Analytical valuation of options on joint minima and maxima

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Analytical valuation of options on joint minima and maxima

Tristan Guillaume

To cite this version:

Tristan Guillaume. Analytical valuation of options on joint minima and maxima. Applied Mathemat- ical Finance, Taylor & Francis (Routledge): SSH Titles, 2002, 8 (4), pp.209-235. �hal-00924237�

ANALYTICAL VALUATION OF OPTIONS ON JOINT MINIMA AND MAXIMA

Tristan Guillaume¹

Revised version, 2004, originally published in Applied Mathematical Finance (2002), 8 (4), 209-235

Abstract

In this paper, we show how to obtain explicit formulae for a variety of popular path- dependent contracts with complex payoffs involving joint distributions of several extrema.

More specifically, we give formulae for standard step-up and step-down barrier options, as well as partial and outside step-up and step-down barrier options, involving multiple integrals of dimensions ranging between three and five. Our method can be extended to other exotic path- dependent payoffs as well as to higher dimensions. Numerical results show that the quasi random integration of these formulae involving multivariate distributions of correlated Gaussian random variables provides option values more quickly and more accurately than Monte Carlo simulation.

Keywords : option, barrier, step barrier, pricing, dimension

1 Université de Cergy-Pontoise, Laboratoire Thema, 33 boulevard du port, F-95011 Cergy-Pontoise

Introduction

Barrier options are one of the oldest types of options. They are also one of the most popular, due to their low cost and their strong leverage effect, as well as to the precision and the flexibility with which they can adapt to investors’ needs or views about the market. As a result, barrier options are now heavily traded, particularly in the foreign exchange markets. They are also embedded in a lot of popular structured derivatives in equity and interest rate markets.

There has been quite extensive research dealing with the analytical pricing of barrier options. A closed form formula for a down-and-out European call option was published by Merton as early as 1973 (Merton, 1973). An exhaustive list of formulae for standard barrier options was published by Rich (1994). In the mid-90’s, contributions started to focus on more exotic types of barrier options, such as partial barrier options (Heynen & Kat, 1995 ; Carr, 1995), outside barrier options (Bermin, 1995), double barrier options (Kunitomo and Ikeda, 1992 ; Bhagavatula and Carr, 1995 ; Geman and Yor, 1996). Heynen and Kat (1995) derive formulae for partial barrier options in a partial-differential-equation framework. Carr (1995) includes American-type rebates. Bermin (1995) develops a decorrelation approach to reduce the dimension of the partial outside barrier option pricing problem. Kunitomo and Ikeda (1992) draw on Levy’s seminal works (1948) to provide a formula for a double barrier option with curved boundaries. Bhagavatula and Carr (1995) deal with double barrier options by means of Fourier series, whereas Geman and Yor (1996) apply Laplace transformation.

So far, there are no available formulae for a special type of barrier option that attracts strong interest among practitioners : step barrier options. These contracts feature sequences of constant one-sided barriers. They are called step barrier options, because barriers are step functions of time. When barriers are monotonically decreasing or increasing, we have a standard step barrier option. There can be gaps during which there is no barrier monitoring. In this case, the step barrier option is said to be partial. When the asset involved in the payoff is not the same as the one involved in barrier monitoring, we have an outside step barrier option. The main

difficulty associated with an analytical valuation of such contracts, in a risk-neutral framework, is that they involve joint distributions of extrema over various time intervals that are currently not explicitly known. Furthermore, the dimension of the pricing problem quickly increases.

Consequently, it is assumed that these contracts have to be numerically priced. When the dimension of the problem is greater than three, Monte Carlo simulation is thought to be the only tractable method, as its convergence rate is independent of dimension, contrary to a grid-based approach.

The purpose of this paper is to show that this valuation problem can be analytically solved, by providing explicit formulae for all kinds of step barrier options, whether they are standard, non- standard, partial or outside contracts, in dimensions that range between three and five, knowing that the method we use can be applied to higher dimensions (sections 1, 2, 3). Then, by using simple low discrepancy sequences to perform quasi random integration, this paper aims at showing that the numerical implementation of these exact formulae is both more efficient and more accurate than alternative advanced or plain Monte Carlo simulation methods (section 4).

1. Analytical framework

It is assumed that the underlying asset follows a geometric brownian motion :

t t t t

dS mS dtsS dW (1)

The parameters _m and _sare constant, and _s is positive. The option life starts at time 0 and that its expiry is T. W_t is Brownian motion defined on a probability space ^{,F Pt,} ^{, where}

 ^, 

t s

F s W st is the natural filtration of W_t.

According to the risk-neutral valuation approach, at the contract inception t0, the no- arbitrage value of a European call option with path-dependent conditions A A₁, ₂,...,A_m is given by the discounted expectation of its payoff under the equivalent martingale measure, conditional on the information available at time t0 :

 , , , , , 1,...,    _{ 1},..., _ 0

m

rT Q

m T A A

e C S T K rs dA A E  S K ^1 S  (2)

where Kis the strike price, ris the riskless rate, _dis a dividend rate,E^Qis the expectation operator under the equivalent martingale measure, Q, and 1_{ .} is the indicator function taking value 1 if the conditions inside the brackets are met and value zero otherwise.

Equivalently, we have :

 , , , , , 1,...,    _ 1,..., , _ 0

m

rT Q

m T A A K

e C S T K r A A E S K S

ST

s d    1 _ 

   

2

1 1

2

0 ,..., , ,..., ,

T

m m

r T W

Q Q

A A K A A K

E S e KE

T T

S S

1 1

d s s

 

 

    

 

 

 

 

 

   

    

   

2

1 1

2 ,..., , ,..., ,

T

m m

T W

Q Q

A A K A A K

FE e KE

T T

S S

1 1

s s

 

 

 

   

 

    

(3)

where FS e₀ ^{ rdT}is the risk-neutral forward price.

Let us define a new measure Qsuch that :

2

2 ^T

T

T W

F

dQ e

dQ

s s

 

 

(4)

Then, according to Girsanov theorem, we have :

 ^{, , , , , 1,...,}  ^.  ^{1,..., ,}  ^.  ^{1,..., ,} 

rT m m T m T

e C S T K rs dA A F Q A A X  k K Q A A X k (5) with : X_T lnS_T/S0,klnK S/ 0

Similarly, we have the following expression for a path-dependent put option :

 ^{, , , , 1,...,}  ^.  ^{1,..., ,}  ^.  ^{1,..., ,} 

rT m m T m T

e P S T K r s d A A K Q A A X  k F Q A A X k

Hence, it suffices to calculate probabilities under the Qmeasure : a simple change of drift from m  r d s²/ 2 to m  r d s²/ 2 will provide the corresponding probabilities under the

Qmeasure.

In general, the greater the number of path dependent conditions, the higher the dimension of the solution will be.

Low-dimensional problems can be solved in terms of univariate or bivariate standard Gaussian distribution functions, ^{N a} ^{and N2}^{a b, ;r}respectively. This is the case for standard barrier and lookback options, as well as for partial or outside barrier options. If we want to find closed form formulae for more complex path-dependent options, we need to use higher-dimensional Gaussian distributions.

In general, if ^{X }^X1,...,Xnis a vector of n joint standardized Gaussian random variables with a symmetric, positive-definite ^{n n}  matrix of variances and covariances _, then the density of

Xis given by :

 

   

1 2

1,..., / 2

2

XT X

n n

f x x e

p Det





  (6)

where X^Tis the transpose of X, ^Det  is the determinant of _and _^1is the inverse of _. For example, if we denote by  ₁₂, ₂₃and _r13the correlation coefficients between three standardized Gaussian random variables X₁,X₂and X₃, it is easily shown, by applying formula (6), that the joint density of X X₁, ₂and X₃ is given by :

 

   

2 2 2

1 2 3 4 5 6

1 2 2 2

2

12 23 13 3/ 2

( , , ; , , )

2

a b c ab bc ac

f a b c e

Det

l l l l l l

r p

     

  

 (7)

with :

       

2 2 2

23 13 12 13 23 12

1 2 3 4

1 1 1

Det Det Det Det

r r r r

l     l     l    l    ^{ l5r13 12Det } ^{23l6  12 23Det} ^r13

(8) and

 

Det  1 12² r²23r13²   212 23 13r (9) However, using these general multinormal expressions becomes analytically cumbersome and computationally inefficient as the dimension of the integral rises. Actually, simplified expressions can be used when dealing with the finite-dimensional distributions of geometric Brownian motion GBM. Let ^Q ^. refer to the equivalent martingale measure, let ^{X ln}^{S /S} ^,

and let T₀, T₁, T₂, T₃ be three different dates during the option life ^{T T0, 3}, such that :

0 1 2 3

T   T T T .

By using the Markov property of Brownian motion, as well as the independence of Brownian increments, one can write :



^{T1 , T2 , T3}



Q X a X b X c =

 

 

^{ }

2 2

1 1 2 2

12

1 1 2 2

2 12

2 2 1

3 2

2 1 2 122 3 2

2 1

x T x T y T y T

T T T T

a b

c T T y

e N dydx

T T T T

m m m m

s s s s

r m

ps r s

          

      

      

      

      

      

      

 



    

 

  

 

  

 

⁽¹⁰⁾

where  ₁₂ T T₁/ ₂ and ^N ^. is the univariate standard Gaussian distribution.

Manipulating this triple integral and standardizing it yields :



^{T1 , T2 , T3}



Q X a X b X  c

 

 

^



^



 

  

2 2

2 12 23

3

1 2

2 2

3

1 2 12 23

2-2 1 2 1

3/2 2 2

12 23

2 1 1

y x z y

c T x

a T b T T T T

e dzdydx

m

m m

s

s s r r

p r r

 



   

 

  

  

  ⁽¹¹⁾

which can be written in a more compact form as :

3

1 2

3 12 23

1 2 3

, ,c T ,

a T b T

N T T T

m

m m

s s s

    

   

 

 

  (12)

where  ₂₃ T₂/T₃

Similarly, ifT₄T₃, we have :



^{1 2 3 4}



3

1 2 4

4 12 23 34

1 2 3 4

, , , ; , ,

, , ,

T T T T

c T

a T b T d T

N

T T T T

Q X a X b X c X d

m

m m m

s s s s

     

    

 

 

 

    

 

 

^



^



^



^



 

   

2 2 2

2 12 23 34

3

1 2 4

2 2 2

3

1 2 4 12 23 34

2-2 1 2 1 2 1

2 2 2 2

12 23 34

2 1 1 1

x w y x z y

c T w

a T b T d T

T

T T T

e dzdydxdw

m

m m m

s

s s s r r r

p r r r

  



     

  

   



   

   ⁽¹³⁾

with :  ₃₄ T₃/T₄

More generally, it is easily shown that given ^{T T T0, 1, 2,...,Td} ^{with : T0   T1 T2 ... Td, and} d, the density of the dcorrelated Gaussian random variables ^{XTi ln}



^{STi /S0}



is given by :

 

   

 

  ^{ }

^{ }

1 2

2 2

2 2 2

1,2 1 1,2 1 2 2 2 , 1 1 , 1

1 2

1 arccos 2 1 arccos

2 2

, 1

, ,...,

exp 2 arcsin exp

d

d

k k k k k k

T T T d

x x x x x x

k k

P X dx X dx X dx

r r r r

p r

  

 

 

    

 

 

 

     

    

             

             

       

  

  

   

  

 

  

1 2 d k

 





(14) where r_{a b,}  T_a/T_b,T_aT_b and the notation  .refers to the first-order derivative of  . Alternatively, the finite-dimensional distribution of order d of arithmetic Brownian motion with drift _m and volatility _s, is given by :



^{T1 1, T2 2,..., Td d}



Q X x X x X x

1 1 2 2

12 23 1,

1 2

, ,..., ^{d d} , ,...,

d d d

d

x T

x T x T

N T T T

m

m m r

s s s ^

    

 

    

 

 

 

    

2 1 1 , 1 2

1

1 1 2 2

1 2

1 2 2 2 1 , 1

1 1

/2 2 2 2

12 23 1,

... ...

2 1 1 ... 1

d k k k k

d d

d k k k

y y

x T y

x T x T T

T T

d d d

d d

e dy dy dy

  

m

m m

s

s s r

p r r r

  

 

 

   



    





  

   ⁽¹⁵⁾

We will now show, in sections 2 and 3 of this paper, how to use these finite-dimensional distributions to value step barrier options.

2. Standard step-up and step-down barrier options

The analytical valuation of a ddimensional standard step-up barrier option involves the calculation of the distribution of dincreasing maxima and terminal value of a geometric Brownian motion. Likewise, valuing a ddimensional standard step-down barrier option involves the calculation of the distribution of ddecreasing minima and terminal value of a geometric Brownian motion.

Let us focus on an up-and-out put option with four different barriers over four intervals of the option life :^{T T0, 1} ^{,T T1, 2} ^{,T T2, 3} ^{,T T3, 4}^{, where T0} is the beginning of the option life and T₄ is the expiry. We name our four barriers : H H H H₁, ₂, ₃, ₄. Since we are dealing with a standard step-up barrier option, we have : H₁H₂H₃H₄(but we shall see in section 3 that we could choose another combination).

To find a formula for the value of this option, we need to know the joint probability, under the equivalent martingale measure, that the maximum value of the underlying asset during

^{T T0, 1}will not hit H₁ and that the maximum value during ^{T T1, 2} will not hit H₂ and that the maximum value during ^{T T2, 3} will not hit H₃ and that the maximum value during ^{T T3, 4} ^will not hit H₄ and that the value of the underlying asset at expiryS₄will be lower than the strike price K. That is, we need to know the value of :



^{01 1, 12 2, 23 3, 34 4, 4}



Q M h M h M h M h X k (16)

where ^{XT ln}^{ST /S0}^,kln^{K S/ 0}^{, hiln}^Hi/S0 ^and

  

, sup

a b

ab t

t T T

M X

 

Graph 1, in appendix 1, gives an illustration of this problem.

The formula for : ^{Q M}



^{10h X1, T1 k}



is a classical result that is assumed to be known (see,e.g., Karatzas & Shreve [1991]).

Using the Markov property of brownian motion and the independence of brownian increments, the second maximum can be dealt with in the following way :



^{01 1, 12 2, T2}



Q M h M h X k =

   

^{   }

1

1 2 1

1 0 1 1 2

1 , 2 ,

T T

h

Q Q

X h M h X k M h T

E 1 _{ } E 1 _{ } X u u du



  

     

  

 



(17) where :

 

    ^{ }  

2 2

1 1 1

1 2 1 1

2 2

1 1 2

2 2 2

1 1

2

2 1 2 2 1

2 1 2 1

2 2

2

u T u h T

T h T

h u

e e

u e

T T

k u T T k h u T T

u N e N

T T T T

m m

s m s

s

m s

s p s p

m m

s s

      

   

 

     



  

          

   

       

(18)

Using identities (A.1), (A.2) and (B.1) in appendix 3, the following solution can be obtained :

 

2 2

1 2 1

2 2

2

1 1 2 1 1 2 2

2 12 2 12

1 2 1 2

2 2

1 1 1 2 1 1 2 1 2

2 12 2 12

1 2 1 2

, , 2

2 2 2

, ,

h

h h h

h T k T h T k h T

N e N

T T T T

h T k h T h T k h h T

e N e N

T T T T

m s

m m

s s

m m m m

s s s s

m m m m

s s s s



        

     

   

   

   

          

   

      

(19)

where  ₁₂ T T₁/ ₂ as in section 2.

The next step is to find : ^{Q M}



^{10h M1, 12h M2, 23h X3, T3k}



. This probability can be formulated as the following conditional expectation :



^{T2 2, 10 1, 12 2}

 

^{T3 , 23 3}



²

Q Q

X h M h M h X k M h

E 1 E 1 X

    

  

  

  

  (20)

Building on our acquired knowledge of ^{Q M}



^{10h M1, 12h X2, T2 k}



, the following integral form of the problem can be written down :



^{T2 2, 01 1, 12 2}

 

^{T3 , 23 3}



²

Q Q

X h M h M h X k M h

E 1 E 1 X

    

  

   

  

 

       

    ^{ }    

1 2 1 2

2 2

1 2 1 2

1 2 1

2 2

2

1 2

2 2

3 4

; ;

; +

h h h h

h

h h h h

h h h

u v v dvdu e u v v dvdu

e u v v dvdu e u v v dvdu

   

   

m s

m m

s s

   

  

   

    

     

   

   

(21)

where :

   ^{3 2} ^{223  3}  ^{3 2}

3 2 3 2

h v 2

k v T T k h v T T

v N e N

T T T T

m

m s m

s s

           

   

        (22)

 

 

 

 

2 2 2 2

2 1 2 2 1

1 1

1 2 1 1 2 1

1 1 2

2 2

1 2

2 2

1 2 1

2 ( )

v u T T v u h T T

u T u T

T T T T T T

e e

u v u v

T T T

m m

m m

s s s s

s p s

   

       

                

   

       

           

      

 2p T T1( 2T1) (23)

 

 

 

 

2 2 2 2

1 2 1 2 1 2 1

1 1

1 2 1 1 2 1

2 2 2

1 1

2 2

3 4

22 1( 2 1)

v u h T T v u h h T T

u T u T

T T T T T T

e e

u v u v

T T T

m m

m m

s s s s

s p

   

       

                  

  

       

           

      

 s p²2 T T1( 2 T1)





Using identities (A.3) and (B.3) in appendix 3, one can obtain the following explicit expression composed of eight terms involving a special form of the trivariate gaussian cumulative distribution :



^{01 1, 12 2, 23 3, T3}



Q M h M h M h X  k

2 2 2

3 2 3

1 1 2 2 1 1 2 2

3 12 23 3 12 23

1 2 3 1 2 3

, ,k T , ^h , ,k 2h T ,

h T h T h T h T

N e N

T T T T T T

m

m s m

m m m m

s s s s s s

           

       

   

   

    (24)

2 1 2

1 3

1 1 2 1 2

3 12 23

1 2 3

2

, 2 , ,

h h T h h T k h T

e N

T T T

m

s m m m

s s s

      

 

    

2 1

2 2

2 1 3

1 1 1 2 2

3 12 23

1 2 3

2 2

,2 , ,

h h h T h h T k h h T

e N

T T T

m

s m m m

s s s

        

    

2 3 2

3 3

1 1 2 2

3 12 23

1 2 3

, , 2 ,

h h T h T k h T

e N

T T T

m

s m m m

s s s

     

 

    

3 2

2 2

3 2 3

1 1 2 2

3 12 23

1 2 3

2 2

, , ,

h h h T h T k h h T

e N

T T T

m

s m m m

s s s

        

    

3 1

2 2

3 1 3

1 1 2 1 2

3 12 23

1 2 3

2 2

, 2 , ,

h h h T h h T k h h T

e N

T T T

m

s m m m

s s s

        

    

3 1 2

2 2

3 2 1 3

1 1 2 1 2

3 12 23

1 2 3

2 2 2

, 2 , ,

h h h h T h h T k h h h T

e N

T T T

m

s m m m

s s s

           

    

where it is recalled that :

 

 

 

^

 

^

 

  

2 2

2 12 23

2 2

12 23

2-2 1 2 1

3 12 23

3/2 2 2

12 23

, , ,

2 1 1

y x z y

x a b c

N a b c e dzdydx

r r

p r r

 

 

 



  

 

  

⁽²⁵⁾

as was shown in section 1. Knowing this probability would be enough to value a three- dimensional standard step up-and-out put option. But, to value a four-dimensional standard step up-and-out put option, we need to add the condition that : M₃⁴h₄, as well as the condition that :

T4

X k. This is done by solving :



^{T3 3, 10 1, 12 2, 32 3}

 

^{T4 , 34 4}



³

Q Q

X h M h M h M h X k M h T

E 1 E 1 X

     

  

  

  

  =