O
S
K
URTOSIS IN THE
U
NDERLYING
D
ISTRIBUTION
KENG-HSIN LO KEHLUH WANG MING-FENG HSU*
Numerical valuation model is extended for European Asian options while considering the higher moments of the underlying asset return distribution. The Edgeworth binomial lattice is applied and the lower and upper bounds of the option value are calculated. That the error bound in pricing Asian options from the Edgeworth binomial model is smaller than the error bound model by Chalasani et al. is shown. The approach is used to price the average rate currency option with different skewness and kurtosis. The numerical results show that this approach can effectively deal with the higher moments of the underlying distri-bution and provide better estimates of option value compared with various stud-ies in literature. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:598–616, 2008 The authors thank the anonymous reviewers and the editor of this journal for their valuable comments and suggestions. Any remaining errors are the authors’ own.
*Correspondence author, Department of Information Management, National Penghu University, 300 Liu-Ho Road, Makung City, Penghu County, Taiwan. Tel: 6-9264115 x3413, e-mail: [email protected]
Received November 2006; Accepted October 2007
■ Keng-Hsin Lo is an Associate Professor at the Department and Graduate School of Business Administration, National Central University, Jhongli City, Taoyuan County, Taiwan.
■ Kehluh Wang is an Associate Professor at the Graduate Institute of Finance, National Chiao Tung University, Hsinchu, Taiwan.
■ Ming-Feng Hsu is a Ph.D. Candidate at the Graduate School of Business Administration, National Central University, Chung-Li, Taiwan, and a Lecturer at the Department of
Information Management, National Penghu University, Makung City, Penghu County, Taiwan.
The Journal of Futures Markets, Vol. 28, No. 6, 598–616 (2008) © 2008 Wiley Periodicals, Inc.
INTRODUCTION
The payoff of an Asian option depends on the arithmetic or geometric price average of the underlying asset during the life of the option. Its value is path dependent, normally without the closed-form solution, and therefore more dif-ficult to calculate than that of a standard option. However, the hedging effect of an Asian option, which is specifically widely used in the foreign exchange market, is better than that of a standard option and offers convenience and lower cost (Hu & Yu, 2000). Nielsen and Sandmann (2003) reported that the open interest of Asian options is in the range of 5–10 billion U.S. dollars on the over-the-counter market.
Many researchers have applied various methods to analyze Asian options: Haykov (1993), Boyle (1977), Grant, Vora, and Weeks (1997), and Kemna and Vorst (1990) have employed Monte Carlo (MC) simulations with variance reduction techniques; Hull and White (1993), Neave and Ye (2003), Chalasani, Jha, and Varikooty (1998), Chalasani, Jha, Egriboyun, and Varikooty (1999), and Reynaerts, Vanmaele, Dhaene, and Deelstra (2006) developed binomial trees, or lattices, with different efficiency enhancements; Dewynne and Wilmott (1995), Rogers and Shi (1995), and Alziary Decamps, and Koehl (1997) applied the partial differential equation approaches; Curran (1994) and Nielsen and Sandmann (1996, 2002, 2003) used general numerical methods; Geman and Yor (1993), Kramkov and Mordecky (1994), and Chacko and Das (1997) applied pseudo-analytic characterizations; and Turnbull and Wakeman (1991) (TW), Levy (1992), Vorst (1992), and Bouaziz, Briys, and Crouhy (1994) all have employed analytic approximations that produce closed-form expressions.
Most of the valuation methods for Asian options assume that the return distribution of the underlying asset is lognormal. However, practitioners and academics are well aware that the finite sum of the correlated lognormal ran-dom variables is not lognormal. It is for this reason that some researchers have tried to investigate other alternatives by considering the number of moments.
TW and Levy (1992) had applied the first two moments to price the aver-age rate currency options and obtained reasonable approximations under low-volatility conditions. They had suggested using higher moments when low-volatility is high. Milevsky and Posner (1998) had used the fundamental method to derive the probability density function of the infinite sum of the correlated lognormal random variables and proved that it is a reciprocal gamma distribu-tion under certain parameter restricdistribu-tions. Fusai and Tagliani (2002) had also used moments to evaluate fixed exercise Asian options and showed that the density of the logarithm of the arithmetic average was uniquely determined. They had verified that entropy decreases significantly when the fourth moments are used, and their approximation is good at low-volatility levels. However, error increases for higher volatility and more moments may be required.
As return distributions in the currency market are usually not normal (Kearns & Pagan, 1997; Tucker & Pond, 1988), incorporating higher moments in the valuation of an Asian currency option should provide better results. In this study, the model developed by Chalasani et al. (1998) is extended for the valuation of European Asian options while considering the higher moments of the underlying asset return distribution. The Edgeworth binomial lattice (Rubinstein, 1998) is applied and the lower and upper bounds of the option value are calculated. The approach is used to price the average rate currency option with different skewness and kurtosis.
When the first two moments are used, the authors’ model obtains a better value for an Asian option with low volatility than those of Levy (1992), Rogers and Shi (1995), and Chalasani et al. (1998). If four moments are used, the authors’ model can provide satisfactory estimates for high-volatility Asian options comparing the results from the discrete Wilkinson approximation, the four-moment approximation, and the MC method.
DEFINITIONS AND THE BASIC BINOMIAL MODEL
An underlying variable, S(t), of an option at time t is generally assumed to sat-isfy the stochastic differential equation in a risk-neutral world:
where the drift m and volatility s are constant, and {B(t)} denotes a Brownian motion process. Assume that the risk-free interest rate r is a constant, and that the option expires at time T.
A binomial tree (Cox, Ross, & Rubinstein, 1979) can approximate the continuous-time function S(t), where one divides the life of the option into n time steps of length t T/n. In each time step, the underlying asset may move up by a factor u with probability pc, or down by a factor d u1 with
probability qc 1 pc, with 0 d 1 u. Firstly, the one-period case is
con-sidered, i.e. time step k 1. The stock price at the end of the period will have two possible values, either up to a value S(0)u with probability pcor down to a
value S(0)u1 with probability 1 pc. These price movements can be
repre-sented in the following diagram:
S(0)
S(0)u1.
S(0)u,
Now consider a call option with two periods (k 2) before its expiry date. The price process of the stock will show three possible values after two periods:
This price process of the stock can be extended to n time steps.
The stochastic differential equation describing this price process, i.e. dS(t) mS(t)ldt sS(t)ldB(t), has the following solution:
where is a standardized normal random variable.
For a binomial random walk to have the correct drift over a time period of t, the following is needed:
namely, pcu (1 pc)d emt. Rearranging this equation the following can be
obtained:
with
Here, let n be a sample space of an experiment including all possible sequences of n upticks and downticks. A typical element of n is presented as v v1, v2. . . vn, where videnotes the ith uptick or downtick. Let {Hk(v)} be an associated family of random variables, where Hk(v) denotes the number of upticks at time k and H0(v) 0 for all v. A symmetric random walk Xkcan be defined, such that for each k 1, Xk Hk (k Hk) 2Hk k, which rep-resents the number of upticks minus the number of downticks up to time k. It is used to define the nodes in a binomial lattice corresponding to the possible positions of the underlying random walk at different times. Specifically, a tree path v is displayed to pass through or reach node (k, h) if and only if Hk(v) h
u es2¢t. pc em¢t d u d pcSu (1 pc)Sd SE[e(m(12)s 2)¢ts£2¢t ] Sem¢t S(t) S(0)e(m(12)s2)ts£2t S(0) S(0), S(0)u1 S(0)u S(0)u2, S(0)u2.
for times k 0,1, . . . ,n and the number of possible upticks h 0,1, . . . ,k. Consequently, the underlying asset price at time k is Sk(k 0,1,. . .,n), where
For example, S3 S0u(223) S0u at node (3, 2) in the lattice diagram of Figure 1. The underlying asset price at node (k, h) is given by S0u2hk, whose average at time k is defined as Ak (S0 S1 c Sk)/(k 1), k 0. Therefore, the payoff of an Asian call with strike price L at time n is Vn (An L) max{An L, 0}. The price of this option is the expected present value discounted to time 0, C E[Vn]/(1 r)n. Note that E[V
n] is a
probabil-ity-weighted average given by kPk(Ak L), where Pkdenotes the risk-neutral probability associated with Akat the expiration date.
EDGEWORTH BINOMIAL MODEL FOR ASIAN OPTION VALUATION
To consider the higher moments, the Edgeworth binomial tree model is first applied (Rubinstein, 1998). Assume that the tree has n time steps and n 1 nodes (h 0,1, . . . , n) at step n. At each node h, there is a random variable yh
[2h n]/n1/2with a standardized binomial density b(y
h) [n!/h!(n h)!](1/2)n.
Giving predetermined skewness and kurtosis, the binomial density is transformed by the Edgeworth expansion up to the fourth moment. The result is
c 1 1 6 g1(g 3 h 3yh) 1 24(g2 3)(y 4 h 6y2h 3)
F(yh) f(yh) b(yh)
Sk S0uXk S0u2Hkk. FIGURE 1
A binomial tree. Node (3, 2) means there are two upticks in any path reaching this node at time 3.
S(0) S0u S0u2 S0u3 S0u S0u1 S0u3 S0u1 S0u2 S0 (0,0) (1,0) (1,1) (2,0) (2,1) (2,2) (3,3) (3,2) (3,1) (3,0)
(1) with f(yh) [1 (1/6)g1(yh3 3yh) (1/24)(g2 3)(yh4 6yh2 3)
(1/72)g1(yh6 15yh4 45yh2 15)], where g1 EQ[yh3] is the skewness and g2
EQ[y
h4] is the kurtosis of the underlying distribution under risk-neutral
meas-ure. Although the sum of F(yh) is not one, F(yh) is normalized by F(yh)/jF(yj)
and denoted as Ph.
The variable yh, which has probability Ph, can be standardized as xh (yh M)/V with M hPhyhand V2 hPh(yh M)2. The variable xhis used later in
Equation (2) to obtain the asset price and the corresponding risk-neutral prob-ability, Ph, for a path to node h.
Consider a tree model of n steps. The asset price at the hth node (h 0, 1, . . . ,n) during the final step, , is
(2) with where S0is the initial asset price, r is the continuously compounded annual risk-free rate, T is the time for expiration of the option (in years), s is the annualized volatility rate for the cumulative asset return, and xhis a random variable from probability distribution Phwith mean 0 and variance 1. Ph is determined by modifying the binomial distribution using the Edgeworth expansion up to the fourth moment of . Finally, m is used to ensure that the expected risk-neutral asset return equals r. Solving backward recursively from the end of the tree, the nodal value, Sn1,h, is
(3) with pe pn,h1/(pn,h1 pn,h) and qe (1 pe), where pn,his Ph/[n!/h!(n h)!].
The path dependence of Asian options is analyzed using the approach by Chalasani et al. (1999). To represent the refined binomial lattice, a new ran-dom variable Wk,hdenoting an area at time k is assigned. Its initial value W0is
zero. For any node (k,h) in the tree, a lowest path reaching (k, h) is defined as the path with k h downticks followed by h upticks, and a highest path reach-ing (k, h) means the one with h upticks followed by k h downticks. The area Wk,h(v) of a path v reaching (k, h) can be defined as the number of
diamond-shaped boxes enclosed between this path v and the lowest path reaching this node. For example, the node (5, 2) means that the paths reaching it have two upticks at time 5. As demonstrated in Figure 2, a path passing through (5, 2) and reaching node (6, 2) is shown by the thick line segments. The area W6,2(v)
Sn1, h [peSˆn,h1 qeSˆn,h]expa rT n b ln(Sˆn,hS0) m r (1T)ln gnh0 Phe s2Txh, Sˆn,h S0emTs2Txh Sˆn,h a1 2b n c n! h!(n h)!d 1 72 g1(y 6 h 15y4h 45y2h 15)d
of this path is the number of diamond-shaped boxes, contained between this path and the lowest path reaching node (6, 2), as shown by the shaded area in the graph. The maximum area of any path reaching (k, h) is the number of boxes between the highest and the lowest paths reaching (k, h), that is, h(k h). The minimum area of any path reaching (k, h) is zero. The set of possible areas of paths reaching node (k, h) is therefore {0,1, . . . ,h(k h)}. Each node of the binomial lattice can be partitioned into “nodelets” based on the areas of the paths reaching this node. Therefore, any path reaching a given nodelet (k, h, a) has an area Wk,h(v) a with h upticks at time k. For instance, Figure 3 shows the nodelets in the nodes (5, 2), (6, 3), and (6, 2). As noted in Chalasani et al. (1999), there is a one–one correspondence between the possible areas and the possible geometric averages of underlying asset prices for paths reaching (k, h). Therefore, (k, h, a) represents all the paths in the binomial tree that reach node (k, h) and has the same geometric average asset price from time 0 to k.
Suppose the area of a path A reaching (k, h) is Wk,h(A) a. If A has an uptick after this point, it reaches node (k 1, h 1) at the next time step. The path A and the lowest path B reaching (k 1, h 1) share the same edge linking (k, h) and (k 1, h 1) in the lattice. Hence, the number of boxes between A and B at time k 1 is the same as the number at time k. In this way the path A reaches nodelet (k 1, h 1, a). On the other hand, if A has a downtick after time k, it will reach node (k 1, h). In this case, the number of boxes at time k 1 between A and the lowest path reaching (k 1, h) will be increased by h to get a h. The path A then reaches nodelet (k 1, h, a h).
FIGURE 2
A binomial lattice. An example of diamond-shaped boxes for node (6, 2) is shown as the shaded area.
(6,3)
(5,2) (6,2)
How the arithmetic average of underlying asset prices over all paths reach-ing (k, h, a) is computed is shown and denoted by ¯¯A (k, h, a) E[AkHk h, Wk,h(v) a], k 0,1,. . .,n, h k. It is simply the average of Ak over these paths. So the arithmetic average of stock prices over all paths reaching nodelet (k, h, a) can be expressed as
(4)
where , , with k 0,1,. . .,
n, h 0,1,. . .,k, a 0,1,. . .,h(k h), m 1,2,. . .,M(k, h, a), and M(k, h, a) is the number of paths reaching (k, h, a) with M(0,0,0) 1. Here, S (k, h, a) is the sum of Sm(k, h, a) over all paths passing through (k, h, a) with S (0, 0, 0)
Sm(k, h, a) g K i0Si,h S–(k, h, a) gMm1Sm(k, h, a) A(k, h, a) S–(k, h, a) (k 1)M(k, h, a) FIGURE 3
Nodelets in the nodes (5, 2), (6, 3), and (6, 2) as circled in Figure 2. This figure exhibits the number of paths M(k, h, a) reaching each nodelet (k, h, a). An example is shown for nodelet
(5, 2, 2), which is updated as in the nodelets (6, 3, 2) and (6, 2, 4).
Node (k, h) (5,2) Area a M(k, h, a) 0 1 1 1 2 2 3 2 4 2 5 1 6 1 Node(k1, h1) (6,3) Area a M(k1, h1, a) 0 1 1 1 2 2 3 3 4 3 5 3 6 3 7 2 8 1 9 1 Node (k1, h) (6,2) Area a M(k1, h, ah) 0 1 1 1 2 2 3 2 4 3 5 2 6 2 7 1
S0, whereas Sm(k, h, a) is the sum of the asset prices along any possible path
passing through (k, h, a) from time 0 to k.
Any path passing through nodelet (k, h, a) and having an uptick will get to nodelet (k 1, h 1, a) at time k 1. Thus, the number of paths reaching nodelet (k 1, h 1, a), namely, M(k 1, h 1, a), should include M(k, h, a) paths through (k, h, a). The sum of the prices from all the paths reaching (k 1, h 1, a), namely, S (k 1, h 1, a), would be S (k, h, a) M(k, h,
a)Sk1,h1 for paths passing through (k, h, a). Likewise, all paths passing
through nodelet (k, h, a) with a downtick will reach nodelet (k 1, h, a h) at time k 1. Similarly, M(k 1, h, a h) should also include M(k, h, a) and S (k 1, h, a h) would be S (k, h, a) M(k, h, a)Sk1,hin the forward induc-tion process.
Next, the authors present how the value of an Asian option after obtaining the arithmetic average of the stock prices from Equation (4) is estimated. The approach used by Rogers and Shi (1995) is applied, where the lower bound and the error bound are calculated for the price of an Asian option. This lower bound on the price of an Asian call option with strike L can be expressed as
(5) where Z (Wn,h, Sn,h) in which the random variable Wn,hdenotes the area at node (n, h), and Sn,h represents the stock price reaching (n, h). The composi-tion in the lower bound, E[AnWn,h, Sn,h], can be expressed as ¯¯A (n, h, a), as in Equation (4). ¯¯A (n, h, a) is the expectation of the average stock price Anat node (n, h), where An (S0 S1,h c Sn,h)/(n 1) on a tree path passing through (n, h, a). All paths through this nodelet have the same probability P(Wn,h, Sn,h), which is M(n, h, a)pehq
enh. Thus, the lower bound can be calculated as
As a result, the error bound is
(6) 1 2 E[[var(An L0Wn,h, Sn,h)] 12] E[E[(An L)0Wn,h, Sn,h] E[(An L)0Wn,h, Sn,h]] E[E[(An L)0Wn,h, Sn,h]] E[E[(An L)0Wn,h, Sn,h]] an h0 a h(nh) a0 M(n, h, a)ph eqneh[A(n, h, a) L]. Cd 0 E[[E(An0Wn,h, Sn,h) L]] a n h0 a h(nh) a0 P(Wn,h, Sn,h)[A(n, h, a) L] E[E[(An0Z) L]]
assuming Vnmin(Z) 0 and V
nmax(Z) 0, where Vn An L.1Note that var(An
L) varlAn EA2n (EAn)2and ¯¯A
2(n, h, a) E[A2
nWn,h, Sn,h]. Let Amin(k, h, a) denote the minimal value of Akand Amax(k, h, a) its maximum over all
paths passing through the nodelet (k, h, a). Thus, the error bound by Equation (6) equals
(7) The ¯¯A (n, h, a) can be derived from Equation (4). Meanwhile, Amin(k, h, a)
Smin(k, h, a)/(k 1) and Amax(k, h, a) Smax(k, h, a)/(k 1), where Smin(k, h, a)
and Smax(k, h, a) are, respectively, the minimum value and maximum value of Sk,hover these paths reaching (k, h, a). ¯¯A2(n, h, a) can also be calculated from the following:
where w(k, h, a) is the sum of and c(k, h, a) is the sum of
2With the lower bound and the error bound, the upper bound
can be obtained.
Suppose one upward probability pe, denoting the probability of the stock price moving up for the next step in the Edgeworth binomial tree, is lower than the other upward probability pc, the probability of the stock price moving up in the binomial tree of Chalasani et al. (1999). The average stock price in a path with upward drift causes higher probability of Amin(k, h, a) L, i.e. higher
probability of zero variance. So the total variance of the average stock price will g0ijk Si,hSij,h.
gk i0 Si,h2 A2(k, h, a) w(k, h, a) 2c(k, h, a) (k 1)2M(k, h, a) M(n, h, a)ph eqneh(A2(n, h, a) A(n, h, a)2)12. 1 2a n h0 a h(nh) a0
Amin(n,h,a)L,Amax(n,h,a) L
P(Wn,h, Sn,h)(A2(n, h, a) A(n, h, a)2)12 1 2a n h0 a h(nh) a0
Amin(n,h,a)L,Amax(n,h,a) L
1The minimum and maximum values of V
n over paths v with Z(v) z is set to be Vnmax(z)
maxv{Vn(v)Z(v) z} and Vnmin(z) minv{Vn(v)Z(v) z}. If Vnmax(zi) 0, then for all paths v with
Z(v) zi, Vn(v) 0 can be deduced, which implies E(Vnzi) 0, and also E(Vnzi) 0, which implies
E(Vnzi) 0. Hence, the error bound is zero. Similarly, if Vnmin(zi) 0, then for all paths v with Z(v) zi,
Vn(v) Vn(v) can be deduced, which implies E(Vnzi) E(Vnzi), and also E(Vnzi) 0, which implies
E(Vnzi) E(Vnzi). Therefore, the error bound is again zero.
2To show how is derived, can be
writ-ten. Because all paths reaching (k, h, a) have the same probability, ¯¯A2(k, h, a) is the average of over these paths.
A2
kaa k i0S
2
i,h 2 a0ijkSi,hSj,hb (k 1)2
(k 1)2A2 kaa k i0Si,hb 2 aki0S2
i,h 2 a0ijkSi,hSj,h
be smaller. According to Equation (7), the error bound of the option price with upward probability pewill be smaller than the error bound with probability pc. It can be shown in the following proposition that this can lead to tighter bounds on the error from approximating E[Vn] if its upward probability is lower. The details are explained in the Appendix.
Proposition: The error bound in pricing a European Asian option from the modified Edgeworth binomial model is tighter than the error bound from the model by Chalasani et al. (1999).
NUMERICAL RESULTS
Valuation of European Asian Options Under Normal Skewness and Kurtosis
Microsoft Visual C is used to program the authors’ algorithm. Considering first the normal skewness and kurtosis, the results are tested and compared with those in the literature. The call option to be valued has the initial stock price S0 100, the maturity T 1 year, and the strike prices L 95, 100, 105, and 110, respectively. The underlying distribution has volatility s 0.05, 0.1, and 0.3, respectively, with normal skewness g1 0 and kurtosis g2 3. The risk-free rate r is set to be 0.05, 0.09, or 0.15. The time step N equals 30, and the computing time and memory space needed in the authors’ algorithm are similar to those of Chalasani et al. (1998). The authors’ simulation results are presented in Tables I and II.
In Table I, the authors’ results are compared with those of Rogers and Shi (1995) and of Chalasani et al. (1998). When the call is in the money, the authors’ valuation in general is smaller than those of Chalasani et al. (1998). However, the range of the authors’ lower and upper bounds is narrower than theirs. For at-the-money and out-of-the-money calls, the authors’ estimates are greater than theirs and closer to those of Rogers and Shi’s, but the distance between the authors’ lower and upper bounds is almost the same as that of Chalasani et al. The difference between the authors’ calculations and those of Rogers and Shi’s is owing to the authors’ numerical approximation compar-ing with their continuous-time integrals.
In Table II, the authors’ results are compared with those of MC simula-tions from Levy and Turnbull (1992) (LT). Because Chalasani et al. (1998) claim that their results are closer to MC estimations than those of Roger and Shi (1995), the authors also list their bounds. As depicted in the table, the authors’ estimates are much closer to the results of MC simulations; hence, the authors’ algorithm in pricing Asian options performs better than that of Chalasani et al. (1998).
TABLE I
Model Comparisons for Asian Options Valuations Under Normal Skewness and Kurtosis
Strike L Vol. s r E-LB E-UB RS-LB RS-UB C-LB C-UB
95 0.05 0.05 7.177 7.177 7.178 7.183 7.178 7.178 100 0.05 0.05 2.712 2.712 2.716 2.722 2.708 2.708 105 0.05 0.05 0.332 0.332 0.337 0.343 0.309 0.309 95 0.05 0.09 8.811 8.811 8.809 8.821 8.811 8.811 100 0.05 0.09 4.306 4.306 4.308 4.318 4.301 4.301 105 0.05 0.09 0.957 0.957 0.958 0.968 0.892 0.892 95 0.05 0.15 11.100 11.100 11.094 11.114 11.100 11.100 100 0.05 0.15 6.799 6.799 6.794 6.810 6.798 6.798 105 0.05 0.15 2.745 2.745 2.744 2.761 2.667 2.667 90 0.10 0.05 11.947 11.947 11.951 11.973 11.949 11.949 100 0.10 0.05 3.635 3.635 3.641 3.663 3.632 3.632 110 0.10 0.05 0.319 0.320 0.331 0.353 0.306 0.306 90 0.10 0.09 13.385 13.385 13.385 13.410 13.386 13.386 100 0.10 0.09 4.909 4.909 4.915 4.942 4.902 4.902 110 0.10 0.09 0.621 0.621 0.630 0.657 0.582 0.583 90 0.10 0.15 15.404 15.404 15.399 15.445 15.404 15.404 100 0.10 0.15 7.024 7.024 7.028 7.066 7.015 7.015 110 0.10 0.15 1.411 1.412 1.413 1.451 1.316 1.317 90 0.30 0.05 13.928 13.936 13.952 14.161 13.929 13.938 100 0.30 0.05 7.924 7.932 7.944 8.153 7.924 7.932 110 0.30 0.05 4.041 4.051 4.070 4.279 4.040 4.049 90 0.30 0.09 14.961 14.968 14.983 15.194 14.964 14.972 100 0.30 0.09 8.811 8.818 8.827 9.039 8.807 8.815 110 0.30 0.09 4.672 4.682 4.695 4.906 4.661 4.671 90 0.30 0.15 16.494 16.500 16.512 16.732 16.499 16.506 100 0.30 0.15 10.197 10.205 10.208 10.429 10.187 10.195 110 0.30 0.15 5.715 5.725 5.728 5.948 5.685 5.696
Note. The European Asian option to be valued has initial stock price S0 100 dollars and option life T 1.0 year. Using time steps
N 30, the lower and upper bounds from the authors’ algorithm are indicated by E-LB and E-UB, respectively, whereas those from Rogers and Shi (1995) are indicated by RS-LB and RS-UB, and those from Chalasani et al. (1998) by C-LB and C-UB. The authors used normal skewness g1 0 and kurtosis g2 3 in their algorithm.
TABLE II
Comparisons with Monte Carlo Simulations Under Normal Skewness and Kurtosis
Strike L Vol. s r Monte Carlo E-LB E-UB C-LB C-UB
95 0.10 0.09 8.91 8.91 8.91 8.91 8.91 100 0.10 0.09 4.91 4.91 4.91 4.90 4.90 105 0.10 0.09 2.06 2.07 2.07 2.03 2.03 90 0.30 0.09 14.96 14.96 14.97 14.96 14.97 100 0.30 0.09 8.81 8.81 8.82 8.81 8.82 110 0.30 0.09 4.68 4.67 4.68 4.66 4.67 90 0.50 0.09 18.14 18.14 18.18 18.15 18.19 100 0.50 0.09 12.98 12.98 13.02 12.99 13.03 110 0.50 0.09 9.10 9.07 9.11 9.08 9.12
Note. The European Asian option to be valued has initial stock price S0 100 dollars and option life T 1.0 year. Using time steps
N 30, the lower and upper bounds from the authors’ algorithm are indicated by E-LB and E-UB, respectively, whereas Monte Carlo
estimates from Levy and Turnbull (1992) are indicated by Monte Carlo, and those from Chalasani et al. (1998) are indicated by C-LB and C-UB. The authors used normal skewness g1 0 and kurtosis g2 3 in their algorithm.
Tables I and II demonstrate the performance of the authors’ algorithm under normal skewness and kurtosis. In the next section, the Asian options under various non-normal skewness and kurtosis are priced and the results are compared with those in the literature.
Valuation of European Asian Options Under Various Skewness and Kurtosis
The valuation performance of the European Asian option in Table III is based on the initial stock price S0 100, the risk-free rate r 0.09, and the maturity T 1 year with varying skewness and kurtosis. The results of the numerical analysis are compared with those of the Edgeworth expansion model by TW, the modified Edgeworth expansion method by LT, and the four-moment approximation model by Posner and Milevsky (1998) (PM). All these models are considered up to the fourth moments in their valuations.
For low-volatility cases (s 0.05 and 0.1) in Table III, the authors’ results from in-the-money or at-the-money calls are very close to MC estimates, which is the benchmark used by LT under lognormal distribution. This is similar to those of LT and TW. For out-of-the-money calls, the authors’ outcomes are the same as theirs under right-skewed conditions. For high-volatility cases (s 0.3 and 0.5), the authors’ outcomes for at-the-money or deep-in-the-money calls approach the results from MC simulation with positive skewness and a slight
TABLE III
Model Comparisons for Asian Option Valuations Under Various Skewness and Kurtosis
Strike L Vol. s g1 g2 MC E-LB E-UB LT TW PM
95 0.05 0 3 8.81 (0.00) 8.81 8.81 8.81 8.81 NAN 100 0.05 0 3 4.31 (0.00) 4.31 4.31 4.31 4.31 NAN 105 0.05 0.03 3 0.95 (0.00) 0.95 0.95 0.95 0.95 NAN 95 0.1 0 3 8.91 (0.00) 8.91 8.91 8.91 8.91 NAN 100 0.1 0 3 4.91 (0.00) 4.91 4.91 4.91 4.91 NAN 105 0.1 0.02 3 2.06 (0.00) 2.06 2.06 2.06 2.06 NAN 90 0.3 0 3 14.96 (0.01) 14.97 14.98 15.00 14.91 14.96 100 0.3 0.01 3 8.81 (0.01) 8.80 8.82 8.84 8.78 8.80 110 0.3 0 3 4.68 (0.01) 4.68 4.70 4.69 4.69 4.67 90 0.5 0.01 3 18.14 (0.03) 18.14 18.21 18.13 17.66 18.14 100 0.5 0 3.02 12.98 (0.03) 12.97 13.03 13.00 12.86 12.97 110 0.5 0 3 9.10 (0.03) 9.09 9.16 9.12 9.22 9.07
Note. E-LB and E-UB indicate the lower and the upper bounds from the authors’ model with various skewness (g1) and kurtosis (g2).
The approximations of Levy and Turnbull (1992) are represented by LT, and of Turnbull and Wakeman (1991) by TW; MC represents the Monte Carlo estimates in the Levy and Turnbull (1992), and PM represents the four-moment approximation by Posner and Milevsky (1998). The simulations assume the option life T 1 year, the domestic interest rate r 0.09, the time steps N 52, and the initial spot price S0 100.
TABLE IV
Model Comparisons for Asian Currency Option Valuations Under Various Skewness and Kurtosis
Strike L Vol. s N g1 g2 MC E-LB E-UB Levy PM
1.8 0.3 4 0.15 3 0.0235 (0.00017) 0.0235 0.0235 0.0231 0.0234 1.65 0.3 4 0.15 3.04 0.0517 (0.00017) 0.0517 0.0517 0.0515 0.0516 1.5 0.3 4 0.10 3.01 0.1034 (0.00017) 0.1034 0.1034 0.1038 0.1034 1.35 0.3 4 0.02 3.02 0.1858 (0.00017) 0.1858 0.1858 0.1864 0.1858 1.2 0.3 4 0.03 3.08 0.2958 (0.00017) 0.2958 0.2958 0.2961 0.2958 1.8 0.3 12 0.03 3 0.0249 (0.00014) 0.0249 0.0250 0.0243 0.0249 1.65 0.3 12 0.03 3.03 0.0540 (0.00014) 0.0540 0.0540 0.0537 0.0538 1.5 0.3 12 0.04 3.01 0.1061 (0.00014) 0.1061 0.1061 0.1067 0.1061 1.35 0.3 12 0.01 3 0.1882 (0.00014) 0.1881 0.1882 0.1890 0.1880 1.2 0.3 12 0 3 0.2966 (0.00014) 0.2965 0.2965 0.2974 0.2967
Note. E-LB and E-UB indicate the lower and the upper bounds from the modified Edgeworth binomial model with various time steps (N), skewness (g1), and kurtosis (g2). Levy represents the discrete Wilkinson approximation. MC represents the Monte Carlo
esti-mates. In addition, PM represents the four-moment approximation by Posner and Milevsky. The simulations assume the option life
T 1 year, the domestic interest rate rd 0.15, the foreign interest rate rf 0.1, and the initial spot price S0 1.5 as units of
domes-tic currency per unit of foreign currency.
leptokurtic. Under lognormal distribution, when the call is deep out of money, the authors’ lower bounds are more accurate than the estimates from all the other methods. The results from MC method are consistent with the authors’ lower and upper bounds. Overall, the authors’ outcomes are better than those of LT and TW, and similar to PM.
The authors’ modified model can be used to price European average rate currency options when r rd rf substitutes for r in Equations (2) and (3). Valuation results are compared with those of Levy (1992), which applies the discrete Wilkinson approximation (Levy) and the MC method, and with those of the four-moment approximation by PM. The impact of the higher moments on the value of the option is explored. Based on Equations (2) and (3), the authors’ simulations are constructed with maturity T 1 year, the domestic interest rate rd 0.15, the foreign interest rate rf 0.1, and the initial spot price S0 1.5, which is in domestic currency per unit of foreign currency.
Various skewness and kurtosis are expressed in Table IV.
Using MC estimates as the authors’ benchmark, it is found in Table IV that for quarterly averaged options, the authors’ valuation results are almost the same under right-skewed and leptokurtic conditions, except in the case of L 1.2.3The authors’ performance is similar to that of the four-moment method, but superior to the discrete Wilkinson approximation. Meanwhile, for monthly
3MC estimates were calculated by averaging 10,000 replications of lnlM(t). Under the null hypotheses of
zero skewness, the asymptotic standard error of skewness with 10,000 replications was 0.0245. See Levy (1992, p. 484).
average Asian options, the authors’ results are also similar to those from MC, just as in the quarterly average cases. However, the difference from Levy is rather huge. Thus, the authors’ valuation method is more accurate than the dis-crete Wilkinson approximation. The pricing model of Asian options should emphasize the higher moments when underlying assets have higher volatility.
CONCLUSION
The modified Edgeworth binomial model to price European-style Asian options with higher moments in the underlying return distribution was developed. Specifically, the values of average rate currency options are simulated under various skewness and kurtosis. Combining the Edgeworth approximation and the averaging algorithm by Chalasani et al. (1998), the authors’ method is faster and more accurate in the sense that the estimates have a smaller error bound. The numerical results show that this approach can effectively deal with the higher moments of the underlying distribution and provide better option value estimates than those found in various studies in the literature.
APPENDIX A: ANALYTICAL EXPLANATION FOR THE PROPOSITION
That the error bound in approximating E[Vn] from a modified Edgeworth
binomial tree model and that from a binomial tree model employed by Chalasani et al. (1999) are proportional to their upward probabilities, respec-tively, in the binomial paths is first shown. For this, a discrete approximation method similar to the lattice approach is used. Let T be the time for expiration of the option. At time T, let Ye(T) denote the variance of the arithmetic average of
the stock prices in the authors’ modified Edgeworth binomial tree with upward probabilities pe, and Yc(T) denote the variance of the average price in a
binomi-al tree from Chbinomi-alasani et binomi-al. with upward probability pc. From Equation (2), the
asset price in the Edgeworth model is affected by the drift with upward trend, resulting in higher average than the case in Chalasani et al. From Equation (7), higher average price increases the probability of Amin(k, h, a) L, i.e. higher probability of zero variance based on explanations in footnote 1. Thus Yc(T) Ye(T) is obtained.
Assume for the moment that peis less than pc. At time t T/3, the
condi-tional expectations of the variances with upward probabilities pe and pcare
given by E3t[Ye(T)] and E3t[Yc(T)], respectively. It can be seen (ignoring the
dis-count factors) that
because
Similarly, at time t T/4,
because
Therefore, as long as the above inequality continuously holds for all time t T/5, the error bound for a tree model with upward probability pewill be
tighter than that for a tree with upward probability pc, given that peis less than pc.
Next, that the upward probability pein an Edgeworth binomial model is
indeed less than the upward probability pcin the model employed by Chalasani
et al. (1998) is shown.
As noted in Equation (3), pn,h Ph/[n!/h!(n h)!] and pn,h1 Ph1/[n!/
(h 1)!(n (h 1))!], where Ph F(yh)/jF(yj) and the Edgeworth-corrected
probability F(yh) f(yh) b(yh), as discussed in Equation (1). The upward
prob-ability peand the downward probability qeare defined as follows:
where f(xh) denotes an Edgeworth expansion function, and xh represents the normalized random variable from yh, i.e. xhequals (yh M)/V with M hPhyh, V2
hPh(yh M)2, and Phis the probability distribution. On the basis of pe
and qeas defined above, two possible cases can be obtained: qe 1 pe pe pn,h1 pn,h1 pn,h 1 1 (pn,hpn,h1) 1 1 ( f(xh)f(xh1)) M(4, h, a)ph e(1 pe)4h(var(A40 a, S4,h))12. a4 h0 a 4 a0 Amax L, AminL M(4, h, a)ph c(1 pc)4h(var(A40 a, S4,h))12 a 4 h0 a 4 a0 Amax L, AminL E4t[Yc(T)]12 E4t[Ye(T)]12 M(3, h, a)ph e(1 pe)3h(var(A30 a, S3,h) )12. a3 h0 a 2 a0 Amax L, AminL M(3, h, a)phc(1 pc)3h(var(A30 a, S3,h))12 a 3 h0 a 2 a0 Amax L, AminL
(1) If f(xh) f(xh1), then 0 pe 0.5, and 0.5 qe 1. (2) If f(xh) f(xh1), then 0.5 pe 1, and 0 qe 0.5.
In case (1), Ph Ph1can be inferred because f(xh) f(xh1). Like the
argu-ment discussed in the first paragraph in this Appendix, the drift with upward trend in an Edgeworth model will affect the average price of the underlying asset. From the error bound in Equation (7), higher average price increases the probability for Amin(k, h, a) L, i.e. higher probability of zero variance.
If f(xh) f(xh1) as in case (2), Ph Ph1is obtained. The underlying asset
price in an Edgeworth model is affected by the drift with downward trend. All average stock prices with non-lognormal distributions are smaller than those with lognormal distributions in an Edgeworth model, but higher than those in a binomial tree from Chalasani et al. when the drifts are greater than zero in Equation (2). So the higher average prices still increase the probability of Amin(k, h, a) L and also the probability of zero variance.
In both cases, if the upward probabilities in an Edgeworth model are lower than or equal to those in a binomial tree from Chalasani et al., then, according to Equation (7), the error bound of an Edgewroth model will be tighter than that of a binomial tree from Chalasani et al.
When the underlying asset return exhibits a lognormal distribution, f(xh) f(xh1) is obtained. The upward probability in the Edgeworth model is then equal
to 0.5 (pe qe 0.5). Meanwhile, the corresponding upward probability, pc, in
the binomial tree model described by Chalasani et al. (1999) is more than 0.5 under s (2r)0.5. Therefore, p
e pcis obtained. If s (2r)0.5, then the upward
probability in the binomial model by Chalasani et al. is less than 0.5. The higher the volatility of the stock price, the greater the total variance of the average stock price. The error bounds of the binomial model by Chalasani et al. become larger when the upward probability is less than 0.5. Similar to the above cases (1) and (2), an upward probability in the Edgeworth model less than or equal to that of Chalasani et al can be set. The drifts with upward trend then will affect the aver-age stock prices in the Edgeworth model; hence, its error bound is smaller.
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