A new PDE approach for pricing arith-
metic average Asian options∗
Jan Večeř†
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh,
PA XXXXXXXXXXEmail: XXXXXXXXXX.
May 15, 2001
Abstract. In this paper, arithmetic average Asian options are studied. It is ob-
served that the Asian option is a special case of the option on a traded account.
The price of the Asian option is characterized by a simple one-dimensional par-
tial differential equation which could be applied to both continuous and discrete
average Asian option. The article also provides numerical implementation of the
pricing equation. The implementation is fast and accurate even for low volatility
and/or short maturity cases.
Key words: Asian options, Options on a traded account, Brownian motion,
fixed strike, floating strike.
1 Introduction
Asian options are securities with payoff which depends on the average of the
underlying stock price over certain time interval. Since no general analytical
solution for the price of the Asian option is known, a variety of techniques have
een developed to analyze arithmetic average Asian options. A number of ap-
proximations that produce closed form expressions have appeared, see Turnbull
and Wakeman [18], Vorst [19], Levy [13], Levy and Turnbull [14]. Geman and
Yor [8] computed the Laplace transform of the Asian option price, but numerical
inversion remains problematic for low volatility and/or short maturity cases (see
Geman and Eydeland [6] or Fu, Madan and Wang [5]). Monte Carlo simulation
works well, but it can be computationally expensive without the enhancement of
variance reduction techniques and one must account for the inherent discretiza-
tion bias resulting from the approximation of continuous time processes through
discrete sampling (see Broadie and Glasserman [3], Broadie, Glasserman and
Kou [4] and Kemma and Vorst [12]).
∗This work was supported by the National Science Foundation under grant DMS XXXXXXXXXX.
†I would like to thank Fredrik Åkesson, Julien Hugonnier, Steven Shreve, Dennis Wong and
Mingxin Xu for helpful comments and suggestions on this paper.
1
In general, the price of an Asian option can be found by solving a PDE in
two space dimensions (see Ingersoll [10]), which is prone to oscillatory solutions.
Ingersoll [10] also observed that the two-dimensional PDE for a floating strike
Asian option can be reduced to a one-dimensional PDE. Rogers and Shi [17]
have formulated a one-dimensional PDE that can model both floating and fixed
strike Asian options. They reduced the dimension of the problem by dividing
K−S̄t (K is the strike, S̄t is the average stock price over [0, t]) by the stock price
St. However this one-dimensional PDE is difficult to solve numerically since the
diffusion term is very small for values of interest on the finite difference grid. The
dirac delta function also appears as a coefficient of the PDE in the case of the
floating strike option. Zvan, Forsyth and Vetzal [21] were able to improve the
numerical accuracy of this method by using computational fluid dynamics tech-
niques. Andreasen [2] applied Rogers and Shi’s reduction to discretely sampled
Asian option. More recently, Lipton [15] noticed similarity of pricing equations
for the passport and the Asian option, again using Rogers and Shi’s reduction.
In this article, an alternative one-dimensional PDE is derived by a simila
space reduction. It is noted that the arithmetic average Asian option (both
floating and fixed strike) is a special case of an option on a traded account. See
Shreve and Večeř [16] and [20] for a detailed discussion about options on a traded
account. Options on a traded account generalize the concept of many options
(passport, European, American, vacation) and the same pricing techniques could
e applied to price the Asian option. The resulting one-dimensional PDE fo
the price of the Asian option is simple enough to be easily implemented to give
very fast and accurate results.
Section 2 of the article
iefly describes options on a traded account. It is
shown in section 3 that the Asian option is a special case of the option on a
traded account. The one-dimensional PDE for the price of the Asian option is
given. Section 4 describes the numerical implementation and compares results
with results of other methods. Section 5 concludes the paper.
2 Options on a traded account
An option on a traded account is a contract which allows the holder of the
option to switch during the life of the option among various positions in an
underlying asset (stock). The holder accumulates gains and losses resulting from
this trading, and at the expiration of the option he gets the call option payoff
with strike 0 on his final account value, i.e., he keeps any gain from trading and
is forgiven any loss.
Suppose that the stock evolves under the risk neutral measure according to
the equation
dSt = St(rdt + σdWt), (2.1)
where r is the interest rate and σ is the volatility of the stock. Denote the option
holder’s trading strategy by qt, the number of shares held at time t. The strategy
qt is subject to the contractual constraint qt ∈ [αt, βt], where αt ≤ βt. It turns
out that the holder of the option should never take an intermediate position, i.e,
2
at any time he should hold either αt shares of stock or βt shares. In the case of
Asian options, αt = βt, so option holder’s trading strategy is a priori given to
him.
In our model the value of the option holder’s account co
esponding to the
strategy qt satisfies
dXqt = qtdSt + µ(X
q
t − qtSt)dt (2.2)
Xq0 = X0.
This represents a trading strategy in the money market and the underlying
asset, where X0 is the initial wealth and µ is the interest rate co
esponding to
einvesting the cash position Xqt − qtSt (possibly different from the risk-neutral
interest rate r). The trading strategy is self-financing when µ = r. The holde
of the option will receive at time T the payoff [XqT ]
+. The objective of the
seller of the option, who makes this payment, is to be prepared to hedge against
all possible strategies of the holder of the option. Therefore the price of this
contract at time t should be the maximum over all possible strategies qu of the
discounted expected value under the risk-neutral probability P of the payoff of
the option, i.e.,
V [α,β](t, St, Xt) = max
qu∈[α,β]
e−r(T−t)E[[XqT ]
+|Ft], t ∈ [0, T ]. (2.3)
Computation of the expression in (2.3) is a problem of stochastic optimal control,
and the function V [α,β](t, s, x) is characterized by the co
esponding Hamilton–
Jacobi–Bellman (HJB) equation
− rV + Vt + rsVs + max
q∈[α,β]
[(µx + q(r − µ))Vx
+ 12σ
2s2(Vss + 2qVsx + q2Vxx)] = 0 (2.4)
with the boundary condition
V (T, s, x) = x+. (2.5)
The maximum in (2.4) is attained by the optimal strategy qoptt .
The case αt = βt = 1 reduces to the European call, the case αt = βt = −1
educes to the European put. The American call and put give the holder of
the option the right to switch at most once during the life of the option to zero
position (i.e., exercise the option), but it does not pay interest on the traded
account while the holder has a position in the stock market. These can be
modelled by setting µ = 0 in (2.2) and allowing only one switch in qt, eithe
from 1 to 0 (American call) or from −1 to 0 (American put). The passport
option has contractual conditions αt = −1, βt = 1, the so-called vacation call
has αt = 0, βt = 1 and the so-called vacation put has αt = −1, βt = 0.
By the change of variable
Zqt =
Xqt
St
, (2.6)
3
we can reduce the dimensionality of the problem (2.3), as we show below. The
same change of variable was used in Hyer, Lipton-Lifschitz and Pugachevsky
[9] and in Andersen, Andreasen and Brotherton-Ratcliffe [1] to price passport
options and in Shreve and Večeř [16] to price options on a traded account.
Applying Itô’s formula to the process Zqt , we get
dZqt = (qt − Z
q
t ) (r − µ− σ2)dt + (qt − Z
q
t ) σdWt. (2.7)
We next define a new probability measure P̃ by P̃(A) =
∫
A
DT dP, A ∈ F , where
DT = e−rT · STS0 = exp
(
σWT − 12σ
2T
)
. (2.8)
Under P̃, W̃t = −σt+Wt is a Brownian motion, according to Girsanov’s theorem.
Notice that
e−rT E[XqT ]
+ = e−rT Ẽ
[
XqT
DT
]+
= S0 · Ẽ
[
XqT
ST
]+
= S0 · Ẽ [ZqT ]
+ (2.9)
and
dZqt = (qt − Z
q
t ) (r − µ)dt + (qt − Z
q
t ) σdW̃t. (2.10)
The co
esponding reduced HJB equation becomes
ut + max
q∈[α,β]
(
(r − µ)(q − z)uz + 12 (q − z)
2σ2uzz
)
= XXXXXXXXXX)
with the boundary condition
u(T, z) = z XXXXXXXXXX)
The relationship between V and u is
V (0, S0, X0) = S0 · u
(
0, X0S0
)
. (2.13)
Closed form solutions and optimal strategies are provided in Shreve and Večeř
[16] for the prices of the option on a traded account for any general constraints
of the type αt ≡ α and βt ≡ β when µ = r.
3 Asian option as an option on a traded account
Options on a traded account also represent Asian options. Notice that d(tSt) =
tdSt + Stdt, or equivalently,
TST =
∫ T
0
tdSt +
∫ T
0
Stdt. (3.1)
After dividing by the maturity time T and rea
anging the terms we get
1
T
∫ T
0
Stdt =
∫ T
0
(
1− tT
)
dSt + S0. (3.2)
4
In the terminology of the option on a traded account, the Asian fixed strike call
payoff (S̄T −K)+ is achieved by taking qt = 1− tT and X0 = S0−K and where
the traded account evolves according to the equation
dXt =
(
1− tT
)
dSt, (3.3)
i.e., when µ = 0 so no interest is added or charged to the traded account. We
have then
XT =
∫ T
0
(1− tT )dSt + S0 −K = S̄T −K. (3.4)
Thus the average of the stock price could be achieved by a selling off one share
of stock at the constant rate 1T shares per unit time.
Similarly, the Asian fixed strike put payoff (K − S̄T )+ is achieved by taking
qt = tT − 1 and X0 = K − S0. For the Asian floating strike call with payoff
(KST − S̄T )+ we take simply qt = tT − 1 + K and X0 = S0(K