Some Relations between Bessel Processes, Asian Options and Confluent Hypergeometric Functions
A closed formula is obtained for the Laplace transform of moments of certain exponential functionals of Brownian motion with drift, which give the price of some financial options, so-called Asian options. A second equivalent formula is presented, which is
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Some Relations between Bessel Processes, Asian Options and Confluent Hypergeometric Functions 1 C.R. Acad. Sci., Paris, Ser. I 314 (1992), 471-474 (with Helyette Geman)
Abstract. A closed formula is obtained for the Laplace transform of moments of certain exponential functionals of Brownian motion with drift, which give the price of some financial options, so-called Asian options. A second equivalent formula is presented, which is the translation, in this context, of some intertwining properties of Bessel processes or confluent hypergeometric functions.
1.
The Asian Options Problem: Statement of Results
From the mathematical point of view, the Asian options problem involves giving a closed formula of the greatest possible simplicity for the quantity:
(1) where k E IR+, v E IR and
A~V) = fat dsexp2(B s + vs), in which (Bs, s 2: 0) denotes a real-valued Brownian motion, starting from o. The uninitiated reader can gain an idea of the questions of financial mathematics associated with Asian options from [2,3], in particular. Here, we give a relatively simple formula for the Laplace transform of C(v)(t, k), that is:
for A sufficiently large, where T).. denotes an exponential variable with parameter A, independent of the Brownian motion B. 1
This note was presented by Paul-Andre Meyer.
M. Yor, Exponential Functionals of Brownian Motion and Related Processes © Springer-Verlag Berlin Heidelberg 2001
.
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3. Bessel Processes, Asian Options and Confluent Hypergeometric Functions
Theorem. Suppose that n 2 0 (not necessarily an integer), and A> 0 and set
f-l = )2A + v 2. Suppose that A> 2n(n+v), which is equivalent to: f-l > v+2n. Then we have, for x> 0:
(2) Moreover, we have
E[(A(v)n T>,
= f(n + l)f(((f-l + v)/2) + l)f(((f-l- v)/2) - n) 2n r((f-l- v)/2)f(n + ((f-l + v)/2) + 1) .
(3)
In the particular case in which n is an integer, this formula simplifies to: (v) n _
E[(AT>,) 1 -
I17=1 (A -
n! 2(P
(4)
+ jv))·
Decomposing the rational fraction on the right-hand side of (4) into simple elements, we obtain the following closed expression for the moments of A~v). Proposition 1. For all a E 1R \ {O}, v E 1R and n EN, we have:
a'n E [
(1' da
exp a(B.
+ va)
r] ~ t, n!
c;"I.) exp ( (a'g'
+ aj
+) (5)
where c;IJ)=2 n
IT k#j
0::; k::; n
In particular, for v = 0, we have:
((B+j)2_(B+k)2)~1.
.
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3.2. Stages in the Proof of the Theorem
where
Pn(z) = (-2)
n {
1 n n!( -z)j } ,. + 22.: ( _ ')'( + ')'
n.
j=1
n
J. n
J.
( -2)n
= -,-{2F(-n, 1,n + l;z) -I}, n.
in which F(o:, (3, "(; z) denotes the hypergeometric function with parameters (0:, (3, "(). Remarks.
1. The formula (6) is in agreement with the following result due
to Bougerol [1]: for fixed t 2 0, sinh(Bt ) d~t.
"(A(O) t
where
bu, u 2
0) is a
real-valued Brownian motion independent of A~O). 2. Using a different approach to that developed in this paper, one can obtain an expression for the Laplace transform in t of the joint distribution of (A~v), B t + vt), in terms of semi groups of Bessel processes, then invert the Laplace tr
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