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10.4 Asian Options 251
The values of f
1
(
φ
), f
2
(
φ
) and M(T) can be computed for a as 1 + i
φ
,i
φ
and
1, respectively. The pricing formula for Asian options with stochastic volatilities is
therefore given by the following form,
C
asian
= e
−rT
S
0
M(T )
n
n
0
F
1
−K
1
F
2
. (10.75)
For these two cases, we have integrated stochastic volatility into the geometric
Asian options. Integrating stochastic interest rates follows a similar line. We now
consider the issue of incorporating jumps into Asian options and demonstrate the
method by applying lognormal jumps. From (7.20) and (10.68), we obtain the fol-
lowing risk-neutral process
d lnGA(t;n)=
ε
1
r −
λμ
J
−
1
2
v
2
dt +(
√
ε
2
v)dW(t)+ln(1+J)
1
n
n
∑
j=1
dY
j
(10.76)
with
Y
j
= Y (T
0
+ hj), lnGA(0;n)=x
0
.
Note that the jump component is independent of the volatility, we only need to
calculate the additional jump terms for CF f
1
(
φ
):
f
PJ
1
= E
exp
ln(1 +J)
1 + i
φ
n
n
∑
j=1
Y
j
−(1+ i
φ
)
λμ
J
t
1
= E
exp
ln(1 + J)
1 + i
φ
n
[nY
0
+
n
∑
j=1
(n − j + 1)
Δ
Y
j
]
−(1+ i
φ
)
λμ
J
t
1
= exp
λ
T
0
[(1+
μ
J
)
(1+i
φ
)
e
1
2
i
φ
(1+i
φ
)
σ
2
J
−1]−(1+i
φ
)
λμ
J
t
1
× E
exp
ln(1 + J)
1 + i
φ
n
n
∑
j=1
(n − j + 1)
Δ
Y
j
,
Using the Markovian property of the Poisson process, we have
f
PJ
1
= exp
λ
T
0
[(1+
μ
J
)
(1+i
φ
)
e
1
2
i
φ
(1+i
φ
)
σ
2
J
−1]−(1+i
φ
)
λμ
J
t
1
× exp
n
∑
j=1
λ
h[(1+
μ
J
)
k
j
exp(
1
2
k
j
(k
j
−1)
σ
2
J
) −1]
= exp
λ
T
0
[(1+
μ
J
)
(1+i
φ
)
e
1
2
i
φ
(1+i
φ
)
σ
2
J
−1]−(1+i
φ
)
λμ
J
t
1
× exp
λ
h
n
∑
j=1
(1 +
μ
J
)
k
j
exp(
1
2
k
j
(k
j
−1)
σ
2
J
) −
λ
(T −T
0
)
252 10 Exotic Options with Stochastic Volatilities
f
PJ
1
= exp
λ
T
0
(1+
μ
J
)
(1+i
φ
)
e
1
2
i
φ
(1+i
φ
)
σ
2
J
−
λ
T −(1+i
φ
)
λμ
J
t
1
×
exp
λ
h
n
∑
j=1
(1+
μ
J
)
k
j
exp(
1
2
k
j
(k
j
−1)
σ
2
J
)
(10.77)
with
k
j
=
(1+i
φ
)(n + j −1)
n
.
For CF f
2
(
φ
) we have a similar result but with another k
j
,
f
PJ
2
= E
exp
ln(1+J)
i
φ
n
n
∑
j=1
Y
j
−i
φλμ
J
t
1
= exp
λ
T
0
(1+
μ
J
)
i
φ
e
1
2
i
φ
(i
φ
−1)
σ
2
J
−
λ
T −i
φλμ
J
t
1
×
exp
λ
h
n
∑
j=1
(1+
μ
J
)
k
j
exp(
1
2
k
j
(k
j
−1)
σ
2
J
)
(10.78)
with
k
j
=
i
φ
(n + j −1)
n
.
10.4.4 Approximations for Arithmetic Average Asian Options
Although Asian options based on the geometric average can be valued analytically
by a closed-form formula, these options are not commonplace in the OTC markets.
The most popular Asian options are based on the arithmetic average of the underly-
ing asset, which, unfortunately, is no longer lognormally distributed. In other words,
the arithmetic average does not follow a geometric Brownian motion. This raises a
serious problem in pricing arithmetic average Asian options. In order to fill this gap,
some approximation methods based on the pricing formula for geometric average
Asian options are suggested. Boyle (1977) was the first to apply the Monte Carlo
control-variate method to get the approximative values of arithmetic average Asian
options. However, the Monte Carlo method is very time-consuming, inefficient and
inaccurate for Asian options as reported in Fu, Madan and Wang (1995).
In this subsection, we briefly review the existing approximation methods that
attempt to reduce the bias between the geometric average and arithmetic average,
hoping that the previously derived formulas can be used. All these approximations
are suggested in the framework of constant volatilities. Nevertheless, the methods
presented below attempt to obtain better values for arithmetic average Asian op-
tions by correcting the moments of the geometric average, which can be calculated
by the CFs. Therefore, the following corrections are also applicable for stochastic
volatilities and stochastic interest rates.
10.4 Asian Options 253
(1). Zhang’s approach (1997).
In his approach, the general mean is the key to associate arithmetic average with
geometric average. The general mean is defined as follows:
GM(
γ
;a)=
1
n
n
∑
i=1
a
γ
i
1/
γ
for
γ
> 0. (10.79)
We need the following two special properties of GM(
γ
;a) for approximation:
GM(1;a)=
1
n
n
∑
i=1
a
i
= AA(a) (10.80)
and
lim
γ
→0
GM(
γ
;a)=lim
γ
→0
1
n
n
∑
i=1
a
γ
i
1/
γ
=
n
∏
i=1
a
i
1/n
= GA(a). (10.81)
Thus, we can consider the arithmetic average and the geometric average as two
special values of the function GM(
γ
;a) at the point
γ
= 1 and
γ
= 0, respectively.
Additionally, GM(
γ
;a) is an increasing continuous function in
γ
. Using Taylor’s
expansion, we have
AA(a)=GM(1) ≈ GM(0)+GM
(0)=GA(a)[1 +
1
2
Var(lna)], (10.82)
where GM
(
ϕ
) is the first derivation of GM. Zhang’s main result can be summarized
as follows:
AA(S)
∼
=
kGA(S) (10.83)
with
k = 1 +
1
2
EV +
1
4
(VV + EV)
2
, (10.84)
where
EV =
(n
2
−1)h
6
1
2
(r −
1
2
v
2
)
2
h +
1
n
v
2
and
VV =
(n
2
−1)(3n
2
−2)
15n
3
1 −
1
2
v
2
2
v
2
h
3
.
Obviously, when n approaches 1, the number k will degenerate to 1. When n
approaches infinity, we obtain the limiting k which is given by
k = 1 +
1
24
(r −
1
2
v
2
)nh
2
+
1
576
(r −
1
2
v
2
)nh
4
. (10.85)
254 10 Exotic Options with Stochastic Volatilities
By using the corrected geometric average, we can calculate the approximative
prices of the arithmetic average Asian options.
(2). Vorst’s approach (1990).
Instead of correcting the geometric average, Vorst suggests a simple method to
correct the strike price by using the difference between E[GA(n)] and E[AA(n)].
The new effective strike price is therefore given by
K
new
= K + E[GA(n)] −E[AA(n)].
Because E[GA(n)] E[AA(n)] , we always have K
new
K. This correction uses
only the first moment whereas Zhang’s correction uses not only the first but also the
second moments. The expectation value E[AA(n)] can be obtained analytically by
the standard tool.
(3). The modified geometric average (MGA).
This method is more accurate than Vorst’s approach as reported by L
´
evy and
Turnbull (1992). The main idea of MGA is that if lnAA(n) is normally distributed
according to N(
μ
,
σ
), then E[AA(n)] = exp(
μ
+
1
2
σ
2
). It immediately follows that
μ
= lnE[AA(n)] −
1
2
σ
2
. In the framework of risk-neutral pricing, MGA implies the
replacement of r by lnE[AA(n)], namely,
r
new
= lnE[AA(n)] . (10.86)
Beside the above mentioned approximation approaches, a number of suggestions
have been proposed to improve the performance of pricing arithmetic average op-
tions based on the available closed-form formula. These works include L
´
evy (1990),
Turnbull and Wakeman (1991).
10.4.5 A Model for Asian Interest Rate Options
In this subsection, we deal with a special interest rate derivative whose payoff at
maturity depends on the average of interest rates in a specified period prior to ma-
turity. Such financial instruments are called Asian interest rate options and, then
are a direct function of interest rates. This feature is different from a bond option
whose payoff is indirectly affected by the interest rate. We can consider Asian in-
terest rate options as an innovation of the interest rate cap or floor with average
attribute. Therefore, Asian interest rate call (put) options are also called cap (floor)
on the average interest rates.
Longstaff (1995) addressed the issue of pricing and hedging this particular option
in the framework of the Vasicek model and found that it is an important alternative
hedging instrument for managing borrowing costs. As Asian stock options do, this
10.4 Asian Options 255
type of derivative is always less costly than the corresponding full cap and less
sensitive to changes in interest rates. Ju (1997) studied Asian interest rate options
by applying the Fourier transform. His idea is that if the density function of the
underlying average is known, then the option based on this average can be given in
an integral form. The unfavorable feature of his solution is that one does not have
a solution form `alaBlack-Scholes and can not get the hedge ratios explicitly. Here
we enhance Ju’s idea and give a closed-form solution for Asian interest rate options
by using the underlying CFs. The following results are valid both for the Vasicek
(1977) model and the CIR (1985b) model.
Without loss of generality, we let the notional amount in the option contract be
equal to one. Let y(T)=
T
0
r(u)du, the average rate is then y(T)/T. We consider a
case where the time to maturity is shorter than or equal to the length of the average
period, i.e., ATP- or ITP-Asian options. Let A denote the sum of the interest rates
from the first day of the average period to the current time. For the call option we
have
C
AIR
= E
exp
−
T
0
r(u)du
1
T
A +
T
0
r(u)du
−K
+
= E
exp(−y(T))
y(T)
T
−K
0
+
, K
0
= K −
A
T
= E
exp(−y(T))
y(T)
T
−K
0
·1
(yTK
0
)
= T
−1
E
e
−y(T)
y(T) ·1
(yTK
0
)
−K
0
E
e
−y(T)
·1
(yTK
0
)
=
V(T )
T
E
e
−y(T)
y(T)
V(T )
·1
(yTK
0
)
−B(T)K
0
E
e
−y(T)
B(0,T)
·1
(yTK
0
)
=
V(T )
T
F
1
−B(0,T)K
0
F
2
, (10.87)
where
B(0,T)=E
exp
−
T
0
r(u)du
= E
e
−y(T)
is a zero-coupon bond price maturing at time T and
V(T )=E
exp
−
T
0
r(u)du
−
T
0
r(u)du
= E
e
−y(T)
y(T)
is a discounted price of the “underlying asset” y(T ). An interesting feature of Asian
interest rate options is that they are discounted by using their own underlying. Obvi-
ously, e
−y(T)
/B(T) and e
−y(T)
y(T)/V (T) define two martingales, respectively, and
guarantee no arbitrage over time. Hence, the characteristic function of F
1
can be
given by
256 10 Exotic Options with Stochastic Volatilities
f
1
(
φ
)=E
e
−y(T)
y(T)
V(T )
e
i
φ
y
= E
y(T)e
(i
φ
−1)y(T)
V(T )
= E
y(T)
V(T )
exp
(i
φ
−1)
T
0
r(u)du
. (10.88)
Similarly,
f
2
(
φ
)=E
e
−y(T)
B(T)
e
i
φ
y(T)
= E
e
(i
φ
−1)y(T)
B(T)
= E
1
B(T)
exp
(i
φ
−1)
T
0
r(u)du
. (10.89)
V(T ) and f
1
(
φ
) can be calculated by the following feature of expectation,
V(T )=−
∂
∂λ
E
e
−
λ
y(T)
|
λ
=1
(10.90)
and
f
1
(
φ
)=−
1
V(T )
∂
∂λ
E
e
−
λ
y(T)
|
λ
=1−i
φ
. (10.91)
Dependent on whether the interest rate is specified either as a mean-reverting
square-root process or as a mean-reverting Ornstein-Uhlenbeck process, we can give
the full formula of the expression −
∂
∂λ
E
e
−
λ
y(T)
.
Case 1: A mean-reverting square root process (CIR, 1985b; Ju, 1997).
−
∂
∂λ
E
e
−
λ
y(T)
= −B(0,T )
×
2
κθ
2
γ
e
−
γ
T
+
κ
1
(1−e
−
γ
T
)
4
γ
3
e
(
κ
−
γ
)T
γ
e
−
γ
T
(
κ
1
−1)+
κ
1
(1−
γ
T)(1 −e
−
γ
T
)+
γ
−
r
1 −e
−
γ
T
γ
+ 0.5
κ
2
(1−e
−
γ
T
)
−
λ
r
σ
2
γ
Te
−
γ
T
−0.5(1−e
−2
γ
T
)
γ
[
γ
+ 0.5
κ
2
(1−e
−
γ
T
)]
2
(10.92)
with
κ
1
=
κ
+
γ
,
κ
2
=
κ
−
γ
.
Case 2: A mean-reverting Ornstein-Uhlenbeck process (Vasicek, 1977).
−
∂
∂λ
E
e
−
λ
y(T)
= −B(0, T )
σ
2
κ
3
−
θ
κ
(e
−
κ
T
−1)−
λσ
2
2
κ
3
(e
−2
κ
T
−1)+
λ
−2
r
κ
.
(10.93)
10.5 Correlation Options 257
By using these two results, we can finally compute f
1
(
φ
). We obtain immediately
F
j
=
1
2
+
1
π
∞
0
Re
f
j
(
φ
)
−i
φ
K
0
T
i
φ
d
φ
, j = 1,2. (10.94)
The put option price is given by using the put-call parity,
Put
AIR
= B(0,T )K
0
F
1
−
V(T )
T
F
2
(10.95)
where
F
j
=
1
2
−
1
π
∞
0
Re
f
j
(
φ
)
−i
φ
K
0
T
i
φ
d
φ
, j = 1,2. (10.96)
The expressions of (10.87) and (10.95) possess a clear structure and the hedge-
ratios can be derived analytically. In comparison with Ju’s expression in which F
1
and F
2
are put together in a single integral form, this pricing formula can be un-
derstood more easily and interpreted more meaningfully. All of these features are
of practical and theoretical importance. Some extensions can also be made, for ex-
ample, for interest rates with stochastic volatility. (Longstaff and Schwarz, 1992).
However, such an extension would be technically rather tedious and cause some
difficulties in implementation.
10.5 Correlation Options
10.5.1 Introduction
Accompanied by a radical development of derivative markets worldwide in the past
decades, more and more trading activities involving different markets or different
products raise an increasing demand for effective new financial instruments for the
purpose of hedging and arbitrage. To meet these needs, correlation options are gen-
erated as a generic term for a class of options such as exchange options, quotient
options and spread options. Usually, the payoffs of correlation options are affected
by at least two underlying assets. Hence, the correlation coefficients among these
underlying assets play a crucial role in the valuation of correlation options and indi-
cate the nomenclature of such particular options. Among correlation options, spread
options have been introduced in several exchanges as their official financial prod-
ucts while quotient options and product options are mainly provided and traded in
the OTC markets. The New York Mercantile Exchange (NYMEX) launched an op-
tion on a crack (fuel) futures spread on October 7, 1994. The New York Cotton
Exchange (NYCE) and the Chicago Board of Trade (CBOT) proposed options on
the cotton calendar spread and on the soybean complex spread, respectively.
While correlation options are drawing more and more attention in risk manage-
ment, their valuation presents a great challenge in finance theory. Particularly, the
258 10 Exotic Options with Stochastic Volatilities
valuation of spread options is a proxy for the complexity of this pricing issue. The
earliest version of the pricing formula for spread options is a simple application of
the Black-Scholes formula by assuming that the spread between two asset prices fol-
lows a geometric Brownian motion, and is then referred to as one-factor model. The
limitations and problems of this extremely simplified model have been discussed by
Garman (1992). The two-factor model where the underlying assets follow two dis-
tinct geometric Brownian motions, respectively, and the correlation between them
is also permitted, requests calculating an integral over the cumulative normal dis-
tribution. Shimko (1994) developed a pricing formula for spread options incorpo-
rating stochastic convenience yield as an enhanced version of a two factor model.
Wilcox (1990) applied an arithmetic Brownian motion to specify the spread move-
ment. The resulting pricing formula, however, is inconsistent with the principle of
no arbitrage. Recently, Poitras (1998) modeled the underlying asset as an arithmetic
Brownian motion and constructed three partial differential equations (PDE) for two
underlying assets and their spread respectively to avoid the arbitrage opportunities
occurring in the Wilcox model. But the drawback of specifying asset price process
as an arithmetic Brownian motion still remains in his model. Hence, we are falling
in a dilemma: specifying asset prices as a geometric Brownian motion leads to dou-
ble integration in the pricing formula, while modeling asset prices as an arithmetic
Brownian motion makes it feasible to obtain a tractable pricing formula, but it is not
coherent with standard models for options. Exchange options can be considered as
a special case of spread options where the strike price is set to be zero. Due to this
contractual simplification, a simple pricing formula `alaBlack-Scholes for exchange
options can be derived (Margrabe, 1978). Product options and quotient options are
two other examples of correlation options and their valuations in a two-factor model
present no special difficulty by applying the bivariate normal distribution function.
However, it has not yet been studied how one can price them in an environment of
stochastic volatilities and stochastic interest rates.
In this section, we attempt to apply the Fourier transform technique to obtain
alternative pricing formulas for exchange options, product options and quotient op-
tions. The favorite feature of the Fourier transform is that it presents not only an
elegant pricing formula allowing for a single integration, but it also accommodates a
general specification of state variables, for example, stochastic volatilities, stochas-
tic interest rates and even jump components. By employing the martingale approach,
we can construct some new types of equivalent martingale measures that ensure the
absence of arbitrage in the risk-neutral valuation of quotient or product options.
However, as in the Black-Scholes world, we can not give a simple tractable pricing
formula for spread options incorporating stochastic volatility and stochastic interest
rates.
To incorporate stochastic volatilities into correlation options, we specify the asset
price process with two independent Brownian motions with stochastic volatilities
simultaneously, that is, the asset price processes take the following form,
10.5 Correlation Options 259
dS
j
(t)
S
j
(t)
= r(t)dt +
σ
j1
V
1
(t)dW
1
(t)+
σ
j2
V
2
(t)dW
2
(t), j = 1, 2. (10.97)
Setting X
1
(t)=ln[S
1
(t)/S
1
(0)] and X
2
(t)=ln[S
2
(t)/S
1
(0)] yields
dX
j
(t)=
r(t) −
1
2
σ
2
j1
V
1
(t) −
1
2
σ
2
j2
V
2
(t)
dt
+
σ
j1
V
1
(t)dW
1
(t)+
σ
j2
V
2
(t)dW
2
(t) (10.98)
with X
1
(0)=0 and X
2
(0)=0. For the purpose of demonstrating how to model the
stochastic volatilities for exchange options, we adopt the Heston model and let the
variance V
j
(t) follow a square root process,
dV
j
(t)=
κ
j
[
θ
j
−V
j
(t)]dt +
σ
j
V
j
(t)dZ
j
(t), j = 1, 2. (10.99)
dW
1
(t)dW
2
(t)=0, dW
1
(t)dZ
1
(t)=
ρ
1
dt, dW
2
(t)dZ
2
(t)=
ρ
2
dt. (10.100)
For an Ornstein-Uhlenbeck process we can proceed in a same manner. The instan-
taneous correlation coefficient between dS
1
/S
1
and dS
2
/S
2
is therefore stochastic
and given by
ρ
(t)=Corr
dS
1
(t)
S
1
(t)
,
dS
2
(t)
S
2
(t)
=
σ
11
σ
21
V
1
(t)+
σ
12
σ
22
V
2
(t)
σ
2
11
V
1
(t)+
σ
2
12
V
2
(t)
σ
2
21
V
1
(t)+
σ
2
22
V
2
(t)
. (10.101)
The uncertainty of
ρ
(t) is caused by the variances V
j
(t). Since V
j
(t) 0, the pa-
rameters
σ
ij
determine whether these two underlying assets are correlated positively
or negatively. By setting V
j
(t) to be deterministic, i.e., the values of
θ
j
,
σ
j
and
κ
j
are nil, we obtain the usual two-factor model. In this case, the processes in (10.97)
are simplified to be
dS
j
(t)
S
j
(t)
= r(t)dt +
σ
j1
dW
1
(t)+
σ
j2
dW
2
(t)
= r(t)dt +
η
j
dW
∗
j
(t), j = 1, 2, (10.102)
where
η
j
=
σ
2
j1
+
σ
2
j2
and the new Wiener processes W
∗
j
(t) are composed of W
1
(t)
and W
2
(t) with the different weights such that dW
∗
1
(t)dW
∗
2
(t)=
ρ
dt. Therefore the
model in (10.98) is more extensive than an usual two-factor model.
260 10 Exotic Options with Stochastic Volatilities
10.5.2 Exchange Options
A European-style exchange option entitles its holder to exchange one asset for an-
other and can be considered as an extended case of a standard option. To value this
type of options, we assume that both assets are risky and follow a geometric Brow-
nian motion. Thus, the strike price is no longer deterministic. In fact, this is the
only point which distinguishes an exchange option from a plain vanilla option. Mar-
grabe (1978) developed a pricing formula for exchange options in the Black-Scholes
framework. Although exchange options are not traded on organized exchanges, they
are implied in many financial contracts, for example, performance incentive fee,
margin account and dual-currency option bonds etc. The valuation of exchange op-
tions is therefore a basis for understanding and valuing these financial contracts. The
payoff of exchange options at expiration is given by
C(T )=
S
1
(T) −S
2
(T) if S
1
(T) > S
2
(T)
0ifS
1
(T) > S
2
(T)
. (10.103)
Obviously, exchange call options on asset 1 for asset 2 are identical to exchange
put options on asset 2 for asset 1. If these two assets are stochastically dependent
on each other, then valuation of exchange options is linked to a joint distribution of
two assets. Additionally, with the price process of one asset, say S
2
, being reduced
to be deterministic, exchange options degenerate to the corresponding standard op-
tions. According to Margrabe (1978), the pricing formula `alaBlack-Scholes can be
expressed as:
C = S
1
N(d) −S
2
N(d −v
√
T) (10.104)
with
d =
ln(S
1
/S
2
)+
1
2
v
2
T
v
√
T
, v =
v
2
1
+ v
2
2
−2
ρ
v
1
v
2
,
where v
i
,i = 1, 2, represents the volatility of the i-th asset and
ρ
is their correlation
coefficient. The purpose of this subsection is to develop a model for exchange op-
tions capturing stochastic volatilities. On the analogy of the standard option pricing
formulas, we can rewrite the pricing formula for exchange options as:
C(S
1
,T;S
2
)=S
1
F
1
(S
1
(T) S
2
(T)) −S
2
F
2
(S
1
(T) S
2
(T))
= S
1
F
1
ln
S
1
(T)
S
2
(T)
0
−S
2
F
2
ln
S
1
(T)
S
2
(T)
0
= S
1
F
1
X
1
(T) −X
2
(T) ln
S
2
(0)
S
1
(0)
− S
2
F
2
X
1
(T) −X
2
(T) ln
S
2
(0)
S
1
(0)
.
(10.105)