Some sequential boundary crossing results for geometric Brownian motion and their applications in financial engineering

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Some sequential boundary crossing results for geometric Brownian motion and their applications in financial

engineering

Tristan Guillaume

To cite this version:

Tristan Guillaume. Some sequential boundary crossing results for geometric Brownian motion and their applications in financial engineering. International Scholarly Research Notices, Hindawi, 2011, 1, pp.1-21. �hal-00924277�

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Some sequential boundary crossing results for geometric Brownian motion and their applications in financial engineering

(published in ISRN Applied Mathematics, 2011, (1), 1-21)

Author : Tristan Guillaume

Mailing address : Université de Cergy-Pontoise, Laboratoire Thema, 33 boulevard du port, F-95011 Cergy-Pontoise Cedex, France

E-mail address : tristan.guillaume@u-cergy.fr Telephone number : + 33 6 12 22 45 88 Fax number : + 33 1 34 25 62 33

Abstract

This paper provides new explicit results for some boundary crossing distributions in a multi- dimensional geometric Brownian motion framework when the boundary is a piecewise constant function of time. Among their various possible applications, they enable accurate and efficient analytical valuation of a large number of option contracts traded in the financial markets belonging to the classes of barrier and lookback options.

1. Introduction

The joint law of the maximum (or the minimum) of a real-valued Brownian motion and its endpoint over a finite time interval is a central result in the study of Brownian motion, particularly with regard to the many applications of the theory in finance, medical imaging, robotics and biology. It can be obtained as a consequence of the « reflection principle », which derives from the strong Markov property of Brownian motion (Freedman [6]). Application of Girsanov’s theorem easily generalizes this seminal result to the case of a geometric Brownian motion (GBM), a frequently encountered diffusion process that is the building block of financial engineering. Alternatively, the law can be derived by a partial differential equation approach, using Kolmogorov’s equation for the transition density function of a diffusion process. The distribution of the first passage time by a one-dimensional GBM to a one-sided or a two-sided straight boundary then follows. A few cases where the boundary is

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curved have been handled (Barba Escriba [2] ; Salminen [20]; Kunitomo and Ikeda [17]). For a comprehensive source of formulae, one may refer to Borodin and Salminen [3].

Various extensions to these results are often needed to solve practical engineering problems. In particular, one may look for joint distributions of the highest or lowest points hit over several time intervals, and one may deal with a multi-dimensional GBM. There are few known explicit results in these more general settings, partly because of the laborious analytical calculations involved, but also due to numerical obstacles : the rapidly increasing dimension of the boundary crossing problem leads to analytical solutions that are expressed in terms of functions that are hard to compute with accuracy.

As far as one-sided boundaries are concerned, formulae have been published for the joint law of a sequence of maxima or minima of a one-dimensional GBM over several time intervals (Guillaume [9]), as well as for the joint law of the maxima or minima of a two-dimensional GBM over one time interval (Iyengar [19] ; He et al. [14]). A formula for the joint law of the exit times of a one- dimensional GBM from two successive two-sided boundaries is also known (Guillaume [11]).

This paper focuses on a sequence of two one-sided straight boundaries conditional on two correlated GBMs, while the value of a third correlated GBM is taken into account at the endpoint of the time interval. The state space is thus three-dimensional. The choice of this particular distribution is motivated both by its usefulness in financial engineering applications and by the fact that it leads to tractable analytical solutions that can be computed with great accuracy and efficiency.

Section 2 presents the two main formulae of this paper. Section 3 deals with applications of these formulae and discusses their numerical implementation.

2. Main formulae

This section contains the two main formulae of the paper. Let ^{S t t}₁ ^, ^⁰ ^and ^{S t t}₂^{ }^, ^⁰ ^be

two geometric Brownian motions with constant drift coefficients, a₁ and a₂ respectively, under a probability measure , constant positive diffusion coefficients s₁ and s₂ respectively, and constant correlation coefficient r_1.2. In other words, the processes S t₁  and S t₂  evolve in time according to the following dynamics :

       

1 1 1 1 1 1

dS t aS t dt sS t dB t (2.1)

       

2 2 2 2 2 2

dS t  aS t dt s S t dB t

where B t₁  and B t₂  are standard real-valued Brownian motions and^{d B B t} ₁^, ₂^{ }^ ^r_1.2^dt

Let H H K K K₁, , , ,₂ ₁ ₂ ₃ be positive real numbers and P . H H K K K t t1, , , , , ,2 1 2 3 1 2 be defined as one of the four following cumulative distribution functions, where t₂   t₁ t₀ 0 :

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  

      ^{ }  

1 1 2

1 2 1 2 3 1 2

up-and-up

1 1 1 1 1 2 1 2 2 2 2 2 3

0

1) P , , , , , ,

sup , , , sup ,

t t t t t

H H K K K t t

S t H S t K S t K S t H S t K

   

 

      

 

 





  

      ^{ }  

1 1 2

1 2 1 2 3 1 2

down-and-down

1 1 1 1 1 2 1 2 2 2 2 2 3

0

2) P , , , , , ,

inf_{t t} , , , inf_{t t t} ,

H H K K K t t

   

 

      

 

 





  

      ^{ }  

1 1 2

1 2 1 2 3 1 2

down-and-up

1 1 1 1 1 2 1 2 2 2 2 2 3

0

3) P , , , , , ,

inf_{t t} , , , sup ,

t t t

H H K K K t t

   

 

      

 

 





  

      ^{ }  

1 2

1

1 2 1 2 3 1 2

up-and-down

1 1 1 1 1 2 1 2 2 2 2 2 3

0

4) P , , , , , ,

sup , , , inf_{t t t} ,

t t

H H K K K t t

S t H S t K S t K _{ } S t H S t K

 

 

      

 

 





Let us introduce the following notations :

 ¹  ²  ³  ¹  ²

1 2 3 1 2

0 0 0 0 0

1 2 2 1 2

ln K , ln K , ln K , ln H , ln H ,

k S t k S t k S t h S t h S t

         

    

              

12

1 1 s2 ,

m a  m₂ a₂s2²²

The next proposition provides an exact formula for the four above-mentioned cumulative distribution functions.

Proposition 1

Let N₃.,.,.; ,q q q1.2 1.3 2.3, denote the joint trivariate cumulative distribution function of three standard normal random variables _{X X X}₁_{, ,}₂ ₃, where _q_{a b}_. is the correlation coefficient between X_a and _X_b,

a b,   1,2,3.

Then, for the up-and-up and the down-and-down distributions, we have :

 .  1 2 1 2 3 1 2

P H H K K K t t, , , , , ,

(2.2)

1 1 1 2 2 1 3 2 2 1 1

3 1.2 1.2

1 1 , 2 1 , 2 2 ; , 2 , 2

k t k t k t t t

N  l smt   l smt   l smt  r t r t 

        

1 1 1 2 1 2 2 1 3 2 2 2

1 1 1.2 2 2 1 2 2

2 22 3

2 1 1

1.2 1.2

2 2

2 , , 2 ;

exp 2

, ,

k t t k t k h t

t t t

h N t t

t t

m m m m

l s r s l s l s

m

s r r

            

       

 

       

   

      

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1 1 1 1 2 2 1 1 3 2 2 1

1.2 1.2

1 1 2 1 1 1 2 2 1 2

1 12 3

1 1 1

1.2 1.2

2 2

2 , 2 , 2 ;

exp 2

, ,

k h t k t h k t h

t t t t t

h N t t

t t

m m m

l s l s r s l s r s

m

s r r

             

         

 

       

   

    

1 2 1.2 2 2

1

2 2

1 1 2 2

1 1 1 1 2 1 2 2 1 1

1.2 1.2

1 1 2 2 1 1 1

3 3 2 2 2 1 1 1

1.2 1.2 1.2

2 2 1 2 2 2

2 4 2

exp

2 2 , 2 ,

2 2 ; , ,

h h

k h t t k t h

t t t

N k h t h t t

t t t t

m m r m

s s s s

m m m

l s r s l s r s

l s m r s r r

  

  

     

       

     

    

       





 

 

where 1

l  if P . H H K K K t t1, , , , , ,2 1 2 3 1 2P_up-and-up_H H K K K t t1, , , , , ,2 1 2 3 1 2

1

l   if P . H H K K K t t1, , , , , ,2 1 2 3 1 2P_down-and-down_H H K K K t t1, , , , , ,2 1 2 3 1 2

And, for the up-and-down and the down-and-up distributions, we have :

 .  1 2 1 2 3 1 2

P H H K K K t t, , , , , ,

(2.3)

1 1 1 2 2 1 3 2 2 1 1

3 1.2 1.2

1 1 , 2 1 , 2 2 ; , 2 , 2

k t k t k t t t

N l s mt l s tm l s tm r t r t

            

 

          

1 1 1 2 1 2 2 1 3 2 2 2

1 1 1.2 2 2 1 2 2

2 22 3

2 1 1

1.2 1.2

2 2

2 , , 2 ;

exp 2

, ,

k t t k t k h t

t t t

h N t t

t t

m m m m

l s r s l s l s

m

s r r

            

       

 

       

   

     

1 1 1 1 2 2 1 1 3 2 2 1

1.2 1.2

1 1 2 1 1 1 2 2 1 2

1 12 3

1 1 1

1.2 1.2

2 2

2 , 2 , 2 ;

exp 2

, ,

k h t k t h k t h

t t t t t

h N t t

t t

m m m

l s l s r s l s r s

m

s r r

             

         

 

       

   

     

1 2 1.2 2 2

1

2 2

1 1 2 2

1 1 1 1 2 1 2 2 1 1

1.2 1.2

1 1 2 2 1 1 1

3 3 2 2 2 1 1 1

1.2 1.2 1.2

2 2 1 2 2 2

2 4 2

exp

2 2 , 2 ,

2 2 ; , ,

h h

k h t t k t h

t t t

N k h t h t t

t t t t

m m r m

s s s s

m m m

l s r s l s r s

l s m r s r r

  

  

     

       

     

   

    

 

 



 

 

 

where 1

l  if P . H H K K K t t1, , , , , ,2 1 2 3 1 2 Pup-and-downH H K K K t t1, , , , , ,2 1 2 3 1 2

1

l   if P . H H K K K t t1, , , , , ,2 1 2 3 1 2Pdown-and-upH H K K K t t1, , , , , ,2 1 2 3 1 2

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Corollary of Proposition 1

The four following joint cumulative distribution functions, that will be useful in Section 3, are deduced from Proposition 1 :

1)       ^{ }  

1 1 1 1 1 1 2 1 2 1 2 2 2 2 2 3

0sup , , , sup ,

t t S t H S t K S t K t t t S t H S t K

   

 

      

 

 



up-and-up 1 2 1 2  up-and-up 1 2 1 2 3

P H H K K, , , , P H H K K K, , , ,

  

2)       ^{ }  

1 1 1 1 1 1 2 1 2 1 2 2 2 2 2 3

0_{ }inf_{t t} S t H S t, K S t, K , inf_{t t t}_{ } S t H S t, K

 

      

 

 



down-and-down 1 2 1 2  down-and-down 1 2 1 2 3

P H H K K, , , , P H H K K K, , , ,

  

3)       ^{ }  

1 1 1 1 1 1 2 1 2 1 2 2 2 2 2 3

0inf , , , sup ,

t t S t H S t K S t K t t t S t H S t K

   

 

      

 

 



down-and-up 1 2 1 2  down-and-up 1 2 1 2 3

P H H K K, , , , P H H K K K, , , ,

  

4)       ^{ }  

1 2

1 1 1 1 1 1 2 1 2 2 2 2 2 3

0sup , , , inf_{t t t} ,

t t S t H S t K S t K _{ } S t H S t K

 

 

      

 

 



up-and-down 1 2 1 2  up-and-down 1 2 1 2 3

P H H K K, , , , P H H K K K, , , ,

  

 Proof of Proposition 1 is provided in the Appendix.

The numerical implementation of Proposition 1 and its corollary is easy using Genz’s algorithm for the computation of trivariate normal cumulative distribution functions (Genz [7]).

In the next Proposition, we introduce a third correlated geometric Brownian motion that will serve as the endpoint of the joint distribution and we show that this can still be analytically valued. Let

^{S t t}₃ ^, ^ ⁰ be a geometric Brownian motion driven by the following dynamics under the initial probability measure  :

       

3 3 3 3 3 3

dS t  aS t dt s S t dB t (2.4)

where a₃ and s₃ are real constants (s₃  0), B t₃  is a real-valued standard Brownian motion and correlations are as follows :

 ₁^, ₃^{ } _1.3

d B B t r dt,^{d B B t} ₂^, ₃^{ }^ ^r_2.3^dt ^(2.5)

Let K₄be a positive real number and P . H H K K K K t t t1, , , , , , , ,2 1 2 3 4 1 2 3 be defined as one of the four following joint cumulative distribution functions, where t₃ t₂:

  

      ^{ }    

1 1 2

1 2 1 2 3 4 1 2 3

up-and-up

1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 4

0

1) P , , , , , , , ,

sup , , , sup , ,

t t t t t

H H K K K K t t t

S t H S t K S t K S t H S t K S t K

   

 

       

 

 





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  

      ^{ }    

1 1 2

1 2 1 2 3 4 1 2 3

down-and-down

1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 4

0

2) P , , , , , , , ,

inf_{t t} , , , inf_{t t t} , ,

H H K K K K t t t

   

 

       

 

 





  

      ^{ }    

1 1 2

1 2 1 2 3 4 1 2 3

down-and-up

1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 4

0

3) P , , , , , , , ,

inf_{t t} , , , sup , ,

t t t

H H K K K K t t t

   

 

       

 

 





  

      ^{ }    

1 2

1

1 2 1 2 3 4 1 2 3

up-and-down

1 1 1 1 1 2 1 2 2 2 2 2 3 3 3 4

0

4) P , , , , , , , ,

sup , , , inf_{t t t} , ,

t t

H H K K K K t t t

S t H S t K S t K _{ } S t H S t K S t K

 

 

       

 

 





Let us introduce the following new notations :

 ⁴

4 ln 3K 0 ,

k S t

 

  

32

3 3 s2

m  a 

The next proposition provides an exact formula for the four above-mentioned joint cumulative distribution functions.

Proposition 2

Let the real function b b b b1 2 3 4 1.2 1.3 1.4 2.3 3.4, , , ;q q q q q, , , , , where b b b b1 2 3 4, , ,  ⁴ and each of the real numbers q q q q q1.2 1.3 1.4 2.3 3.4, , , , is included in _1,1_, be defined as follows :

b b b b1 2 3 4 1.2 1.3 1.4 2.3 3.4, , , ;q q q q q, , , , 

 (2.6)

   

1 2 3

2 2

2 1 1.2 2 3 2.3 2

2 2 2

3 2 1 3 2

2 1 3 2

3 1.3 1

4 1.4 1 3.4 1 2

1.3 3 2 1

4 1.3

1 8 exp 2 2 2

1

b b b

x x x

x x x x

x

x x

b x

N dx dx dx

q q

f f

f f p

q q q f q

  

   

 

    

   

    

   

 

 

  

where N^{ }. is the standard normal cumulative distribution function and the following definitions apply :

2 1 1 1.22 ,

f  q f_{3 2}  1q_2.3² , _{3.4 1} ^3.4 ^{1.3 1.4} _{4 1.3} _1.4² _{3.4 1}²

3 1 , 1

q q q

q  f f  q q

Then, for the up-and-up and the down-and-down distributions, we have :