2.2 Fixed strike Asian option of a continuous arith-
a riskless bond everytime interval (τ, τ + ∆τ) lapses. The total number of units of asset required from t toT for such transfer is
1 T −T0
Z
t T
e−r(T−t)dτ = 1−e−r(T−t)
r(T −T0) . (2.10)
By the no-arbitrage principle, the value of the option is equal to the value of the replicating portfolio. The option value for A(t) ≥ K is given by (Kemma & Vorst 1990, German and Yor 1993) is
V(S, A, t) = (A(t)−K)e−r(T−t)+ 1−e−r(T−t)
r(T −T0) S, A(t)≥K. (2.11) Note that σ does not appear explicitly in the formula, it appears implicity inS and Aand gamma is zero and delta is a function of time only. For A(t)≤K, the option value is governed by equation (2.8), but there is no closed form analytical solution available. The option value can be obtain by the numerical scheme called the finite difference method using the boundary condition.
V(0, A, t) = max(e−r(T−t)(A(T)−K),0) (2.12)
S→∞lim
∂V
∂S(S, A, t) = T −t T −T0
e−r(T−t) (2.13)
V(S, K, t) = 1 T −T0
1−e−r(T−t)
r S.
The last boundary is obtained by setting A = K in (2.11). Since equation (2.8) resembles a two-dimensional convection-diffussion equation but with the diffusion term missing in one of the spatial dimensions, we may encounter severe oscillations in the finite difference solution. Monte Carlo simulations can also be used as an alternative numerical approach. We will now look at the finite difference method and at the approximation methods by Levy, Turnbull & Wakeman and Monte Carlo method.
The finite difference method Consider a PDE equation (1.6) for arithmetic average
∂V
∂t +σ2
2 S2∂V2
∂S2 + (rS−q(S, t))∂V
∂S +S∂V
∂I −rV = 0. (2.14) To price Asian options, we must solve the above PDE, with appropriate final and boundary conditions. We are looking for a numerical solution of the form
Vi,jk ≈V(k∆t, i∆S, j∆I).
The PDE can be solved backward using the finite difference scheme Vi,jk+1−Vi,jk
∆t + (1−θ)(σ2
2 SiVi+1,jk+1 −2Vi,jk+1+Vi−1,jk+1
∆S2 +rSiVi+1,jk+1 −Vi,jk+1
∆S )
+(1−θ)(Si−3Vi,jk+1+ 4Vi,j+1k+1 −Vi,j+2k+1
2∆I −rVi,jk+1) +θ(σ2
2 Si
Vi+1,jk −2Vi,jk +Vi−1,jk
∆S2 +rSi
Vi+1,jk −Vi,jk
∆S )
+θ(Si−3Vi,jk + 4Vi,j+1k −Vi,j+2k
2∆I −rVi,jk) = 0. (2.15)
At each time level the 2-Dimensional problem can be decoupled into a series of 1D problems, by approximating V line by line, in the direction of a decreasing j index.
A reduced one-dimensional PDE derived by Rogers and Shi:
∂H
∂t +σ2
2 R2∂2H
∂R2 −(1
T +rR)∂H
∂R = 0 (2.16)
H(T, R) = max(−R,0),for a fixed call option
= max(R,0),for a put option where
R = I/S
V(t, I, S) = SH(t, K− tI T /S).
Since this PDE depends on the maturityT, this cannot be applied to an American options. Only two-dimensional can be used to solve American fixed strike options.
2.2.1 Arithmetic Rate Approximation (Turnbull and Wake- man)
Turnbull & Wakeman (T&W)’s approximation makes use of the fact that the distri- bution under arithmetic averaging is approximately lognormal. It adjust the mean and the variance in order to be consistent with exact moments of the arithmetic averages. They put forward the first and second moments of the average in order to price the option. The mean and variance are used as inputs in the general Black Scholes formula. The value of a call option is given as
C =Se(b−r)T2N(d1)−Ke−rT2N(d2) and the put option is
P =Ke−rT2N(−d2)−Se(b−r)T2N(−d1) where N(x) is the cumulative
d1 = ln(S/K) + (b+σ22A)T2 σ√
T d2 = ln(S/K) + (b− σ22A)T2
σ√
T =d1−σA√
T (2.17)
where T2 is the time remaining until the maturity. For averaging options which have already begun their averaging period, then T2 is simplyT the original time to maturity, if the averaging period has not yet begun, then T2 is T −τ.
σA =
rln(M2) T −2b.
The adjusted volatility and dividend yield are given as b= ln(M1)
T .
Assume that the averaging period has not yet begun, the first and the second mo- ments are given as
M1 = e(r−q)T −e(r−q)τ (r−q)(T −τ) , M2 = 2e(2(r−q)+σ2)TS2
(r−q+σ2)(2r−2q+σ2)T2 + 2S2 (r−q)T2
1
2(r−q) +σ2 − e(r−q)T r−q+σ2
. We adjust the strike price if the averaging period has already begun as
KA= T
T2K−(T −T2) T2 SA
where T is reiterated as the original time to maturity, T2 as the remaining time to maturity, K as the original strike price and SA is the average asset price. By Haug (1998) if r=q, the formula will not generate a solution.
2.2.2 Arithmetic Rate Approximation (Levy)
This analytical approximation is suggested to give more accurate result than Turn- bull and Wakeman approximation. The analytical approximation to a call option is
C ≈SzN(d1)−Kze−rT2N(d2) and the put option is
P ≈C−Sz+Kze−rT2 where
d1 = 1
√
K[ln(L)
2 −ln(Kz)]
d2 =d1−√
K (2.18)
and
Sz = S
(r−q)T(e−qT2 −e−rT2)
Kz =K −SA
T −T2
T
K = ln(L)−2[rT2+ ln(SZ)]
L= M T2 M = 2S2
r−q−σ2
"
e(2(r−q)+σ2)T2−1 2(r−q) +σ2
#
− e(r−q)T2 −1 r−q .
The price of an Asian call under Turnbull and Wakeman was compared to that of Levy’s approximation. Given the following input:
Asset Price= 100, Average Price= 95, q=5%,r=10%,V=15%,T=0, T1 = 1, T2 = 0.5 andK is the strike price.
TABLE 2.1
K T W Levy Absolute Error
95 3.202859 3.199390 0.0034690 96 2.444752 2.440545 0.0042066 97 1.787605 1.782873 0.0047318 98 1.246971 1.242086 0.0048849 99 0.827130 0.822518 0.0046122 100 0.520494 0.516509 0.0039841 101 0.310270 0.307114 0.0031558 102 0.175088 0.172788 0.0022995 103 0.093529 0.091982 0.0015470 104 0.047316 0.046352 0.0009645 105 0.022689 0.022130 0.0005593
The values are said to be similar since, the absolute differences between these two approximations are very small.
2.2.3 Monte Carlo Simulation
Monte Carlo Simulation is convenient and flexible and is very useful to Asian op- tions which are highly path dependent. It is applicable as long as the underlying follows a Markovian-diffusion. Various methods using Monte Carlo simulation have been developed to price Arithmetic Asian option. For example Levy, (Turnball and Wakeman) and Curran. In the next section we will see an approximation analytical solution for geometric closed form by Kemma and Vorst (1990). This solution has been shown by authors that it can be used as control variate within a Monte Carlo simulation framework. The control variate technique can be used to find more ac- curate analytical solution to a derivative price if there is a similar derivative with a known analytic solution. The arithmetic Asian option can be price using Monte Carlo simulation. It can give a relatively accurate prices for option values. If given the price of a geometric Asian, then one can price the arithmetic Asian option by the equation below.
VA=E( ˜VA−V˜B) +VB (2.19) where ˜VAis the estimated value of the arithmetic Asian through simulation, ˜VBis the simulated value of the geometric Asian, and VB is the exact value of the geometric Asian.