One-state Variable Binomial Models for ... - Dept of Maths, NUS

One-state Variable Binomial Models for European-/American-Style Geometric Asian Options Min Dai Laboratory of Mathematics and Applied Mathematics, and Dept. of Financial Mathematics, Peking University, Beijing 100871, China Email address: [email protected]

Abstract Geometric Asian options are path-dependent options whose payoffs depend on the geometric average of the underlying asset prices. Following Cox et al (1979) arbitrage arguments, We develop one-state variable binomial models for the options on the basis of the idea of Cheuk and Vorst (1997). The models are more efficient and faster than those lattice methods proposed by Hull and White (1993), Ritchken et al (1993), Barraquand and Pudet (1996) and Cho and Lee (1997). We also establish the equivalence of the models and difference schemes.

1

Introduction

Asian options are path-dependent options whose payoffs depend on the average of the underlying asset prices. The dependence of the average on the options’ payoff can be specified in several ways. One such option is the fixed strike Asian option in which case the “underlying asset price” of the payoff is replaced by an average, whereas in the other case the “strike price” of the contract is an average of the underlying asset prices. The latter option is called the floating strike Asian option. Furthermore, the type of averaging may differ. Firstly, the average may be based on discretely sampled prices or on continuously sampled prices. Secondly, averaging may be on an arithmetic or geometric basis. The binomial model, first proposed by Cox et al (1979), has become one of the most popular approaches to pricing options due to its simplicity and flexibility. With interpolation techniques, Hull and White (1993) and Ritchken, et al (1993) extend the model to the valuation of Asian options. Barraquand and Pudet (1996) give a similar lattice algorithm called the forward shooting grid method. Cho and Lee (1997) present a lattice model involving all paths to value geometric Asian options. Generally speaking, the binomial model for (strong) path-dependent options involves two state variables because an additional path-dependent variable is introduced. However, there exist one-state binomial models for some special cases. For instance, Babbs (1992) and Cheuk and Vorst (1997) present one-state binomial models for lookback options. Using such models cuts the amount of computations remarkably. 1

From the view point of PDE, an important reason that Babbs (1992) and Cheuk and Vorst (1997) are able to construct one-state variable binomial models for lookback options is that the corresponding PDEs can be reduced to one-dimensional time-dependent problem by using some transformations. Since similar transformations can also be applied to Asian options [see, for example, Ingersoll (1987), Wilmott et al (1993), Dewynne and Wilmott (1995a), Roger and Shi (1995), etc], it is possible to construct one-state variable binomial models for Asian options. This paper is concerned with the construction of one-state variable binomial models for geometric Asian options. Analytical pricing formulas for the options of European-style are available [see, for example, Wilmott et al (1993), Kwok (1998) and Hull (2000)]. However, there are not any closed forms of American-style prices. It is well known that binomial models can readily deal with American-style options. Our main purpose is to develope one-state variable binomial models both for European- and American-style geometric Asian options with continuous and discrete sampling. Later we will find that it is easy to establish the equivalence of the models and difference schemes. Throughout this paper we confine ourselves to call options because the case of puts is similar. The remaining of this paper is organized as follows. In next section we construct one-state variable binomial models for the European style with continuous sampling. In section 3 we extend the models to cope with the American style and discretely sampled observations. Section 4 is devoted to the equivalence of these models and certain explicit difference schemes. In last section we conclude.

2

Binomial models for European style with continuous sampling

Let T be expiration date, [0, T ] be the lifetime of the option, and r, q and σ represent the interest rate, continuous dividend rate and volatility respectively. If N is the number of discrete time points, we have time points n∆t, n = 0, 1, . . . , N with ∆t = T /N. Let V n (Sn , In ) be the option price at time point n∆t In

with underlying asset price Sn and geometric average value e n+1 , where In =

n X

ln Si . Assume that Sn

i=0

will be either Sn u for an upward movement with probability p or S n d for a downward movement with probability 1 − p at time (n + 1)∆t. Here u = eσ

√

∆t

, d=

1 e(r−q)∆t − d and p = . u u−d

u d , where Consequently, In will become either In+1 or In+1

(

√ u In+1 = In + ln(Sn u) = In + ln Sn + σ√∆t . d In+1 = In + ln(Sn d) = In + ln Sn − σ ∆t

(2.1)

Following Cox, Ross and Rubinstein arbitrage arguments, one has V n (Sn , In ) =

1 u d [pV n+1 (Sn u, In+1 ) + (1 − p)V n+1 (Sn d, In+1 )], ρ

where ρ = er∆t . 2

(2.2)

2.1

Floating strike case

First let us consider the case of floating strike, namely the payoff is given by IN

V N (SN , IN ) = (SN − e N +1 )+ . Suppose we can write V n (Sn , In ) = Sn W n (yn ),

(2.3)

yn = In − (n + 1) ln Sn .

(2.4)

where Due to (2.1), one has √ √ u In+1 − (n + 2) ln(Sn u) = In + ln Sn + σ ∆t − (n + 2) ln Sn − (n + 2)σ ∆t √ = In − (n + 1) ln Sn − (n + 1)σ ∆t √ = yn − (n + 1)σ ∆t. Then we can rewrite and similarly

√ u V n+1 (Sn u, In+1 ) = Sn uW n+1 (yn − (n + 1)σ ∆t),

(2.5)

√ d V n+1 (Sn d, In+1 ) = Sn dW n+1 (yn + (n + 1)σ ∆t).

(2.6)

It follows from (2.2)-(2.6) that W n (yn ) =

√ √ 1 [puW n+1 (yn − (n + 1)σ ∆t) + (1 − p)dW n+1 (yn + (n + 1)σ ∆t)] ρ

(2.7)

At expiry one has IN

yN e N +1 + ) = (1 − e N +1 )+ . W (yN ) = (1 − SN To simplify the notation, we denote √ W n (j) = W n (jσ ∆t).

N

(2.8)

Note that I0 = ln S0 and then y0 = 0. In order to compute V 0 (S0 , ln S0 ) = S0 W 0 (0), at time n∆t we should give the values of W n at the nodes in the interval (−

n−1 X

n−1 √ X √ n(n + 1) √ n(n + 1) √ σ ∆t, σ ∆t). (k + 1)σ ∆t, (k + 1)σ ∆t) = (− 2 2 k=0 k=0

In virtue of (2.7) and (2.8), option prices can be calculated by using the following backward induction  1   W n (j) = [puW n+1 (j − n − 1) + (1 − p)dW n+1 (j + n + 1)],   ρ  

for 0 ≤ n ≤ N − 1, j = −n(n + 1)/2, −n(n + 1)/2 + 2, . . . , n(n + 1)/2 √

j jσ ∆t    W N (j) = (1 − e N +1 )+ = (1 − u N +1 )+   

for j = −N (N + 1)/2, −N (N + 1)/2 + 2, . . . , N (N + 1)/2 3

(2.9)

with V 0 (S0 , ln S0 ) = S0 W 0 (0). This is the one-state variable binomial model for the European-style floating strike geometric Asian option with continuous sampling.

1.0000

1.0352

0.0151

0.9885

0.0267

1.0116

0.0224

1.0116

0.0402

0.9885

0.0352

0.9660

0.0531

0.0345

1.1224

0

1.0968

0

1.0717

0

1.0717

0

1.0473

0

1.0473

0

1.0234

0.0132

1.0234

0

1.0000

0.0261

1.0000

0

0.9772

0.0388

0.9772

0.0228

0.9549

0.0511

0.9549

0.0451

0.9331

0.0728

0.9331

0.0669

0.9118

0.0882

0.8909

0.1091

Figure 1: Continuously sampled floating strike option with N = 4. As an illustration of the model, we present an example. We suppose that S 0 = 100, σ = 0.2, r = 0.09, q = 0., T = 1/3 and N = 4. Then ∆t = 0.0833, u = 1.0594, d = 0.9439 and p = 0.5507. The movements of the state variable y are represented by a four-step binomial tree which is shown in Figure 1. The calculations are also shown in the figure. The left number at each node is the value of the state variable yn . The right number at each node shows the value of the option (in underlying asset price units) computed from the backward procedure (2.9). Thus, the estimated dollar value of the option at time zero is 0.0345 ∗ 100 = 3.45.

2.2

Fixed strike case

Let us move on to the fixed strike option whose payoff at expiry is IN

V N (SN , IN ) = (e N +1 − X)+ ,

(2.10)

where X is the strike price. In this case we let V n (Sn , In ) = W n (yn ) and yn = In + (N − n) ln Sn . Similarly to previous arguments, one can derive from (2.1), (2.2) and (2.10) W n (yn ) =

√ √ 1 [pW n+1 (yn + (N − n)σ ∆t) + (1 − p)W n+1 (yn − (N − n)σ ∆t)] ρ

and

y

W N (y) = (e N +1 − X )+ . 4

(2.11)

Denote

√ W n (j) = W n (y0 + jσ ∆t).

Here y0 = I0 + N ln S0 = (N + 1) ln S0 . In order to compute V 0 (S0 , ln S0 ) = W 0 (y0 ), at time n∆t it is enough to give the values of W n at the nodes in the interval (y0 −

n−1 X

n−1 √ X √ n(2N − n + 1) √ n(2N − n + 1) √ σ ∆t, y0 + σ ∆t). (N −k)σ ∆t, y0 + (N −k)σ ∆t) = (y0 − 2 2 k=0 k=0

Hence, option prices can be calculated by using the following backward induction  1   W n (j) = [pW n+1 (j + N − n) + (1 − p)W n+1 (j − N + n)],   ρ  

for 0 ≤ n ≤ N −√1, j = −n(2N − n + 1)/2, −n(2N − n + 1)/2 + 2, . . . , n(2N − n + 1)/2

y0 +jσ ∆t j   N +  N +1 N +1 − X)+ , W (j) = (e − X) = (S u  0  

(2.12)

for j = −N (N + 1)/2, −N (N + 1)/2 + 2, . . . , N (N + 1)/2

with V 0 (S0 , ln S0 ) = W 0 (y0 ). The tree is shown in Figure 2. One can observe that the tree is different from that of the floating strike case. The reason is that we adopt different transformations for different cases, that is, (2.3-2.4) for floating strike and (2.11) for fixed strike. Thus the resulting state variables are not same. This leads to distinct recursive algorithms as well as different number of nodes of trees at each time-step.

Figure 2: Continuously sampled fixed strike option with N = 4.

2.3

Numerical experiments

To verify the validity of the above models, we compare them with the lattice algorithm of Cho and Lee (1997) as well as with analytical formulas in Table 1. Since both our models and Cho and Lee’s 5

algorithm involve all paths on the basis of Cox, Ross and Rubinstein arguments, their results should be same, which is verified by Table 1. Note that at time n∆t the total numbers of nodes of our models are n(n + 1) n(2N − n + 1) + 1 and + 1 for the floating strike case and the fixed strike case, respectively, 2 2 n3 + 5n while the corresponding number of Cho and Lee’s algorithm is + 1. Therefore our models 6 prevail evidently. In Table 2 and Table 3 option prices computed by the models are given for several values of N and σ. Table 1: European floating strike call and fixed strike call with continuous sampling (X = 95 (for fixed strike), S = 100, σ = 0.2, r = 0.09, q = 0., N = 60) T (year) 1/6

1/4

1/3

Analytical Our Models C-L Algorithm Analytical Our Models C-L Algorithm Analytical Our Models C-L Algorithm

Floating Strike 2.301 2.300 2.300 2.940 2.936 2.936 3.514 3.510 3.510

Fixed Strike 5.849 5.845 5.845 6.316 6.311 6.311 6.761 6.755 6.755

Table 2: European floating strike call with continuous sampling (S = 100, r = 0.09, q = 0.02, T = 1/3 year) N 4 12 60 120 240 Analytic

σ = 0.1 1.948 1.974 1.984 1.986 1.986 1.986

σ = 0.2 3.248 3.286 3.300 3.302 3.302 3.303

σ = 0.3 4.601 4.650 4.669 4.670 4.670 4.671

Table 3: European fixed strike call with continuous sampling (S = 100, X = 95, r = 0.09, q = 0., T = 1/3 year) N 4 12 60 120 240 Analytic

σ = 0.1 6.300 6.313 6.319 6.319 6.320 6.320

σ = 0.2 6.691 6.734 6.755 6.758 6.760 6.761 6

σ = 0.3 7.485 7.538 7.571 7.576 7.578 7.581

3

Extensions to American style and discretely sampled observations

In this section we will extend the one-state variable binomial models to cope with the American style and discretely sampled observations.

3.1

American style

It is straightforward to extend the model to value the American style floating strike option. We can roll back through the tree in the same way with European style except that the early exercise is considered, that is,    j 1  n n+1 n+1 + n+1  [puW (j − n − 1) + (1 − p)dW (j + n + 1)], (1 − u W (j) = max ) ,   ρ  

for 0 ≤ n ≤ N − 1, j = −n(n + 1)/2, −n(n + 1)/2 + 2, . . . , n(n + 1)/2

j    W N (j) = (1 − u N +1 )+   

(3.1)

for j = −N (N + 1)/2, −N (N + 1)/2 + 2, . . . , N (N + 1)/2

with V 0 (S0 , ln S0 ) = S0 W 0 (0). In Table 4 we compare our model with the forward shooting grid method (FSGM) of Barraquand and Pudet (1996). In FSGM the grid quantization parameter (for the spacing of the arithmetic averages) η = 1 and the quadratic interpolation is adopted. The latter makes the algorithm unconditionally stable [see Jiang and Dai (2000)]. It can be observed that our model is superior to FSGM remarkably. Table 4: American floating strike call with continuous sampling (S = 100, r = 0.09, q = 0.02, T = 1/3 year)

N 60 120 240 480 60 120 240 480 60 120 240 480

Our model FSGM Price CPU (sec) Price CPU (sec) σ = 0.1, correct solution ' 2.39 2.333 0.06 2.313 1.6 2.359 0.3 2.352 12.9 2.375 2.7 2.370 103 2.383 22 2.381 771 σ = 0.2, correct solution ' 4.25 4.137 0.06 4.114 1.7 4.191 0.3 4.175 12.9 4.222 2.7 4.212 103 4.239 23 4.236 768 σ = 0.3, correct solution ' 6.14 5.967 0.06 5.930 1.6 6.046 0.3 6.016 12.9 6.092 2.7 6.077 103 6.117 23 6.112 770

7

Unfortunately, the one-state variable binomial model for the European-style fixed strike option cannot be extended to price its American-style counterpart, because the transformation (2.11) fails to apply to the case of American-style due to early exercise. We may recall the paper of Cheuk & Vorst (1997) in which one-state variable binomial models of American-style fixed strike lookback options are not available either because of the same reason. As a matter of fact, from the PDE view point, one cannot reduce the pricing models of American-style fixed strike (Asian or lookback) options to one-dimensional time-dependent problems.

3.2

Discretely sampled observations

All above models can be extended to cope with discretely sampled observations. First let us take the European-style floating strike case for example. For simplicity, let the interval between observing instants be regular and ti = iH∆t, i = 0, 1, . . . , L, be the observing instants. Here H = N/L ∈ Z, that is, there are H + 1 time points between two subsequent observing instants. Define In =

X

ln SiH .

0≤iH≤n

Once again we let yn = In − (i + 1) ln Sn , for n ∈ [iH, (i + 1)H), i = 0, 1, . . . , L − 1

(3.2)

V n (Sn , In ) = Sn W n (yn ).

(3.3)

and If n = (i + 1)H − 1 in which case (2.1) remains valid, then we can similarly get by (3.2), (3.3) and (2.2) W n (yn ) =

√ √ 1 [puW n+1 (yn − (i + 1)σ ∆t) + (1 − p)dW n+1 (yn + (i + 1)σ ∆t)] ρ

(3.4)

u d , it is not hard to check that (3.4) still remains valid If n ∈ [iH, (i + 1)H − 1), noting In = In+1 = In+1 due to the transformations (3.2) and (3.3). As before we let

√ W n (j) = W n (jσ ∆t). Similar to previous arguments, we arrive at the binomial model for the European-style floating strike option with discrete sampling as follows:  1   W n (j) = [puW n+1 (j − i − 1) + (1 − p)dW n+1 (j + i + 1)],   ρ        

for n = iH + k, i = 0, 1, ..., L − 1, k = 0, 1, ..., H − 1 and j = −(iH + 2k)(i + 1)/2, −(iH + 2k)(i + 1)/2 + 2, ..., (iH + 2k)(i + 1)/2 j

W N (j) = (1 − u L+1 )+ , j = −N (L + 1)/2, −N (L + 1)/2 + 2, . . . , N (L + 1)/2,

8

(3.5)

with V 0 (S0 , ln S0 ) = S0 W 0 (0). Consequently, the binomial model for the American-style floating strike option with discrete sampling is    j 1 n+1 n+1 +  n i+1  , W (j) = max [puW (j − i − 1) + (1 − p)dW (j + i + 1)], (1 − u )   ρ        

for n = iH + k, i = 0, 1, ..., L − 1, k = 0, 1, ..., H − 1 and j = −(iH + 2k)(i + 1)/2, −(iH + 2k)(i + 1)/2 + 2, ..., (iH + 2k)(i + 1)/2

(3.6)

j

W N (j) = (1 − u L+1 )+ , j = −N (L + 1)/2, −N (L + 1)/2 + 2, . . . , N (L + 1)/2,

with V 0 (S0 , ln S0 ) = S0 W 0 (0). Note that for H = 1 the binomial models of discretely sampled observations reduce to those of continuously sampled observations. The tree with L = 3 and H = 2 is given by Figure 3.

Figure 3: Discretely sampled floating strike option with L = 3, H = 2. For the European-style fixed strike option with discrete sampling, we may let yn = In + (L − i) ln Sn , for n ∈ [iH, (i + 1)H), i = 0, 1, . . . , L − 1, V n (Sn , In ) = W n (yn ) and denote

√ W n (j) = W n ((L + 1) ln S0 + jσ ∆t)

to get the following one-state variable binomial model:  1   W n (j) = [pW n+1 (j + L − i) + (1 − p)W n+1 (j − L + i)],   ρ  

for n = iH + k, i = 0, 1, ..., L − 1, k = 0, 1, ..., H − 1

  j = −iH(2L − i + 1)/2 − k(L − i), −iH(2L − i + 1)/2 − k(L − i) + 2, ..., iH(2L − i + 1)/2 + k(L − i)    j  N L+1 +

W (j) = (1 − u

) , j = −N (L + 1)/2, −N (L + 1)/2 + 2, . . . , N (L + 1)/2 9

(3.7)

with V 0 (S0 , ln S0 ) = W 0 (0). Figure 4 shows the tree with L = 3 and H = 2. Just like the continuously sampled case, the model cannot be extended to price the American-style fixed strike option.

Figure 4: Discretely sampled fixed strike option with L = 3, H = 2. Some numerical experiments are given by Table 5-7. Table 5: European floating strike call with discrete sampling (S = 100, r = 0.09, q = 0.02, σ = 0.2, T = 1/3 year) L 4 12 24 120

1 3.248 3.286 3.294 3.302

2 3.207 3.272 3.287 3.300

H 5 3.193 3.263 3.283 3.299

Analytic 10 3.188 3.261 3.281 3.299

20 3.186 3.259 3.281 3.299

3.184 3.258 3.280 3.299

Table 6: American floating strike call with discrete sampling (S = 100, r = 0.09, q = 0.02, σ = 0.2, T = 1/3 year) L 4 12 24 120

1 3.543 3.865 4.009 4.191

2 4.180 4.181 4.203 4.242

H 5 4.737 4.453 4.336 4.271

10

10 4.911 4.523 4.391 4.282

20 5.005 4.573 4.417 4.287

Table 7: European fixed strike call with discrete sampling (S = 100, X = 95, r = 0.09, q = 0.0, σ = 0.2, T = 1/3 year) L 4 12 24 120

4

1 6.691 6.734 6.747 6.758

H 5 6.667 6.726 6.743 6.757

2 6.675 6.728 6.744 6.758

Analytic 10 6.665 6.725 6.742 6.757

20 6.665 6.724 6.742 6.757

6.664 6.724 6.742 6.757

Relation to explicit difference schemes

It is well known that the binomial models of vanilla options are equivalent to certain explicit difference schemes [see Hull (2000) or Jiang and Dai (1999)]. For lookback options, such equivalence has also been noted by Dai (2000). In this section we will establish such equivalence for geometric Asian options. First let us recall the continuous-time partial differential equation (PDE) models. Suppose the underlying asset price St follows the lognormal process in the risk-neutral world, namely dSt = (r − q)dt + σdWt , St where Wt is the standard Brown movement. In order to use a unifying framework to deal with both discretely sampled and continuously sampled model, we introduce the following variable It = Here

Z

t 0

p(τ ) ln Sτ dτ.

   1, for continous sampling L X p(t) = ,  δ(t − ti ), for discrete sampling  i=0

ti , i = 0, 1, ..., L are the observing instants, and δ is the Dirac delta function. The governing equation for the European-style geometric Asian option price V (S, I, t) is given by [see Wilmott et al (1993)] ∂V 1 ∂2V ∂V ∂V + p(t) ln S + σ 2 S 2 2 + (r − q)S − rV = 0, ∂t ∂I 2 ∂S ∂S 0 ≤ t < T, 0 < S < ∞, − ∞ < I < ∞

(4.1)

with the final conditions

V (S, I, T ) =

 I     (S − exp( R T

I     (exp( R T 0

0

p(τ )dτ

p(τ )dτ

))+ for floating strike call

) − X)+ for fixed strike call

Note that S, I and t are mutually independent in the PDE. 11

.

(4.2)

For the case of floating strike, by transformation y=I−

Z

t

p(τ )dτ ln S, W (y, t) = 0

V (S, I, t) , S

(4.3)

(4.1) and (4.2) reduce to  Z Z t 2 ∂W 1 2 t σ2 ∂W  2∂ W  + σ ( p(τ )dτ ) + (q − r − ) p(τ )dτ − qW = 0,   2  ∂t 2 ∂y 2 0 ∂y 0   y    W (y, T ) = (1 − exp( R T 0

p(τ )dτ

))

+

(4.4)

0 ≤ t < T, − ∞ < y < ∞

For the case of fixed strike, by transformation y=I+

Z

T

p(τ )dτ ln S, W (y, t) = V (S, I, t),

(4.5)

t

(4.1) and (4.2) reduce to  Z Z T 2 ∂W 1 2 T σ2 ∂W  2∂ W  + σ ( p(τ )dτ ) + (r − q − ) p(τ )dτ − rW = 0,   2  ∂t 2 ∂y 2 t ∂y t   y    W (y, T ) = (exp( R T 0

p(τ )dτ

) − X)

0 ≤ t < T, − ∞ < y < ∞

+

.

(4.6)

Theorem 1 By neglecting a high order of ∆t, the binomial models (2.9) and (3.5) for European-style floating strike geometric Asian options (or (2.12) and (3.7) for the fixed strike case) are equivalent to certain explicit difference schemes of (4.4) (or 4.6), respectively. The proof is put in the appendix where we will see that the explicit difference schemes are of varying mesh with time. The governing equation for the American-style geometric Asian option price V (S, I, t) is given by [see Wilmott et al (1993)] (

)

∂V ∂V 1 ∂2V ∂V min − − p(t) ln S − σ 2 S 2 2 − (r − q)S + rV, V − Λ(S, I, t) = 0, (4.7) ∂t ∂I 2 ∂S ∂S 0 ≤ t < T, 0 < S < ∞, − ∞ < I < ∞ with the same final conditions (4.2), where

Λ(S, I, t) =

 I     (S − exp( R t

I     (exp( R t 0

0 p(τ )dτ

p(τ )dτ

))+ for floating strike call

) − X)+ for fixed strike call

12

.

For the American-style floating strike case, using transformation (4.3) may still reduce (4.7) and (4.2) to the one-dimension problem: ) (  Z Z t  1 2 t ∂2W σ2 ∂W y ∂W  2 +  )) = 0, min − − σ ( p(τ )dτ ) − (q − r − ) p(τ )dτ + qW, W − (1 − exp( R t    ∂t 2 ∂y 2 2 0 ∂y 0 0 p(τ )dτ   y     W (y, T ) = (1 − exp( R T 0

p(τ )dτ

))

0 ≤ t < T, − ∞ < y < ∞

+

(4.8)

Theorem 2 By neglecting a high order of ∆t, the binomial models (3.1) and (3.6) for American-style floating strike geometric Asian options are equivalent to certain explicit difference schemes of (4.8), respectively. The proof is similar to that of Theorem 1. However, in the case of American-style fixed strike, the transformation (4.5) cannot reduce (4.7) and (4.2) to the one-dimension problem. This is why we cannot construct a one-state variable binomial models in such case.

5

Conclusion

In this paper one-state variable binomial models for geometric Asian options are developed on the basis of Cox et al (1993) arbitrage arguments. As compared with those lattice methods proposed by Hull and White (1993), Ritchken et al (1993), Barraquand and Pudet (1996) and Cho and Lee(1997), the models allow more efficient and faster computation of option values. The models also allow us to consider discretely sampled observations and especially early exercise (for floating strike) where the analytical pricing formulas are not available. In addition, the equivalence of the models and explicit difference schemes is established. For possible future works, it may be interesting to explore the one-state variable binomial models of arithmetic Asian options for which it seems that interpolation technique adopted by Hull and White (1993), Ritchken et al (1993) and Barraquand and Pudet (1996) would have to be employed.

References [1] Babbs, S. (1992) Binomial valuation of lookback options. Working Paper, Midland Global Markets. [2] Barraquand, J., and T. Pudet (1996) Pricing of American path-dependent contingent Claims, Mathematical Finance, 6, 17-51. [3] Cox, J.C., S.A. Ross, and M. Rubinstein (1979) Option pricing: a simple approach, Journal of Financial Economics, 7, 229-263. 13

[4] Cheuk, T.H.F., and T.C.F. Vorst (1997) Currency lookback options and observation frequency: a binomial approach, Journal of International Money and Finance, 16:2, 173-187. [5] Cho, H.Y., and H.Y. Lee (1997) A lattice model for pricing geometric and arithmetic average options, The Journal of Financial Engineering, 6:3, 179-191. [6] Dai, M. (2000) A modified binomial tree method for currency lookback options, Acta Mathematica Sinica, 16:3, 445-454. [7] Dewynne, J.N., and P. Wilmott (1993) Partial to the exotic, Risk magazine, 6:3, 38-46. [8] Dewynne, J.N., and P. Wilmott (1994) Untitled, MFG Working Paper, Oxford University. [9] Dewynne, J.N., and P. Wilmott (1995a) A note on average rate options with discrete sampling, SIAM Journal on Applied Mathematics, 55:1, 267-276. [10] Dewynne, J.N., and P. Wilmott (1995b) Asian options as linear complementarity problems: analysis and finite-difference solutions. Advances in Futures and Options Research, 8, 145-177. [11] Hansen, A.T., and P.L. Jorgensen (1998) Analytical valuation of American-style Asian options. Working paper. [12] Hull, J. (2000) Options, Futures and their Derivatives, Fourth Edition, Prentice Hall International Inc. [13] Hull, J., and A. White (1993) Efficient procedures for valuing European and American pathdependent options, Journal of Derivatives, 1, 21-31. [14] Ingersoll, J.E. (1987) Theory of Financial Decision Making, Rowman & Littlefield. [15] Jiang, L., and M. Dai (1999) Convergence of binomial tree method for American options, in Partial Differential Equations and their Applications, edited by H. Chen and L. Rodino, World Scientific Publishing Co. Pte. Ltd., 106-118. [16] Jiang, L., and M. Dai (2000) Convergence analysis of binomial tree method for American-type path-dependent options, in Free boundary problems: theory and applications, I (Chiba, 1999), 153-156, GAKUTO International Series: Mathematical Sciences and Applications, edited by N. Kenmochi. [17] Kwok Y.K. (1998) Mathematical Models of Financial Derivatives, Springer, Singapore. [18] Ritchken, P., L. Sankarasubramanian, and A.M. Vijh (1993) The valuation of path dependent contracts on the average, Management Science 39, 1202-1213. [19] Rogers, L., and Z. Shi (1995) The value of an Asian option. Journal of Applied Probability, 32, 1077-1088. 14

[20] Wilmott, P., J. Dewynne, and S. Howison (1993) Option Pricing: Mathematical Models and Computation, Oxford Financial Press. [21] Wu, L., Y.K. Kwok, and H. Yu (1999) Asian option with the American early exercise feature, International Journal of Theoretical and Applied Finance, 2:1, 101-111.

6

Appendix: proof of Theorem 1

We only take the European-style floating strike case for example. Given mesh size ∆y, ∆t > 0, N ∆t = T, let Q = {(j∆y, n∆t) : j ∈ Z, 0 ≤ n ≤ N } stand for the lattice, and let V jn represent the value of numerical approximation at (n∆t, j∆y). First we will consider the continuously sampled case, for which (4.4) reduces to  2 2   ∂W + 1 σ 2 t2 ∂ W + (q − r − σ )t ∂W − qW = 0, 0 ≤ t < T, − ∞ < y < ∞ ∂t 2 ∂y 2 y 2 ∂y  

(6.1)

W (y, T ) = (1 − e ) T

Making use of the approximations

n+1 n+1 n+1 n+1 n+1 ∂W ∂ 2 W . Wj+n+1 + Wj−n−1 − 2Wj . Vj+n+1 − Vj−n−1 = , = ∂y 2 (j∆y,(n+1)∆t) (n + 1)2 ∆y 2 ∂y (j∆y,(n+1)∆t) 2(n + 1)∆y





(6.2)

for space and taking the explicit difference for time at the node (j∆y, (n + 1)∆t), we have n+1 n+1 Wjn+1 − Wjn 1 2 Wj+n+1 + Wj−n−1 − 2Wjn+1 2 2 + σ (n + 1) ∆t ∆t 2 (n + 1)2 ∆y 2 n+1 n+1 Wj+n+1 − Wj−n−1 σ2 − qWjn = 0 +(q − r − )(n + 1)∆t 2 2(n + 1)∆y

Taking ∆y = σ∆t3/2 , we get Wjn = where

1 n+1 n+1 [aWj−n−1 + (1 − a)Wj+n+1 ], 1 + q∆t √ 1 ∆t σ2 a= − (q − r − ). 2 2σ 2

(6.3)

(6.4)

The final condition is given by Wjn = (1 − e

j∆y T

)+ = (1 − e

jσ

√ ∆t N

)+ .

(6.5)

It is not hard to check that a 1 1−a 1 pu = + O(∆t3/2 ), (1 − p)d = + O(∆t3/2 ). ρ 1 + q∆t ρ 1 + q∆t 15

(6.6)

Then the binomial model (2.9) is equivalent to the explicit difference scheme (6.3) and (6.5) by neglecting a high order of ∆t. We now move on to the case of discretely monitoring. As in Section 3.2, we assume that the observing instants are regular, namely t i = iH∆t, i = 0, 1, ..., L. Noticing that Z

t 0

p(τ )dτ = i + 1, for t ∈ [ti , ti+1 ), i = 0, . . . , L − 1.

(4.4) reduces to  ∂W 1 ∂2W σ2 ∂W    + σ 2 (i + 1)2 + (q − r − )(i + 1) − qW = 0,  2

∂t

   

2

W (y, T ) = (1 − e

∂y

y L+1

)+

2 ∂y for t ∈ [ti , ti+1 ), i = 0, . . . , L − 1, − ∞ < y < ∞

(6.7)

We employ the approximations n+1 n+1 n+1 n+1 n+1 ∂ 2 W ∂W . Wj−i−1 + Wj+i+1 − 2Wj . Wj+i+1 − Wj−i−1 = , = ∂y 2 (j∆y,(n+1)∆t) (i + 1)2 ∆y 2 ∂y (j∆y,(n+1)∆t) 2(i + 1)∆y





(6.8)

for space and explicit difference for time at the node (j∆y, (n + 1)∆t), n ∈ [iH, (i + 1)H), and take √ ∆y = σ ∆t,

to get Wjn =

1 n+1 n+1 [aWj−i−1 + (1 − a)Wj+i+1 ], n ∈ [iH, (i + 1)H), i = 0, . . . , L − 1, 1 + q∆t

(6.9)

where a is given by (6.4). The final condition is j∆y

WjN = (1 − e L+1 )+ = (1 − e

√ jσ ∆t L+1

)+ .

(6.10)

Due to (6.6), the binomial tree method (3.5) is equivalent to the explicit difference scheme (6.9) by neglecting a high order of ∆t. Here we give a brief explanation about why the approximations (6.2) and (6.8) should be taken. Indeed, both (6.3) and (6.9) belong to the varied mesh difference method. Since (4.4) is a convectiondiffusion equation, in the view point of numerical PDE, (6.2) and (6.8) lead to that the coefficient of the diffusion term predominates over that of the convection term, which guarantees the stability of the above schemes.

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