Michael Schroeder, Peter Carr
Asian options are widely used financial derivatives
which provide non-linear payoffs on the arithmetic
average of an underlying asset. Their valuation
has intrigued finance theorists for over a decade
by now, even in the Black-Scholes context.
Pursuing the Laplace transform approach proposed
by Yor and Geman in 1993, this talk discusses our
derivation of series and asymptotic expansions
for computing benchmark prices for Asian options.
And our point of view is that these formulas
are consequences of the manifold interrelations
this valuation problem has with other central
parts of mathematics.
The punchline of the talk so is as follows. Since the
Yor-Geman Laplace transforms are not those of the
Asian option prices and, moreover, have been
derived only under the restriction that the
riskneutral drift is not less than half the
variance, we re-address the notion of Laplace
transforms of Asian option prices in a first
part of the talk. While we will not discuss how
in joint work with Peter Carr we have been able to
lift the second of these restrictions, we will
discuss how ideas of Peter Carr's have salvaged
the relevance of the Yor-Geman results for
valuing Asian options.
In a second part of the talk we will sketch the
ideas, based on methods from complex analysis
and special functions, of our analytic inversion
of the Laplace transform, and compare the
resulting integral for the Asian option price
with Yor's 1992 triple integral for it.
In a third part, we will discuss series and
asymptotic expansions for computing
our integral. And we will indicate some of the
background ideas from the analysis on the
Poincare upper half plane and modular forms