.

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Calculations for the bond-type M -binary :Inputs :

V T (X ) = B(X , T ; S, a ) = I n (S X >S a )D = [N, M, n, n ]P = [0, a , S, I n ]

[ , , ] = [(r 12 2i ) k , diag(i k ), ij i j min( k , l)]where i, j I N , k , l I M and k = T k t.

Outputs :

= e r ; = T tD =

d = [log(x i /a ik ) + ( r 12 2i ) k ]/ i k = dik (say)C = R = ij kl

Hence, the present value of this M -binary given by Eq(17), is

B(x , t ; S, a ) = e

r N n (Sd ik ; SRS ). (23)For the corresponding asset binary, the payoff is

V T = A j (X , T ; S, a ) = X jM I n (S X >S a )

We assume that X jM is also one of the components of the exercise condition;it is a simple matter to handle the alternative case, by making an appropriatechange in the exercise condition matrix A. This has payoff index vector = 1 jM , otherwise all other input parameters are as for the bond binary

above. A similar calculation then leads to the expression

A j (x , t ; S, a ) = x j N n (Sd ik ; SRS ) (24)where dik = dik + ij j k .

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(a) Multi-asset, Single-period bond and asset binariesWhen there is only a single period (M = 1) we may drop all the time indicesk, l and the above formulae reduce to

B(x , t ; S, a ) = e r N n (Sd i ; SRS ) (25)A j (x , t ; S, a ) = x j N n (S [di + ij j ]; SRS ) (26)

where = ( T t), di = [log(xi /a i ) + ( r 12 2i ) ]/ i and R = ij .

(b) Single-asset, multi-period bond and asset binariesWhen there is only a single asset (N = 1) we may drop all the asset indicesi, j to get

B(x, t ; S, a ) = e r N n (Sd k ; SRS ) (27)A(x, t ; S, a ) = x N n (Sd k ; SRS ) (28)

with [dk , dk ] = [log(x/a k ) + ( r 12 2) k ]/ k and R = kl .

(c) Single-asset, single-period bond and asset binariesIn the simplest case where N = M = 1 we reduce to 1-asset, 1-period binarieswhich are the well known Black-Scholes components of standard Europeancall and put options:

A(x, t ; s, a ) = x N (sd) and B(x, t ; a) = e r N (sd ) (29)

respectively with s = 1 and [d, d ] = [log(x/a ) + ( r 12 2) ]/ .

(d) Compound Options

The bond and asset type binaries derived above can be used to price relatedderivatives. We illustrate this idea with a standard call-on-call compoundoption. Let c denote the strike price of the compound option exercised atT 1, and a2 the strike price of the underlying call option exercised at T 2 > T 1.Let x = a1 denote the solution of: C (x, T 2 T 1; a2) = c where the left side

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of this equation denotes the standard Black-Scholes price of a strike a2 call

option with time ( T 2 T 1) remaining to expiry. The compound option willbe exercised at T 1 if and only if X 1 > a 1. Hence the nal payoff of thecompound option at time T = T 2 can be written in the form 7

V (x, T 2) = [(X 2 a2)+ cer ( 2 1 )] I (X 1 >a 1)

= ( X 2 a2)I (X 1 >a 1)I (X 2 >a 2) cer ( 2 1 ) I (X 1 >a 1). (30)

This is a portfolio of asset and bond type binaries considered in parts (b)and (c) above with m = 1 and m = 2. We can therefore write down the

present value of this portfolio in terms of uni-variate and bi-variate normals:

V (x, t ) = x N (d1, d2; ) a2e r 2 N (d1, d2; ) ce

r 1 N (d1) (31)where = 12 = 1/ 2 and the notation is that of Eqs(27) and (28). Thisformula is the well-known result rst derived by Geske (1979).(e) Multiple strike reset optionsThese options have recently been priced in the Black-Scholes framework by

Liao and Wang (2003). A reset call option involves a single-asset and amulti-period payoff of the form V T = [X (T ) K ]+ where

K =k0 if X min > a 1ki if a i+1 < X min a i ; i = 1 , 2, , ( p1)k p if X min a p

(32)

X min = min( X 1, X 2, , X n 1) and X i = X (T i ) are the monitored assetprices at n 1 pre-determined reset dates T i with T 1 < T 2 < < T n 1 < T .The ki are the reset strike prices and the a i are a decreasing set of discrete

ladder prices. It follows that

V T = p

i=0

(X T ki )+ I (a i+1 < X min a i )7 The exponential multiplying c is the future value factor at time T 2 for a cash payment

at time T 1 .

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with a0 =

and a p+1 = 0. Let T n = T and dene X as the n

dimensional

payoff vector with components X i , i I n . Then it is a relatively straight

forward matter to express this payoff as a portfolio of single-asset, multi-period asset and bond type binaries dened in part(b) above. This is readilyachieved using the identity

I (a i+1 < X min a i )I (X T >k ) = I n (X > a i ) I n (X > a i ) (33)where a i = ( a i , a i , . . . , a i , ki ) and a i = ( a i , a i , . . . , a i , ki 1) .

Eqs(27) and (28) then give the price of the multiple reset call option as thepresent value of this portfolio. We do not write the explicit formula downbut remark that the representation we get appears to be considerably simplerthan the one stated by Liao and Wang (2003).

5.2 Discrete Geometric Mean Asian Options

Prices for these options in the Black-Scholes framework are also well known,for both discrete and continuous time monitoring. We shall derive here twoformulas for discretely monitored geometric mean Asian call options (one axed strike, the other a oating strike). These examples provide strong evi-dence of the utility of the fundamental formula for pricing exotics. Further-more, they illustrate perhaps the simplest cases where the exercise conditionmatrix A is not an identity matrix.

While it is common to have the monitoring times equally spaced, this is byno means essential. We therefore assume there is a single asset and that the

n monitored time periods T k , (1 k n) are arbitrarily spaced. Geomet-ric mean Asian options have payoffs at time T T n which depend on thegeometric mean

Gn = n X 1X 2 X n (34)

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where X k = X (T k ) is the observed asset price at time T k . Since t < T 1 in

our set-up, the rst monitored price has yet to be observed. It is not difficultto include monitored prices prior to t, and this leads to only slightly morecomplicated expressions.

(a) Fixed Strike CallThe payoff for a xed strike k, geometric mean Asian call option is given byV T (X ) = ( Gn k)+ . This expression can be decomposed into the differenceof two M binaries : V 1 = Gn I (Gn >k ) and V 2 = kI (Gn >k ). We investigateV 1 in detail.

Inputs :

V 1(X , T ) = Gn I (Gn >k )D = [1,n,n, 1]

P = [1n

1, k, 1,1n

1 ]

[ , , ] = [(r 12 2) k , diag( k ), 2 min( k , l)]; k = T k t

Outputs :

= e r ( )12

2 ( ) ; =1n

k

k ; =1

n2k,l

min( k , l)

D = d = d + ; d = [log(x/k ) + ( r 12 2) ]/

C = 1

V 1(x, t ) = x N (d).

The calculation for the second term V 2(x, t ) is very similar, with now beingreplaced by a zero vector. This results in V 2(x, t ) = ke

r N (d ). The nal18


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expression for the xed strike, geometric mean Asian call option price is

therefore:V (x, t ) = xe r ( )

12

2 ( ) N (d) ke r N (d ). (35)

(b) Floating Strike Call 8

The oating strike geometric mean Asian call option has an expiry T T npayoff given by V T = ( X n Gn )+ . Hence the geometric mean acts as thestrike price of an otherwise standard call option. This too can be decomposedinto two M binaries :

V T = X n I (Gn X 1n < 1) Gn I (Gn X

1n < 1).

Of particular interest here are the associated expressions for the parametersS, A and a so that the exercise condition is equivalent to: S X A > S a . Wealso nd: S = 1, A = ( 1n 1 1n ) , a = 1 and x = x1. With these pa-rameters it is evident that the term log( x A / a ) vanishes identically. Straightforward calculations then lead to the expression:

V (x, t ) = x N (d) e r ( n )+ 12

2 ( )

N (d ) (36)where

d = ( r + 12 2)( n )/ s; d = [( r 12 2)( n ) + 2( )]/ s

and s = ( n + 2 ).

5.3 Quality Options

We dene a quality option as any derivative whose payoff depends on themaximum or minimum of multiple asset prices at a xed expiry T . Assumethere are n correlated assets and a single period T coinciding with the expirydate of the option. In particular, a quality binary option Qs p(x , t ; k) pays one

8 This particular derivative does not seem to have been priced in the literature.

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unit of asset p (say), with p

I n , provided this is the maximum (if s = 1),

or the minimum (if s = 1), of all the asset values at time T and a givenamount of cash k. In order to price these binaries Theorem 2 is very useful,as it shows how to construct the required exercise condition matrix A andassociated exercise price vector a .

Theorem 2Let X V n and let k be a positive scalar. Then for any integer p I n , thestatement X p = max / min(X , k) is equivalent to sX A p > s a p where s = +1

corresponds to the maximum, s = 1 to the minimum and A pAnn , a pV n have components

Aij =1 if j = p

1 if i = j = p0 otherwisea i =

k if i = p1 if i = p i, j I n (37)

The proof of this result is also relegated to the Appendix.

Theorem 2 allows us to write the quality binary payoff in the form of a simpleM -binary :

Qs p(X , T ; k) = X p I n (sX A p >s a p). (38)

The most interesting part of the calculations is that for the correlation matrix.With = ij i j , C p = D 1 p (A pA p)D

1 p and D 2 p = diag( A pA p) we obtain

the symmetric correlation matrix C p with components

C ij =1 for i = j

p ip i ip for i = j = p

2ip + 2jp

2ij2 ip jp for i = j = p

(39)

and 2ij =

2i +

2 j 2ij i j for all i, j .

Then with the choices = 1 p and S = sI n , the general binary option formula(17), for any t < T , results in the expression

Qs p(x , t ; k) = x p N n (sd p; C p) (40)20


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where d p is the vector with components

di =log(x p/k ) + ( r + 12

2 p) / p for i = p

log(x p/x i ) + 12 2ip / ip for i = p.

(41)

(a) Best and Worst OptionsTwo well known quality options are the best and worst options on n assetsand k units of cash, with expiry payoff:

Qs (X , T ; k) = max( X , k) if s = 1min(X , k) if s =

1. (42)

Each of these payoffs can also be written as a portfolio of quality binariesand a bond type M -binary (considered in Section 5.1):

Qs (X , T ; k) =n

p=1

X pI n (sX A p >s a p) + kI n (sX


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Hence for any t < T we can write down from Eq(44) the pricing formula

C max (x , t ; k) =n

p=1

Q+ p (x , t ; k) + kB (x , t ;I n , k1) ke r . (46)

Similarly, the corresponding results for a put option on the minimum are:

P min (X , T ; k) = [k min( X )]+ = k min( X , k) (47)and

P min (x , t ; k) = ke r

n

p=1

Q p (x , t ; k) kB (x , t ; I n , k1). (48)These expressions of course agree with previously published formulations of the problem, albeit in a very different notation, and obtained by a very dif-ferent method.

Formulae for a call option on the minimum and a put option on the maximumare easily obtained by a similar construction, so we are content to skip thedetails.

5.4 Discrete lookback options

Suppose there is a single asset, whose price X i = X (T i ) is monitored atthe increasing sequence of times T i for i I n . Derivatives whose payoffs atexpiry T T n depend on the maximum or minimum value of X i are calleddiscrete lookback options . It is clear that these options are the single-asset,multi-period counterparts of the multi-asset, single-period quality optionsconsidered in the previous sub-section. We shall consider here the case of

a binary option that pays X p provided X p = max( X , k) where k is a xedamount of cash 9. The payoff for this lookback binary is then

L(x, T ; k) = X p I n (X A p > a p) (49)9 The xed cash k can also be thought of as the currently ( i.e. time t) observed maxi-

mum asset price.

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where A p, a p are dened in Theorem 2 with indices referring to times rather

than assets.

Since the analysis using Eq(17) of the Main Theorem is very similar to thatfor the quality binary, we leave out the details and simply state the result.The main difference occurs in the covariance matrix which must now bereplaced by = 2 min( i , j ). We ultimately obtain the result:

L(x, t ; k) = xe r ( p ) N n (d p; C p) (50)where d p is the vector with components

di =log(x/k ) + ( r + 12

2) p / p for i = p1 (r +

12

2) p i for i < p1 (r 12 2) i p for i > p

(51)

and C p is the symmetric correlation matrix with components

C ij =

1 for i = j(1 i / p)

12 for i < j = p

[( i p)/ ( j p)]1

2 for p < i < j[( p j )/ ( p i )]

12 for i < j < p

0 for i < p < j

(52)

Related lookback options, such as xed strike lookback calls and puts cannow be priced in terms of portfolios of these lookback binaries.

5.5 Discrete Barrier Options

As a nal illustration of our fundamental pricing formula Eq(17) we consider

a multi-asset, multi-period down-and-out discrete barrier call option. At ex-piry T the strike a option is assumed to pay [max {X i (T )} a]+ providedX ik > bi for a given set of discrete barrier prices bi . Thus if any of the assetprices X ik = X i (T k ) (i I N , k I M as usual) falls below the barrier level,the contract is immediately knocked out; if X ik stays above the barrier level

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for all monitoring times T k the contract pays a call option on the maximum

asset price recorded at the expiry date T > T M .

This option is certainly not a simple one and its pricing would, under stan-dard methods of analysis, present a great challenge. The fundamental for-mula derived in this paper allows its calculation in just a few steps.

First we write its expiry payoff in the form

V (X , T ; a, b) = [max

{X i (T )

} a]+ I m (X i (T k ) >b i ) (53)

where m is the barrier exercise dimension. The Main Theorem tells us thatthe only task is to write this expression as the sum of M binaries and iden-tify the associated parameters.

In order to do this, dene the payoff vector X as a collection of the masset prices X b = X i (T k ) used for barrier monitoring and the N pricesX T = X i (T ) that determine the call option payoff. The partitioned pay-off vector X = [X b, X T ] has dimension n = ( m + N ).

Using the identity (max X T a)+ max( X T , a) a and the results of Section 5.3 on pricing quality options, we can write the payoff (53) in therequired form as

V (X , T ; a, b) =N

p=1

X

p I n (X A

p > a p) + a I n (S X >S c ) aI m (X b > b)

(54)where

p =01 p

; A p =I m 00 A p

; a p =b

a p

S = I m 00 I N ; c = ba1

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and b = bik = bi for all k. The rst term in Eq(54) is the sum of generalM binaries considered in the Main Theorem, while the second and thirdterms are bond type M binaries considered in Section 5.1. The presentvalue of this payoff can therefore be obtained by employing the principle of static replication and results in the expression

V (x , t ; a, b) =N

p=1

x p N n (d

p; C

p ) + aB (x , t ; S , c ) aB (x , t ; I m , b) (55)

where (d p, C

p ) are determined from Theorem 1 with ( A, a ) replaced by

(A

p, a

p); and the bond type B terms are given explicitly by Eq(23). Littlewould be served by expanding the details of this representation any further,so we are content to leave the solution in the stated form of Eq(55).

6 Conclusion

Theorem 1 stated in Section 4 is the main result of this paper. It providesa versatile formula for pricing a very general multi-asset, multi-period exoticbinary option (our M -binary ) which we have demonstrated is a fundamen-tal component of many of the exotics published in the literature. The basicidea, very much in the spirit of Ingersoll (2000), is to decompose an optionsexpiry payoff into portfolios of M binaries and invoke the principle of staticreplication to obtain an arbitrage-free price for the present value of the op-tion. Each M -binary expiry payoff has the very specic 4-parameter form:X I m (S X A > S a ). These parameters, [ , A, a , S ] together with the dualasset-time correlation matrix R of Eq(11) and monitoring time structure T k ,provide sufficient scope to match a great variety of exotic option payoffs.

The exercise condition matrix A plays a particularly important role. If itis different from an identity matrix, it is responsible for generating changesin the correlation structure, which in turn has an important bearing on theeventual pricing formula. An interesting observation is that the dimensionof the multi-variate normal for each component binary is determined only by

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the exercise dimension m, and not by the payoff dimension n or the number

of assets N or time periods M .

The pricing formula itself Eq(17), is quite complex and may in certain cir-cumstances involve considerable calculation. However, although we haveconcentrated in this paper on analytical solutions, the formula can readilybe coded up for numerical computation, particularly with matrix orientedprogramming languages such as Matlab and Mathematica.

To illustrate the power and versatility of the formula, we have priced manywell known exotic options and also a new one of very great complexity:namely, a discretely monitored call barrier option on the maximum of N assets. This option includes both a multi-asset payoff and multi-period mon-itoring.

Extensions of the general formula Eq(17) are underway to allow similar pric-ing of continuously monitored multi-asset barrier and lookback options.

7 References

1. Boyle P 1977 Options: a Monte Carlo approach Journal of Financial Economics 4 323-338

2. Buchen P 2001 Image options and the road to barriers Risk Magazine 14(9) 127-130

3. Buchen P 2003 The pricing of dual expiry options, Journal of Quanti-tative Finance , IOP (to appear)

4. Carr P 1988 The valuation of sequential exchange opportunities Journal of Finance , 43 1235-1256

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5. Conze A and Viswanathan 1991 Path dependent options: the case of

lookback options Journal of Finance 46 1893-1907

6. Geske R 1979 The valuation of compound options Journal of Financial Economics 7 63-81

7. Harrison J and Pliska S 1981 Martingales and stochastic integrals inthe theory of continuous trading Stochastic Processes and their Appli-cations 11 215-260

8. Heynen R and Kat H 1994 Partial barrier options, Journal of Financial Engineering 3 253-274

9. Heynen R and Kat H 1996 Brick by brick Risk Magazine 9(6)

10. Ingersoll J 2000 Digital contracts: simple tools for pricing complexderivatives The Journal of Business 73(1) 67-88

11. Johnson H 1987 Options on the maximum or minimum of several assetsJournal of Financial and Quantitative Analysis 22 277-283

12. Kemna A and Vorst A 1990 A pricing method for options based onaverage asset values Journal of Banking and Finance 14 113-129

13. Liao S-Z and Wang C-W 2003 The valuation of reset options withmultiple strike resets and reset dates Journal of Futures Markets 23(1)87-107

14. Longstaff F 1990 Pricing options with extendable maturities: analysisand applications Journal of Finance 45(3) 935-957

15. Margrabe W 1978 The value of an option to exchange one asset foranother Journal of Finance 33 177-186

16. Merton R 1973 Theory of rational option pricing Bell Journal of Eco-nomics and Management Science 4 141-183

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17. Reiner E and Rubinstein M 1991 Breaking down the barriers Risk Mag-

azine 4(8)

18. Rich D and Chance D 1993 An alternative approach to the pricing of options on multiple assets Journal of Financial Engineering 2 271-285

19. Rich D 1994 The mathematical foundation of barrier option pricingtheory Advances in Futures and Options Research 7 267-311

20. Rubinstein M 1991 Options for the undecided Risk Magazine 4(4) 43

21. Rubinstein M 1991 Double trouble Risk Magazine 5(1)

22. Skipper M 2003 Inside exotic barrier options Sydney University School of Mathematics & Statistics Research Report

23. Stulz R 1982 Options on the minimum or the maximum of two riskyassets Journal of Financial Economics 10 161-185

24. Zhang P 1995 Correlation digital options Journal of Financial Engi-neering 4 75-96

A Appendix

In this Appendix we prove the two Theorems stated in Sections 4 and 5.3.

Theorem 1 (Section 4)The proof of the Main Theorem depends on the three Lemmas listed below.The rst deals with option pricing theory; the second and third deal with

specic properties of Gaussian random vectors.

Lemma 1 In a complete, arbitrage-free, multi-asset market the discountedprice of any derivative contract is a martingale with respect to the

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risk-neutral measure. This leads to the well-known pricing formula

(Harrison and Pliska 1981)

V (x , t ) = e r (T t ) E {V T (X )|X (t) = x }. (A1)Lemma 2 Let cV n be a constant vector, Z Gn (R) a Gaussian random

vector with positive denite correlation matrix R and let F (Z ) be ascalar function of n variables. Then

E {ec Z F (Z )}= e12 c R c E {F (Z + Rc )}. (A2)

We like to term this result the Gaussian Shift Theorem because themean has been shifted from zero to Rc . The proof is fairly elementaryand involves little more than the following identity for multi-variateGaussian pdfs:

(z Rc ) ec z 1

2 c R c (z )

where (z ) = exp( 12 z R 1z ) and K is the usual normalising constant.This Lemma can also be shown to be equivalent to expectation undera change of measure.

Lemma 3 For any matrix B Amn of rank m n, a vector bV m and aGaussian random vector Z Gn (R) with correlation matrix R,

E {I m (B Z < b)}= N m (D 1b; D 1(BRB )D 1) (A3)

where D 2 = diag( BRB ). This Lemma is also easily proved notingthat BZ Gm with covariance matrix BRB . The diagonal matrix Dprovides the normalisation to an mdimensional correlation matrix.

Proof of Main TheoremFrom the payoff function Eq(16) for an M -binary and Lemma 1 we have

V (x , t ) = e r E {X I m (S X A >S a )|X (t) = x }; = T t= e r E {X I m (SA log X >S log a )|X (t) = x }.

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For log-normal asset price dynamics described by Eq(13) we rst obtain

X = x exp{ + ( ) Z }Then using Lemma 2 with c = , the last equation for V (x , t ) can bewritten as

V (x , t ) = x e r + E {ec Z I m (SA(log x + + Z ) >S log a )}= x E {I m (SA( Z + R ) > S [log(x A / a ) + A ])}= x E {I m ((SA) Z s a p

sA p log X > s log a p

s

1 01 I n 1

log X plog X p > s

log k0

slog X p

log X p1 log X p> s log k0

sX p

X p1> s kX p

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The last line demonstrates that X p = max( X , k) if s = 1 and X p =

min(X , k) if s = 1 and Theorem 2 is proved.oOo

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