The Arithmetic Average Case
of Asian Options:
Reducing the PDE to the
Solomon M. Antoniou
Scientific Knowledge Engineering
and Management Systems
37 Ȁoliatsou Street, Corinthos 20100, Greece
We consider the path-dependent contingent claims where the underlying asset
follows an arithmetic average process. Considering the no-arbitrage PDE of these
claims, we first determine the underlying Lie Point Symmetries. After
determination of the invariants, we transform the PDE to the Black-Scholes (BS)
equation. We then transform the BS equation into the heat equation and we
provide some general solutions to that equation. This procedure appears for the
first time in the finance literature.
Keywords: Path-Dependent Options, Asian Options, Arithmetic Average Path-
Dependent Contingent Claims, Lie Symmetries, Exact Solutions.
A path-dependent option is an option whose payoff depends on the past history of
the underlying asset. In other words these options have payoffs that do not depend
on the asset’s value at expiry. Two very common examples of path-dependent
options are Asian options and lookback options.
The terminal payoff of an Asian option depends on the type of averaging of the
underlying asset price over the whole period of the option’s lifetime. According to
the way of taking average, we distinguish two classes of Asian options: arithmetic
and geometric options.
A lookback option is another type of path-dependent option, whose payoff
depends on the maximum or minimum of the asset price during the lifetime of the
The valuation of path-dependent (Asian) European options is a difficult problem
in mathematical finance. There are only some simple cases where the price of
path-dependent contingent claims can be obtained in closed-form (refs. -).
If the underlying asset price follows a lognormal stochastic process, then its
geometric average has a lognormal probability density and in this case there is