Valuation of American partial barrier options

Transcription

1 Rev Deriv Res DOI 0.007/s Valuation of American partial barrier options Doobae Jun Hyejin Ku Springer ScienceBusiness Media LLC 0 Abstract his paper concerns barrier options of American type where the underlying asset price is monitored for barrier hits during a part of the option s lifetime. Analytic valuation formulas of the American partial barrier options are provided as the finite sum of bivariate normal distribution functions. his approximation method is based on barrier options along with constant early exercise policies. In addition numerical results are given to show the accuracy of the approximating price. Our explicit formulas provide a very tight lower bound for the option values and moreover this method is superior in speed and its simplicity. Keywords Partial barrier option American option Hitting time Barrier approximation JEL Classification G3 C65 Introduction Barrier options are widely traded in over-the-counter marets because they are more flexible and cheaper than vanilla options. hese options either cease to exist or come into existence when some pre-specified asset price barrier is hit during the option s life. Merton 973 has derived a down-and-out call price by solving the corresponding partial differential equation with some boundary conditions. Rubinstein and Reiner 99 published closed form pricing formulas for various types of single barrier D. Jun H. Ku B Department of Mathematics and atistics Yor niversity oronto ON M3JP3 Canada hu@mathstat.yoru.ca D. Jun dbjun@su.edu 3

2 D. Jun H. Ku options. Rich 994 also provided a mathematical framewor to value barrier options. Due to their popularity in a maret more complicated structures of barrier options have been studied by a number of authors. Kunitomo and Ieda 99 derived a pricing formula for double barrier options with curved boundaries as the sum of an infinite series.geman and Yor 996 followed a probabilistic approach to derive the Laplace transform of the double barrier option price. In these papers the underlying asset price is monitored for barrier hits or crossings during the entire life of the option. On the other hand Heynen and Kat 994 studied partial barrier options where the underlying price is monitored during only part of the option s lifetime. Partial barrier options have two classes. One is forward starting barrier options where the barrier appears at a fixed date strictly after the option s initial starting date. he other is early ending barrier options where the barrier disappears at a specified date strictly before the expiry date. hey can be applied for various types of options according to the clients needs as controlling the starting or ending time of the monitoring period. Also they can be used as components to synthetically create other types of exotic options. Heynen and Kat 994 gave valuation formulas for partial barrier options in terms of bivariate normal distribution functions. As a natural variation on the partial barrier structure window barrier options have become popular with investors particularly in foreign exchange marets. For a window barrier option a monitoring period for the barrier commences at the forward starting date and terminates at the early ending date. We refer to Guillaume 003. In the case of American options which give their holders the additional flexibility of early exercise an exact and closed-form pricing solution has not existed because the option price and the early exercise boundary must be determined simultaneously. Consequently the literature of American options has proposed only numerical solution methods and analytical approximations. he numerical methods include the finite difference method by Brennan and Schwartz 977 and Parinson 977 and the binomial model of Cox et al hese numerical methods are quite flexible and simple to implement. However even after employing enhancement techniques such as control variates or convergence extrapolation they are very time consuming. here are many approximation schemes developed to reduce this time consuming tas. Johnson 983 expressed the put value as an approximate function of its parameters.gese and Johnson 984 approximated the American option price through an infinite series of multivariate normal distribution functions. Barone-Adesi and Whaley 987 used Merton s 973 solution for perpetual American options and the quadratic method of MacMillan 986. Despite its high efficiency and the accuracy improvements this method is not convergent because there is no control parameter to adjust to improve the accuracy. Longstaff and Schwartz 00 adapted Monte Carlo simulation methods to deal with the American put problem. hey addressed the optimal stopping problem in a Monte Carlo framewor by comparing the conditional expected value of continuing with the value of immediate exercise if the option is currently in the money. Sullivan 000 approximated the option value function through Chebyshev polynomials and employed a Gaussian quadrature integration scheme at each discrete exercise date. Although the speed and accuracy of the proposed numerical approximation can 3

3 Valuation of American partial barrier options be enhanced via the Richardson extrapolation its convergence properties are still unnown. Kim 990 Jaca 99 and Carr et al. 99 obtained an analytic integral-form solution for American options where the formulas represent the early premium of an American option as an integral which has the early exercise boundary. Broadie and Detemple 996 provided tight lower and upper bounds for American call prices based on the assumption that the early exercise boundary is a constant. Ju 998 approximated the early exercise boundary as a multipiece exponential function and substitute it by the early exercise premium integral derived by Kim 990 to price American options. Ingersoll 998 described another approximation method of American options based on barrier options: he exercise policy is approximated by a simple class of functions and the best policy within that class is selected by standard optimization techniques. he advantages of this method are its simplicity and speed even when used in general-purpose computer programs such as spreadsheets. Concretely he dealt with a constant barrier approximation and an exponential barrier approximation for American put. Chung et al. 00 derived the essential formulae for solving the lower bound and the optimal exercise boundary. For the American barrier option problem Gao et al. 000 suggested an approximation method for American barrier options. hey applied the approximation techniques of a standard American option to an American barrier option and proposed two approximation methods using Huang et al. 996 and Ju 998 to approximate an American barrier s exercise boundary. Dai and Kwo 004 provided an analytic formula for noc-in options and showed that the in-out barrier parity relationship for American barrier options could not be obtained unlie the case of European barrier options. Ingersoll 998 presented American up-and-in put price by an approximation method based on barrier options using constant and exponential exercise policies. his paper concerns the barrier option of American type where the barrier appears at a fixed date strictly after the option s initial starting date. o the best of our nowledge the literature of American exotic option suggests no approximation formula for American partial barrier options. Moreover the numerical methods such as Monte Carlo method and Lattice method for these options demand much time. his paper extends the approximation method of American barrier option suggested in Ingersoll 998 to the case of partial barrier options of American type. he constant functions are considered for early exercise boundaries. By our method American partial barrier option can be valued in a simple and speedy way. his article is organized as follows. Section presents a review of valuing American barrier option using barrier derivatives. Section 3 provides the analytic approximation of American partial barrier option. his section is divided into two subsections. he first subsection covers the case that up-barrier is greater than or equal to strie price. he second one presents valuation formulas for the digitals when up-barrier is less than strie price. Finally Sect. 4 provides the conclusion. American barrier option using barrier derivatives: a review In this section we present a brief review of the valuation for American barrier option using barrier derivatives described in Ingersoll 998. his method provides 3

4 D. Jun H. Ku a good approximation to the option price with the advantages of its simplicity and speed. Let r be the ris-free interest rate q be a dividend rate and >0 be a constant. We assume the price of the underlying asset S follows a geometric Brownian motion = S 0 expt W t where = r q and W t is a standard Brownian motion under the ris-neutral probability P. An American up-and-in put option will be exercised when it is sufficiently in the money but only after the stoc price has risen to the noc-in barrier or instrie. o value this contract it will be convenient to introduce the following digitals: Let DS t ; A be the value at time t of receiving one dollar at time if and only if the event A occurs and DSS t ; A be the value at time t of receiving one share of stoc at time if and only if the event A occurs. he D is said to be a digital or binary option and the DS is said to be a digital share. he quantity EK τ ; A denotes the value at time t of payment X K τ at the first time τ that the stoc price S hits the barrier K τ provided the event A occurs before time where X is a strie price. he E is said to be a first-touch digital. Consider an American up-and-in put expiring with strie price X. Let us denote by the up-barrier and by Kt the optimal exercise policy. Let τ B denote the first time the stoc price is equal to B and τ B B denote the first time after τ B that the stoc price is equal to B. Let A ={t<τ < τ K > S <X} be the event of exercise at maturity under the optimal policy and A ={t<τ τ K <} be the event of early exercise under the optimal policy. hen the value of the up-and-in put can be written as IP =X DS t ; A DSS t ; A EK t ; A he barrier approximation for this put taes the maximum value within a class of restricted policies. For example for constant exercise policies IP IP const = max [X DS t ; A 3 SS t ; A 3 E ; A 4 ] where A 3 ={t<τ < τ > S <X}A 4 ={t<τ τ <} and τ is the first time the stoc price hits the constant policy barrier after hitting the barrier. he values for these digitals are given by DS t ; A 3 = e r t { [ N 3 h ] N h X [ N h ] } N h X

5 Valuation of American partial barrier options DSS t ; A 3 = e q t { [ N h ] N h X [ N h ] } N h X E ; A 4 = X [ b β β N g bβ β N g ] where N is the standard normal distribution function ln z t h z = ln z t h z = t t = r q = r q b = and β = 3 Analytic approximation for American partial barrier options g z = ln z β t t b r. In this section we consider the partial barrier option of American type. American options give their holders the flexibility of early exercise. An American up-and-in put option can be exercised before the expiration time when it is in the money but only after the stoc price rises above the noc-in barrier. We consider the up-and-in put where the barrier appears at a specified time strictly after the option s initiation. hat is if the underlying asset price hits the up-barrier over the time period between and expiration then the put option can be exercised before or at time with strie price X. If the asset price never crosses the up-barrier between and expiration this option pays off zero. In order to obtain the approximation to valuing American partial barrier option using barrier derivatives under exercise policies we use the digital opions D ; A DSS t ; A and EK τ ; A for t< defined in Sect.. We denote by τ the first time that the stoc price reaches the barrier after time.forτ it is the first time that the stoc price falls to the exercise policy after τ.letk denote the optimal exercise policy. Let A 5 ={τ <τ K >S <X} be the event of exercise at maturity under the optimal policy and A 6 ={τ K <} be the event of early exercise under the optimal policy. hen the value of this partial up-and-in put is written as PIP =X DS t ; A 5 DSS t ; A 5 EK t ; A 6 For the barrier approximation of this option we consider a class of all constant exercise policies. We let A 7 ={τ <τ >S <X} be the event of 3

6 D. Jun H. Ku exercise at maturity under a constant policy and A 8 ={τ <} be the event of early exercise under policy. hen we can express the option price as PIP const = max K c [X DS t ; A 7 DSS t ; A 7 E ; A 8 ] 3. If the set of policies considered contains all continuous functions then the resulting put value will be exact. Since the set K c is the set of all constant functions then the resulting value will be an approximation providing a very tight lower bound to the put price. We first present a useful Lemma to calculate the values D DS and E of digital digital share and first-touch digital. We recall that the standard normal density function and distribution function nx = π e x and Nx = and the bivariate standard normal distribution function N a b; ρ = a b π ρ exp where ρ is the coefficient of correlation. Lemma 3. For any real aαβγ and δ a a t γ δ n a γ Nα βtdt = N δ δ δ n t γ Nα βtdt = N Proof Letting u = t γ δ a δ t γ δ n Nα βtdt = δ x ntdt x ρxy y γ a δ a γ δ ρ α βγ β δ ; α βγ β δ ; αβδuγ dxdy βδ β δ βδ β δ uv e dvdu. π Change the variables and define a coefficient of correlation ρ as follows: x = u y = v βδu β δ ρ = βδ β δ. 3

7 Valuation of American partial barrier options hen a = t γ δ n Nα βtdt δ a γ δ αβγ β δ a γ = N δ π ρ exp α βγ β δ ; x ρxy y ρ βδ. β δ dxdy For the integral a t γ δ n δ Nα βtdt we can get the above result by a similar method. Let us introduce a process X t = ln S 0. hen X t is a Brownian motion with drift. Define τ u and τ ul by stopping times for this process defined as the first time that X t = u>x 0 after time and the first time after τ u that X t = l<u respectively. Lemma 3. For x l the probability that the process X t crosses u after time and then hits l before expiration and X is greater than x is Pτ ul X >x X 0 = 0 u = exp l u N l u x ; l u exp N l x ; Proof Pτ ul X >x X 0 = 0 = PX <u τ ul X >x X 0 = 0 PX u τ ul X >x X 0 =

8 D. Jun H. Ku Since u>l { X u τ ul } is equivalent to { X u τ l }. hen Pτ ul X >x X 0 = 0 = PX <u τ ul X >x X 0 = 0 PX u τ l X >x X 0 = 0 = u u π e e π x Pτ ul X >x X = x dx x Pτ l X >x X = x dx sing Lemma in the Appendix of Ingersoll 998 we have Pτ ul X >x X 0 = 0 u = e x e l u x l u x N dx π u = e l u e l e π u u x e l x N l x x dx e x x l u x N dx π e π Applying Lemma 3. we obtain x N l x x dx Pτ ul X >x X 0 = 0 u = exp l u N l u x ; l u exp N l x ; Lemma 3.3 he probability that the process X t crosses u after time and then falls below l before time is 3

9 Valuation of American partial barrier options Pτ ul X 0 = 0 u = exp l u N l u ; l u exp N l ; u u exp N l u ; u N l ; Proof We note that Pτ ul X 0 = 0 = Pτ ul X >l X 0 = 0 Pτ u X l X 0 = since { τ ul X l } = { τ u X l }. he first probability of 3.3is given by Lemma 3. with x = l and the second one can be calculated by a similar method to the proof of Lemma 3.. Pτ u X l X 0 = 0 u = e x Pτ u X l X = x dx π u e π x Pτ u X l X = x dx When X >u the event { τ u X l } is equivalent to {X l}. hus Pτ u X l X 0 = 0 u = e x Pτ u X l X = x dx π = u u e π x PX l X = x dx e x e l u u x x N dx π 3

10 u = e u e x l x N dx π u u e x l u x N dx π e π Applying Lemma 3. again to obtain x N l x dx D. Jun H. Ku Pτ u X l X 0 = 0 u u = exp N l u ; u N l ; Formulas for the option with barrier greater than strie price We assume X. he valuation formulas for the digitals in 3. are DS t ; A 7 = e r t [G X G ] e r t [G X G ] e r t DSS t ; A 7 [G 3 X G 3 ] e r t[ G 4 X G 4 ] = e q t [ G X G ] e q t [ G X G ] e q t E ; A 8 [ β b = X 3 [ G 3 X G 3 ] e q t[ G 4 X G 4 ] β b βb H 3 βb H β b H 4 ] βb H

11 Valuation of American partial barrier options where G i X G i X and H i i =...4 are given in heorems 3. and 3.3. Remar 3. When the barrier appears immediately after the option s initiation i.e. converges to 0 it can be checed that the above formulas for D DS and E become the values of these digitals for American barrier option given in Sect.. Lemma 3.4 For l x u the probability that the process X t crosses u after time and then does not fall below l before expiration and its value at time is less than x is where Pτ u < τ ul > X x X 0 = 0 u l = exp [F x F l] exp [F x F l] exp l u [F 3 x F 3 l] F 4 x F 4 l F x = N u F x = N u F 3 x = N u F 4 x = N u x u ; l x ; l u x ; x ;. Proof Pτ u < τ ul > X x X 0 = 0 = Pτ u < X x X 0 = 0 Pτ u < τ ul X x X 0 =0 = Pτ u < X x X 0 = 0 Pτ ul X x X 0 = 0 = Pτ u < X x X 0 = 0 Pτ ul X 0 = 0 Pτ ul X >x X 0 = 0 he first probability is obtained from 3.4 with l = x. he second and third probabilities are calculated by Lemmas 3.3 and 3.. 3

12 D. Jun H. Ku We now consider the digital options for American up-and-in put where the underlying asset price is monitored over the time period between and maturity under a constant exercise policy. he values of these options are determined from the above Lemmas. heorem 3. For X the values of a digital option and a digital share at time t< for the event A 7 = { τ < τ > S X } are DS t; A 7 = e r t [G X G ] e r t [G X G ] e r t DSS t; A 7 [G 3 X G 3 ] e r t [G 4 X G 4 ] = e q t [ G X G ] e q t [ G X G ] e q t [ G 3 X G 3 ] e q t[ G 4 X G 4 ] where G X = N G X = N G 3 X = N h 3 G 4 X = N h 3 h 3 h X h 3 h X h X h X t t t t t t t t ; ; ; ; and ln z t h z = h 3 z = ln z t t. t G i X is the same as G i X except = r q in replacement of for i =

13 Valuation of American partial barrier options Proof Apply Lemma 3.4 with letting u = ln S t l= ln S t and x = ln S X t derive the ris-neutral probability of exercise at maturity. hen to Pτ < τ > S X = [G X G ] [G X G ] [G 3 X G 3 ] G 4 X G 4 where G i X for i = 3 4 are defined as above. hen the value of digital option DS t; A 7 at time t DS t; A 7 = e r t Pτ < τ > S X is obtained as desired. he digital share DSS t; A 7 can be valued by changing to = r q and replacing the discount factor e r t by e q t See for example Ingersoll 000. heorem 3. he value of a digital option and a digital share at time t for the event A 8 ={τ <} are DS t; A 8 = e r t [ G DSS t; A 8 = e q t [ G ] G G 3 G ] G G 3 G 4 Proof Apply Lemma 3.3 with u = ln S t l= ln S t and x = ln S X t the ris-neutral probability of early exercise. hen to derive Pτ = hus the value of digital option at time t G G G 3 G 4 DS t; A 8 = e r t Pτ is obtained. he digital share DSS t; A 8 can be valued as in heorem 3.. 3

14 D. Jun H. Ku nder a constant exercise policy the up-and-in put option will be exercised early prior to maturity for X if the stoc price hits the up-barrier after and then falls to after τ before maturity. Now we consider the value of a first-touch digital for time τ. We examine the case when there is no dividend on the stoc first. Lemma 3.5 If the stoc does not pay dividends the value of a first-touch digital for the event A 8 ={τ <} is E; A 8 [ = X r S t G r r ] G G 3 G 4 where G i is the same as G i except = r in replacement of for i = 3 4. Proof he first-touch digital pays X at time τ. his money can be used to purchase X shares of the stoc at that time. Since the shares do not pay dividends it is worth X S at maturity i.e. E; A 8 = X DSS t; A 8 where DSS t; A 8 is the value when q = 0in3.5. heorem 3.3 he value of the first-touch digital for the event A 8 is [ β b E; A 8 = X β b βb H 3 βb H β b H 4 ] βb H where H = N H = N H 3 = N g H 4 = N g g g ; g g ; g ; g ; t t t t t t t t 3

15 Valuation of American partial barrier options and g z = ln z β t t g z = ln z β t. t Proof When the stoc price pays dividends the asset price follows the continuous diffusion process d = r q dt dw. o eliminate the dividend term in the process we set V t = S β b t where hen by Ito s lemma b = and β = b r. dv t = rv t dt β bv t dw t. 3.6 We may apply Lemma 3.5 to the process V t since 3.6 does not contain the dividend term. he barriers for V t corresponding to and are β b and β b. Furthermore the volatility is replaced by β b. hen the value of the first-touch digital for the event A 8 is EVt β b ; A 8 = X β b V t β b V t r β b H β b β b r β b β b H 3 H 4 V t r β b H where = r β b H = N ln β b V t t ln β b V t t β b β b t β b t ; t t H = N ln β b V t t ln β b β b V t t t β b t ; t t H 3 = N ln β b V t t ln β b V t t β b β b t β b t ; t t 3

16 H 4 = N ln β b V t t β b t ln β b D. Jun H. Ku V t t β b t ;. t t hus E; A 8 β b = X β b r β b [ β b = X β b βb H 3 β b r β b H 3 H 4 βb H H β b H 4 ] βb H β b r β b H where H = N H = N H 3 = N g H 4 = N g g g ; g g ; g ; g ; t t t t t t t t and g z = ln z β t t g z = ln z β t. t he following graph Fig. illustrates the American up-and-in put prices using the approximation 3. with different values of initial spot S 0 and barrier s starting time. Also Fig. shows the option prices with different values of up-barrier and. 3

17 Valuation of American partial barrier options S Fig. PIP const result varying S 0 and when X option parameters: = 05 X = 00 r = 0.05 = 0.3 and = Fig. PIP const result varying and when X option parameters: S 0 = 00 X = 00 r = 0.05 = 0.3 and = 3. Formulas for the option with barrier less than strie price We assume <X. he valuation formulas for the digitals in 3. are DS t; A 7 =e r t [G G G 5 X G 5 ] e r t [G X G ] e r t [G 3 X G 3 ] e r t[ ] G 4 G 4 G 6 X G 6 3

18 D. Jun H. Ku DSS t; A 7 = e q t [ G G G 5 X G 5 ] e q t [ G X G ] e q t [ G 3 X G 3 ] e q t [G 4 G 4 G 6 X G 6 ] E ; A 8 [ β b = X β b βb H 3 βb H β b H 4 ] βb H where G i X G i Xi =...6 and H i i =...4 are given in heorems and 3.4. Lemma 3.6 For x>u the probability that the process X t crosses u after and then does not fall below l before time and its value at time is less than x is Pτ u < τ ul > X x X 0 = 0 u l = exp [F u F l F 5 x F 5 u] exp [F x F l] exp l u [F 3 x F 3 l] F 4 u F 4 l F 6 x F 6 u where F 5 x = N u F 6 x = N u x u ; x ;. Proof Pτ u < τ ul > X x X 0 = 0 = Pτ u < X x X 0 = 0 Pτ ul X 0 = 0 Pτ ul X >x X 0 = 0 = Pτ u < X u X 0 = 0 Pτ u < u<x x X 0 = 0 Pτ ul X 0 = 0 Pτ ul X >x X 0 = 0. 3

19 Valuation of American partial barrier options he third and fourth probabilities above are calculated by Lemmas 3.3 and 3.. he first probability comes from 3.4 with a replacement of l by u. hus we only need to prove the second probability. Pτ u < u<x x X 0 = 0 u = e x Pτ u < u<x x X = x dx π = = u u u u N e π x Pτ u < u<x x X = x dx e x Pu < X x X = x dx π e π x Pτ u < u<x x X = x dx e x [ x x N π u x ] dx [ x u x N N = N u e u x ; [ u N N u u e π x e u x u x ] dx u N u ; x u ; u ; heorem 3.4 For X> the values of a digital option and a digital share at time t< for the event A 7 = { τ < τ > S X } are ] DS t; A 7 =e r t [G G G 5 X G 5 ] 3

20 D. Jun H. Ku e r t [G X G ] e r t [G 3 X G 3 ]e r t[ ] G 4 G 4 G 6 X G 6 DSS t; A 7 = e q t [ G G G 5 X G 5 ] e q t [ G X G ] e q t [ G 3 X G 3 ] e q t[ ] G 4 G 4 G 6 X G 6 where G 5 X = N G 6 X = N h 3 h h 3 h X X ; ; t t. t t G i X is the same as G i X except = r q in replacement of for i =...6. Proof Apply Lemma 3.6 with having u = ln S t l = ln S t and x = ln S X t. hen we obtain the result similarly as in the proof of heorem 3.. In the following Fig. 3 illustrates the American up-and-in put prices using 3. with different values of initial spot S 0 and when <X. Also Fig. 4 shows the option prices with varying up-barrier and. We next present the values of American partial up-and-in put option by our formulae and compare them with those by Monte Carlo method with an Antithetic Variate See for example Glasserman 003 and by the rinomial lattice model using the adaptive mesh model AMM. able shows the values of American partial up-and-in put option whose monitoring period begins at predetermined time with varying initial price S 0 and strie price X. he parameter values that we used are = 05 = 0.3 = 0.5 = 0. r= 0.05 and q = 0. he values of S 0 vary from 96 to 04 and the values of X from 95 to 05. he values PIP const in able are calculated by the formulae in Sect. 3.. able shows the values of American partial up-and-in put option with different levels of upper barrier and time. he parameter values in this computation are S 0 = 00 X= 05 = 0.3 = 0.5 r= 0.05 he adaptive mesh method Figlewsi and Gao 999 sharply reduces nonlinearity error by grafting one or more small sections of fine high-resolution lattice onto a tree with coarser time and price steps. 3

21 Valuation of American partial barrier options S Fig. 3 PIP const result varying S 0 and when < X option parameters: = 05 X = 0 r = 0.05 = 0.3 and = X Fig. 4 PIP const result varying and when < X option parameters: S 0 = 00 X = 0 r = 0.05 = 0.3 and = and q = 0. he values of vary from 0 to 08 and the values of from 0. to 0.4. he values PIP const in able are computed by using the formulae in Sects. 3. and 3.. VNis an option value of PIP const using barrier options with constant policy barriers in 3.. N is the element number of constant policy set K c to see the best policy where policies are evenly spaced from 0 to X. Since the American put option comes into action only if the up-barrier is hit after the option price PIP const decreases as the initial stoc price gets farther apart from the up-barrier. We notice that as the number N of constant exercise policies increases the option value VN converges to a constant very quicly as shown in Fig. 5. 3

22 D. Jun H. Ku able Comparison of American partial barrier put option values PIP with varying S 0 and strie price X S 0 X V0 V30 V50 V00 V500 MC AMM Option parameters: = 05 = 0. = 0.5 = 0.3 r = 0.05 q = 0. VN is an option value of PIP const where N is the number of constant policy barriers. MC is a result of simulation using the Antithetic Variates a Variance Reduction Method of Monte Carlo simulation. AMM5 is a result of rinomial lattice method by the AMM with level 5. is the optimal policy barrier for V0000 MC is a result of simulation using the Antithetic Variates a Variance Reduction Method of Monte Carlo simulation. For the American partial barrier option using policy barriers Monte Carlo method requires much larger amount of computer time because a large number of sample paths and policy barriers and a large enough monitoring frequency must be needed. For the Monte Carlo approximation in ables and the computer time is more than 0000 times as long as for our formulae method to obtain the similar results under the same policy numbers. For the MC results in ables and a monitoring frequency is 000 the number of sample paths is 5000 and the number of policy barriers evenly spaced from 0 to X is 00. AMM5 is an outcome of rinomial lattice model by the adaptive mesh model presented in Figlewsi and Gao 999. his is the approach for constructing a lattice- 3

23 Valuation of American partial barrier options able Comparison of American partial barrier put option values PIP with varying and V0 V30 V50 V00 V500 MC AMM Option parameters: S 0 = 00 X= 05 = 0.3 = 0.5 r= 0.05 q= 0. VNis an option value of PIP const where N is the number of constant policy barriers. MC is a result of simulation using the Antithetic variates a Variance Reduction Method of Monte Carlo simulation. AMM5 is a result of trinomial lattice method by the AMM with level 5. is the optimal policy barrier for V Value Policy Number Fig. 5 PIP const result varying policy barrier number N option parameters: S 0 = 00 X= 00 = 05 r= 0.05 = 0.3 = 0. and = 0.5 based valuation model that allows the user to vary the resolution in different parts of the tree. While the binomial tree for American barrier options is not as efficient as it is for standard American options this adaptive mesh method can provide a more 3

24 D. Jun H. Ku efficient benchmar for comparison with our explicit formulas. he AMM for barrier options with level 5 is used in ables and. We note that the last column is the optimal policy barrier when N = and the best constant policy depends of course on option parameters such as initial stoc price strie price upper barrier and. 4 Conclusion his paper studies the valuation problem of American partial barrier option. Because a wide variety of traded options are American type the problem of valuing American options has been an important topic in financial economics. he literature of American option has proposed good numerical solution methods and anlytic approximations. However American partial barrier options are much more difficult to price. o the best of our nowledge the literature suggests no approximation formula for American partial barrier options. his paper adopts the barrier approximation method under constant exercise policies and provides an analytic approximation as the finite sum of bivariate normal distribution functions. Due to this contribution one can calculate the American partial barier option prices in a simple and speedy way. References Barone-Adesi G. & Whaley R An efficient analytic approximation of American option values. Journal of Finance Brennan M. J. & Schwartz E. S Saving bonds retractable bonds and callable bonds. Journal of Financial Economics Broadie M. & Detemple J American options valuations: New bounds approximations and a comparison of existing methods. Review of Financial udies Carr P. Jarrow R. & Myneni R. 99. Alternative characterizations of American puts. Mathematical Finance Chung S. L. Hung M. W. & Wang J. Y. 00. ight bounds on American option prices. Journal of Baning & Finance Cox J. C. Ross S. A. & Rubinstein M Option pricing: A simplified approach. Journal of Financial Economics Dai M. & Kwo Y. K Knoc-in American options. Journal of Futures Marets Figlewsi S. & Gao B he adaptive mesh model: A new approach to efficient option pricing. Journal of Financail Economics Gao B. Hunag J. & Subrahmanyam M he valuation of American options using the decomposition technique. Journal of Economic Dynamic & Control Geman H. & Yor M Pricing and hedging double-barrier options: A probabilistic approach. Mathematical Finance Gese R. & Johnson H. E he American put option valued analytically. Journal of Finance Glasserman P Monte carlo methods in financial engineering. New Yor: Springer-Verlag. Guillaume Window double barrier options. Review of Derivatives Research Heynen R. C. & Kat H. M Partial barrier options. Journal of Financial Engineering Huang J. Subrahmanyam M. & Yu G Pricing and hedging American options: A recursive integration method. Review of Financial udies Ingersoll J. E. Jr Approximating American options and other financial contracts using barrier derivatives. Journal of Computational Finance 85. Ingersoll J. E. Jr Digital contracts: Simple tools for pricing complex derivatives. he Journal of Business

25 Valuation of American partial barrier options Jaca S. D. 99. Optimal stopping and the American put. Mathematical Finance 4. Johnson H. E An analytic approximation for the American put price. Journal of Financial and Quantitative Analysis Ju N Pricing an American option by approximating its early exercise boundary as a multipiece exponential function. Review of Financial udies Kim I. J he analytic valuation of American options. Review of Financial udies Kunitomo N. & Ieda M. 99. Pricing options with curved boundaries. Mathematical Finance Longstaff F. A. & Schwartz E. S. 00. Valuing American options by simulation: A simple least-squares approach. Review of Financial udies MacMillan L. W An analytical approximation for the American put prices. Advances in Futures and Options Research Merton R. C heory of rational option pricing. Bell Journal of Economics and Management Science Parinson M Option pricing: he American put. Journal of Business Rich D he mathematical foundations of barrier option pricing theory. Advances in Futures and Options Research Rubinstein M. & Reiner E. 99. Breaing down the barriers. Ris Sullivan M. A Valuing American put options using Gaussian quadrature. he Review of Financial udies 3Spring