Pricing American-style Basket Options By Implied Binomial Tree

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1 Priing Amerian-style Basket Options By Implied Binomial Tree Henry Wan * Draft: Marh 00 Abstrat It is known that the most diffiult problem of priing and hedging multi-asset basket options are those with both high dimensionality and early exerise. This artile proposes a numerial algorithm by reduing multivariate distributions of a portfolio into a single variable and modeling that as a univariate stohasti proess in the form of an implied binomial tree. It is demonstrated that the method provides a fast and flexible way of priing and hedging high dimensional multi-asset basket options with early exerise. * This artile is a report to the Applied Finane Projet of University of California at Berkeley. The author is grateful for omments of traders at Wahovia Seurities, Jens Jakwerth at University of Konstanz, Germany, and Mark Broadie at Columbia University. The author is partiularly grateful for the extensive and insightful suggestions of Mark Rubinstein for advising the projet.

2 Introdution A basket option is an option on a portfolio of several underlying assets. With growing diversifiation in investor's portfolio, options on suh portfolios are inreasingly demanded. One example of the demand is purhasing a protetive put option on a portfolio suh that investor's down side risk is limited. However, Amerian-style basket options ontinue to be hallenging in terms of both algorithm omplexity and omputational burden. The modeling and mathematis of priing basket options are similar to those of options on a single asset exept that there are orrelated random walks and multi-variate Ito's lemma that need to be applied. In very few speial ases, suh as exhange options, losed form solutions an be found. More often, numerial methods suh as numerial integration, finite-differene methods and simulations are neessary for low or medium size problems. When dimension is higher, it is relatively heaper to use simulations sine its omputational ost does not inrease exponentially as other methods. The ease exists only for European-style options. It is however known that the most diffiult problems of priing and hedging multi-asset basket options are those with both high dimensionality, for whih we would like to use simulation, and with early exerise, for whih we would like to use either binomial tree or finite differene methods. "There is urrently no numerial method that opes well with suh a problem" (Wilmott, 998). There have been many artiles published on topis of multi-dimensional Amerian-style options. They an be ategorized by either lattie-based or simulation-based approahes. Lattie-based approahes, suh as binomial trees, trinomial trees and finite differene methods, are widely used for options on a single asset. When dimension is higher, usually up to four, extensions of binomial and trinomial trees from the univariate binomial tree (Cox, Ross and Rubinstein 979) an be applied. Rubinstein (994a) models twoasset rainbow options using "binomial pyramids". It generalizes one-dimension up and down trees into twodimensional squares. The similar approah an be further extended into higher dimensions. Appendex A generalizes this approah into four dimensions. The finite differene methods an also be extended into multiple dimension ases by using Alternating Diretions Impliit (ADI) method (Clewlow and Strikland, 998). Although these lattie-based approahes are generally easy to deal with early exerise by a bakward algorithm, their memory and omputation requirements explode exponentially as the dimension of problem inreases. On the other hand, without exponentially growing omputation effort as most lattie-based methods do for higher dimension options, simulation enjoys high flexibility and modest omputational ost independent of dimensions of a problem. The embedded forward simulation algorithm, however, underlies its diffiulty in priing options with early exerise features, suh as Amerian options. These options generally require a bakward algorithm to determine the optimal exerise poliy. Sine the laim written by Hull (997) that "A limitation of the simulation approah is that it an be used only for European-style derivatives", many papers have devoted to overome this hallenge. As the reviewing work done by Broadie and Glasserman (997b), there are three main tehniques in applying simulation into multi-asset Amerian options. In many ases, multiple of suh tehniques are applied into an algorithm. One ategory of tehniques is based on parametrization of exerise boundary and uses simulation to maximize the expeted payoff within the parametri family. Different optimization methods and families of urve fitting funtions have been proposed. The seond approah is based on finding both upper and lower bounds for option prie and giving valid onfidene intervals for the true value. The simulated trees method and the stohasti mesh algorithm proposed by Broadie and Glasserman (997a, 997) and the reent primal-dual simulation algorithm by Andersen and Broadie (00) belong to this ategory. In the simulated tree method, eah single node generates its own non-ombining independent subtree from samples. The priing is then arried out by a bakward reursive proedure. Although the method is linear in the problem dimension, it is exponential in terms of the number of steps. The stohasti mesh method has the advantage of linear in the number of exerise opportunities, but interloks among nodes in the mesh reate onsiderably ompliation in the algorithm. Reently there are developments of applying duality to ompute an upper bound of true prie from the speifiation of

3 some arbitrary martingale proess. The tightness of the upper bound depends ritially on the hoie of the martingale proess. Andersen and Broadie (00) signifiantly improve the performane of the alulation. The third ategory of tehniques, whih is also the most widely used one in ombining with other approahes, is the dynami programming style bakward reursion in determining the optimal exerise poliy. Tilley (993) used bakward algorithm and bundling tehnique for approximating the prie and the optimal exerise deision into the simulation. It is effiient and easy in priing one-dimension Amerian options. Sine then artiles were published on the improvement of this basi onept. Longstaff and Shwartz (00) used least-squares regression on polynomials to approximate the holding value and optimum to exerise. Barraquand and Martineau (995) applied a state aggregation tehnique and redued a high-dimensional problem into a single dimension one sine the exerise deision only depends on the single variable, the intrinsi value. The resulting one-dimension problem an be solved quikly by standard bakward dynami programming and provides signifiant omputation saving for high-dimensional problems. Although it is extremely fast in alulating an approximation of prie, the auray primarily depends on how well the optimal exerise poliy an be represented by the single payoff variable. In some multi-dimensional ases, the option payoff is not suffiient to determine the optimum to exerise. As Broadie and Glasserman (997) indiated, "the stratifiation algorithm will not onverge to the orret value." The approah to priing basket options in the artile is similar in spirit to the onept of Barraquand and Martineau (995) by reduing a high-dimensional problem into a single variable one. In the ase of basket options, the basket itself is the variable in determining its holding and exerise values. Instead of using simulated histogram, our method uses a set of European-style standard options to infer the risk-neutral probabilities and hene the stohasti proess of the basket represented by an implied binomial tree. We will show the results of this approah onverge to the orret value with modestly inreasing the number of steps. The Approah The method we use to prie a basket option involves four steps. First, we value European-style standard all and put options with different strikes, whose underlying is the multi-asset basket, by simulations. Seond, we infer the risk-neutral probabilities of the basket from the set of European-style options on the basket, the assoiated market prie of the basket, and the risk-less interest rate bond. Third, we reover the fully speified stohasti proess of the basket, in the form of an implied binomial tree, from its risk-neutral probabilities. Finally, with the implied tree, we an alulate the value and hedging parameters of any derivative instruments on the basket maturing with or before the European options. simulation in valuing European-style standard basket options The first step is valuing European-style standard all and put options of the basket. As we know, an - variate binomial tree beomes exponentially ompliated and omputationally expensive when both the number of assets and the number of moves are getting larger. For example, when a tree has 00 moves and deals with only 4 assets simultaneously, the number of nodes at the end will be (00+) 4, or about 00 million. Suh numerous omputations and memory requirement prevents us from using the lattie-based method to value high-dimensional Europen-style basket options. Another set of approahes to value multi-asset basket options is by simulations. Sine the value of a European-style option is the risk-neutral expetation of its disounted payoff at the time of maturity, simulations obtain an estimate of suh a risk-neutral expetation by omputing the average of a large number of disounted payoffs. Beause the omputation effort of simulation is independent of the number of random fators, it is heap to value European-style multi-asset basket options with high dimensionality in terms of omputation and memory requirements. The simulation

4 is also flexible to inorporate other proesses and fators, suh as random volatility, random interest rates, jumps in asset pries, and other more realisti market onditions. Our approah, however, is not valuing Amerian-style basket options by simulations. The purpose of using simulation is to utilize its flexibility in modeling multiple random fators or more realisti random proesses, as a tool to obtain the risk-neutral distribution of the underlying basket. The approah involves three steps in applying simulations : () Simulate the ending asset pries in a basket under the risk-neutral measure. A way of simulating multi-dimensional stohasti proess is desribed in Appendix B, where a orrelation matrix among different proess is assumed given. In general, a European option an be valued as the expeted risk-neutral payoff at expiry. M s Cˆ ( ) = exp( rt ) Payoff T () M s= where M is the number samples. () Estimate the mean and standard deviation of the return on the basket. Assuming there are M simulation samples, eah simulation ends up with a set of asset pries at the expiration T. The asset pries, therefore, determine a value of the basket and hene the return on the basket from the time zero. With M samples of basket returns, one an alulate the unbiased estimations on both mean and standard deviation of basket returns under the risk-neutral measure: ( s) 0 M M ( s) ( s) [ R ] ˆ σ = ( R ˆ μ ) s= M s= ( s) ( s) ( s) [ n S + n S + L + n S ] log[ S ] ˆ μ = M R S log n S,0, T + n S,0, T + L + n S,0, T 0 (3) Value the European-style standard all and put options based on a set of equally log-spaed strike pries. Consider we want to value m+ options with different strikes, the strikes we hoose are: j m ( ˆ μ + x ˆ ) x, j = 0,,, m K j = S0 exp jσ j L, m The options we hoose are all out-of-money options. That is, we value the European put options for the exp μˆ K j exp μˆ. strikes K j < ( ), and the European all options for those strikes ( ) S 0 The valuation for the set of alls and puts are fast beause we only need to proeed the time-onsuming simulation in step () one. After the M outomes have been simulated, the valuation is simply applying Equation () in omputing expeted payoffs. The effiieny of all simulation an be improved by using variane redution tehniques. In the interest of this artile, the desription of these tehniques is not emphasized. S 0

5 Implied Ending Risk-eutral Probabilities The seond step of the approah is to infer the implied ending risk-neutral probabilities of the univariate basket from the set of European-style alls and puts on the basket. Consider a state seurity, or Arrow-Debreu seurity, whih pays one dollar if a speifi future state is realized and zero otherwise, the prie of suh a seurity an be used to build more ompliated payoffs at the expiration. For example, if a seurity pays X, X,, and X n if the states,,, and are realized respetively. The prie of this seurity at time zero will be the sum of those payoffs weighted by state pries assoiated with the orresponding states. C = (X *q + X * q + + X n * q n ) / r ow, if we disretize the ontinuous prie spae into disrete states similar to a binomial tree at the end of the expiration date, the payoffs of a standard European all option with strike prie K under those states will be: C = max { 0, ST j K } j = 0,,,., m The prie of a European all option is: C = j π j max{0, ST j K} Similarly, the prie of a European put option will be: P = j π j max{0, K ST j } As we mentioned, we have obtained a set of European-style all and put options on the basket through the simulation. We an use them to imply the state pries and hene the risk-neutral probabilities of the univariate basket. Assuming there are m+ disrete states at the expiration. We want to find out the state pries for them. The state pries an be omputed by the pries of standard European options, the urrent value of the basket, and the prie of zero-oupon risk-free bond. If the states are hosen to be the set of strikes we hoose for the standard options in the previous setion, where K 0 < K < < K m, we an reate a payoff table as shown below for the standard alls, puts, the underlying asset and the risk-free bond. States Seurity Pries Strike K 0 K K K j- K j K m- K m- K m Put P(K ) K K K 0 Put P(K ) K K K 0 K K Put P(K 3 ) K 3 K 3 K 0 K 3 K K 3 K Put P(K j- ) K j- K j- K 0 K j- K K j- K 0 Call C(K j ) K j 0 K m- K j K m- K j K m K j Call C(K m-3 ) K m-3 K m- K m-3 K m- K m-3 K m K m-3 Call C(K m- ) K m- K m- K m- K m K m- Call C(K m- ) K m- K m K m- Asset S 0 0 e d*t K 0 e d*t K e d*t K e d*t K j- e d*t K j e d*t K m- e d*t K m- e d*t K m Bond e -r*t Table : Payoff table of standard alls and puts

6 Consider the lowest strike of the put option P(K ), it pays K K 0 when the state is K 0 and zero for all the states above K 0. Therefore, the state prie for K 0 is nothing but P(K )/(K K 0 ). P(K ) = (K K 0 ) π 0 π 0 = P(K ) / (K K 0 ) ow onsider the put options with strike K, its prie is given by: P(K ) = (K K 0 ) π 0 + (K K ) π There is only one unknown π, sine π 0 has been solved previously. We an ontinue this proess forward by applying the prie of put option and previously solved state pries π k, k = 0,,, i- to solve for the state prie π i- : P(K i ) = k=0,,i- (K i K k ) π k + (K i K i- ) π i- To avoid building up numerial errors in states with high pries, we use all options and start with the high end of strikes. For example, the state prie for K m an be omputed by C(K m- )/(K m K m- ) sine the all option C(K m- ) only pays K m K m- when the state is K m and zero elsewhere. Other state pries an be omputed progressively by applying the following: C(K i ) = (K i+ K i ) π i+ + k=i+,,m (K k K i ) π k The above proedure an be used to ompute most of the state pries exept two states adjaent to the forward pries of the underlying asset. In our ase, they are strikes K j- and K j where the puts and alls are met in our payoff table. To solve for state pries for these two states, another two equations are introdued to satisfy no-arbitrage onditions for the forward prie of underlying asset and the prie of risk-free bond. Aording to the payoffs of these two seurities, we have, S 0 exp(-dt) = k=0,,j- K k π k + K j- π j- + K j π j + k=j+,,m K k π k, and exp(-rt) = k=0,,j- π k + π j- + π j + k=j+,,m π k Two equations and two unknowns, we an solve for both π j- and π j. After we obtain all the state pries, the risk-neutral probabilities are nothing but the state pries inflated by interest rate. q i = π i exp(rt), for all i = 0,,, m Implied Binomial Tree of the basket After retrieving the ending risk-neutral probabilities, the next step of our approah is to reover the implied stohasti proess of the basket, in the form of its implied binomial tree. The idea behind this is reognizing that the pries of standard European options of the basket embed information about the underlying basket portfolio. If there exists a diffusion proess to ahieve its values of European options, the Amerian-style or other exoti options should follow the same proess. Here, the standard European-style options are treated as fundamental seurities, whose pries are given. In our ase, the option pries are through simulations. Rubinstein (994, 998) presents a method in onstruting an implied binomial tree by using the ending risk-neutral distributions. Other approahes in onstruting implied trees, for example implied trinomial trees, an be found from the overview literature by Jakwerth (999). Implied trees are not unique without making speifi assumptions on how the underlying asset prie reahes the terminal distribution. The main assumption of Rubinstein's method is that the path probabilities for all paths reahing the same terminal state are equal. Here is a summary of the method. Assuming that an asset may either move up or down at

7 any given time, and assuming two paths with up-then-down and down-then-up lead to the same outome of the asset in the end. These assumptions lead a ombining binomial tree as shown in Figure. At the end of n-th period, there are total n+ possible outomes or nodes on the tree. Eah has a node probability P node (n,j) whih is the disrete probability mass of the outome at node (n,j). Figure also shows a ommon bell-shaped node probabilities. With the assumption of ombining binomial tree, there are n-hoose-j leading paths that ause the same outome at node (n,j). We further assume all the paths leading to the same outome (n,j) have the same path probability p path (n,j). Therefore, p path ( n, j) = P node n C j ( n, j) = P node ( n, j) n! j!( n j)! Ending risk-neutral dist. Probability Tree P(n-,j) q -q P(n,j), p(n,j) P(n,j+), p(n,j+) Figure : Implied Binomial Tree Moving bakward from the n-th period into "n-"-th period, for a node (n-,j), it has two paths to move ahead, either up to the node (n, j) or down to the node (n,j+). Eah path has the path probability of p path (n,j) and p path (n,j+). So the node probability of this node (n-,j), whih is the probability of getting to there from the root of the tree, is the sum of two path probabilities leading to the next period. P node ( n, j) = p ( n, j) + p ( n, j + ) path path The loal up-move probability q is the probability of the movement to (n,j) given the ondition that the movement is started from node (n-,j). Therefore, the onditional down-move probability is -q. p q = P node ( n, j) ( n, j) path = p path p path ( n, j) ( n, j) + p ( n, j + ) path If we are given a risk-neutral probability distribution at the end of expiry, the implied probability tree an be onstruted, by working bakward from the period to 0 by applying the above equations reursively. At the same time, trees for both asset and the option are also onstruted bakward. Options payoff Asset Tree q S(n-,j) -q S(n,j) S(n,j+) Options Tree q V(n-,j) -q V(n,j) V(n,j+) Figure : Implied Asset and Option Trees

8 In the ase of an Amerian put option on a basket, whose ending risk-neutral probabilities are given, the reursive formula for both the value of basket and the put option are as following: S(n-,j) = [q * S(n,j) + (-q) * S(n, j+)] * exp[-(r-d)h] P(n-,j) = max { K S(n-,j), [q * P(n,j) + (-q) * P(n,j+)] * exp(-rh) } Amerian all options an be proeeded similarly. With the implied tree on the basket, we are able to both prie and hedge the basket options by standard tehniques used in binomial tree. The following setion demonstrates numerial results of implementing our approah in omparing to other methods. Results Single asset options Our first example is a speial ase of single asset options. We will show how to bak out the implied binomial tree and obtain the exat same results as those alulated by the ordinary binomial tree for both European and Amerian-style single asset options. Consider options on single underlying stok: Time to Expiration: T =.0 year Interest Rate: r = 0% Dividend Yield: δ = 5% Volatility: σ = 0% Spot Prie: S 0 = 00 Strike Pries: K = 60, 70, 80, 90, 00, 0, 0, 30, 40. We first onstrut a simple two-step binomial tree to illustrate our approah. In the example, we have two moves, i.e., m =, h = 0.5. After omputing the multipliative fators for both upward and downward moves, where u = exp[(rδ½σ )h + σ h] =.693, and d = exp[(rδ½σ )h σ h] = 0.88, We have the binomial tree as shown in Exhibit. The tree is onstruted suh that there is an equal probability of one-half for the asset prie to either move up or move down. ½ ½ 6.93 ½ ½ 88. ½ ½ Exhibit : Standard Two-step Binomial Tree This tree an be used to prie both European and Amerian-style options. The omputation results for various strikes, ranging from 60 to 40, are shown in the 4 th and 5 th olumns in Exhibit 5 for all options and in the 0 th and th olumns for put options. In omparison, we also list the ' pries for

9 European-style options. It is believed that binomial tree is the disretization of the ontinuous-time model. ow, we want to show that we an reover the above binomial tree by using a set of European-style standard all and puts, the urrent asset prie and the risk-free bond. Consider the three ending outomes on the tree. Their node probabilities are ¼, ½, and ¼ respetively. Therefore, the mean and standard deviation of their log-returns are 0.03 and = ¼ * log(36.73/00) + ½ * log(03.04/00) + ¼ * log(77.66/00) 0. = {¼ * [log(36.73/00)-0.03] + ½ * [log(03.04/00)-0.03] + ¼ * [log(77.66/00)-0.03] } ½ As we indiate previously, we sample the ending state spae by using the formula S 0 *exp[mean + std * (j - m)/ m], j=0,,, m. In our ase, this formula leads to sampling states at 77.66, 03.04, and ext, we show we are able to reover the entire binomial tree by given European-style option pries struk at these sampling states. For example, a European all option struk at 03.05, is valued at 7.6 by using the above two-step binomial tree. The valuation tree is shown in Exhibit. ½ ½ 6.0 ½ ½ 0 ½ ½ 0 Exhibit : Binomial Tree for a Call Option By using this European all option, together with the underlying asset and risk-free bond, we an ompute the risk-neutral probabilities of the three states. The payoff table for the three seurities under three different states and the alulated ending risk-neutral probabilities are listed in Exhibit 3. States Seurity Pries Strike Call Asset 00 0 e 0.05 *77.66 e 0.05 *03.05 e 0.05 *36.73 Bond states probability Exhibit 3: Payoff table and Risk-neutral probabilities By using the ending risk-neutral probabilities, we reover the implied binomial tree as shown in Exhibit 4. In this trivial ase, the implied tree is exatly the same as the original binomial tree. Therefore, the priing results by using binomial tree and the implied tree should be the same for both European and Amerianstyle options. Exhibit 5 shows suh omparison between results by standard binomial tree and those by implied tree with the same number of moving steps.

10 ½ ½ 6.93 ½ ½ 88. ½ ½ Exhibit 4: Implied Binomial Tree (One-Asset) Binomial Tree ( steps) (500000) Binomial Tree ( steps) (500000) Differenes to the results from -D Binomial Tree Binomial Tree ( steps) (500000) Differenes to the results from -D Binomial Tree Binomial Tree ( steps) (500000) Exhibit 5: Prie Comparison (-step trees) Exhibit 5 ompares the results of the implied tree to those of ' model and simulations. Although both binomial tree and implied tree value options slightly different with those of ontinuous model, the differene will be orreted by inreasing number of steps as shown in Exhibit 6, where 00 steps of trees are used. Implied Tree ( steps) Implied Tree ( steps) Strike European European Amerian European European Amerian Strike European European Amerian European European Amerian Implied Tree ( steps) Implied Tree ( steps) Strike European European Amerian European European Amerian Strike European European Amerian European European Amerian Binomial Tree (00 steps) (500000) Implied Tree (00 steps) Binomial Tree (00 steps) (500000) Implied Tree (00 steps) Strike European European Amerian European European Amerian Strike European European Amerian European European Amerian Differenes to the results from -D Binomial Tree Differenes to the results from -D Binomial Tree Binomial Tree (00 steps) (500000) Implied Tree (00 steps) Binomial Tree (00 steps) (500000) Implied Tree (00 steps) Strike European European Amerian European European Amerian Strike European European Amerian European European Amerian Exhibit 6: Prie Comparison (00-step trees)

11 Two-asset options Assuming the options payoff depend on the value of basket whih is defined as n S + n S. Information about the basket options and the underlying assets are listed as below. Time to Expiration: T =.0 year Interest Rate: r = 5% Dividend Yield: δ = 5%, δ = 5% Volatilities: σ = 0%, σ = 0% Correlation: ρ = 0.5 Spot Pries: S,0 = 50, S,0 = 50 # of shares: n =, n = Strike Pries: K = 60, 70, 80, 90, 00, 0, 0, 30, 40. The benhmark valuation model is the two-dimensional binomial tree, or the so alled the "binomial pyramids." As a simple example, we start with a two-step two-dimensional binomial tree. For eah node, there are four possible moves in the -dimensional prie spae. The four multipliative moving vetors an be alulated as: (u, A), (u, B), (d, C) and (d, D), where u = exp[(rδ ½σ )h + σ h] =.404 d = exp[(rδ ½σ )h σ h] = A = exp[(rδ ½σ )h + σ h (ρ + ρ )] =.00 B = exp[(rδ ½σ )h + σ h (ρ ρ )] = C = exp[(rδ ½σ )h σ h (ρ ρ )] =.046 D = exp[(rδ ½σ )h σ h (ρ + ρ )] = 0.86 By the end of two steps, there are 3-by-3 nodes in the two dimensional spae. Exhibit 7 gives all the nine nodes and their assoiated node probabilities. The undersored numbers are values of the basket (65.03, 7.) (65.03, 56.46) (65.03, 44.9) (49.0, 6.6) (49.0, 49.0) (49.0, 38.36) (36.94, 54.36) (36.94, 4.55) (36.94, 33.30) Ending odes odal Probabilities Exhibit 7: Ending odes and ode Probabilities on a Two-Dimension Tree As we see, the two-step bivariate binomial model effetively reates nine different states for the basket, and basket options are nothing but options on the single-variate basekt. We will show that a single-variate implied tree reovered by the risk-neutral probabilities of the basket an be used to prie and hedge suh options.

12 Our first step is to sample out three disrete states from the 3-by-3 possible ending values of the basket in order to build a two-steps implied tree. This is done by the equal log-spaed sampling method desribed in the previous example of single-asset options. We start with alulating the mean and standard deviation of the basket's log-returns. Knowing the 3-by-3 possible ending values of the basket and the orresponding nodal probabilities, we an ompute the mean and standard deviation of log-returns on the basket. In this ase, they are and 0.733, respetively. We sample three states out for onstruting a two-steps implied tree, based on the formula: S 0 *exp[mean + std * (j - m)/ m], j=0,,, m, where m =. The states are 77.0, 98.5 and The seond step is to retrieve risk-neutral probabilities of the three states. They are obtained by reating a no-arbitrage payoff table using the pries of standard European options, urrent prie of the basket and the risk-free bond. The proess is similar to that of in the single asset example. Exhibit 8 shows the "binomial pyramids" in priing a two-asset all option struk at Its prie is 7.3. The resultant payoff table aross states as well as the risk-neutral probabilities is in Exhibit Exhibit 8: Two-Dimension Binomial Tree for a Call Option States Seurity Pries Strike Call Asset 00 0 e 0.05 *77.0 e 0.05 *98.5 e 0.05 *5.87 Bond states probability Exhibit 9: Payoff table and Risk-neutral Probabilities Our third step is using the risk-neutral probabilities to onstrut implied binomial tree. The single-variate implied tree for the basket is obtained in Exhibit 0. The option tree for priing an Amerian all option with strike at 80 is also shown in Exhibit, where the undersored 33.8 indiate an early exerise at that node Exhibit 0: Implied Binomial Tree (Two-Asset basket)

13 Exhibit : Option Tree for Amerian Call with Strike at 80 In Appendix C, various ases are tested with inreasing number of steps in demonstrating the onvergene between the results from implied tree and those obtained from multi-dimension binomial tree. In omparison, we also implement three other methods, whih are: () the standard Blak ' formula for European options using the return volatility obtained from the ending risk-neutral probability, () simulation for European options, and (3) the Least-squares simulation for Amerian options. In most ases, we keep all five methods with the same steps exept some of the three- and four-asset ases where neither the multi-dimensional binomial trees nor simulations are able to handle 00 steps or more. Conlusion This artile ontributes the ongoing hallenge of priing Amerian-style multi-asset basket options using a ombination of simulation and implied binomial tree. The method provides a quik valuation and further hedging parameters via the generated implied tree. At the mean time, the appliation of simulation brings the possibility of inorporation of other fators suh as random volatility and orrelation, random interest rate, and jumps.

14 Appendix C: Test Results Two-asset options (Case ) Time to Expiration: T =.0 year Interest Rate: r = 5% Dividend Yield: δ = 5%, δ = 5% Volatilities: σ = 0%, σ = 0% Correlation: ρ = 0.5 Spot Pries: S,0 = 50, S,0 = 50 # of shares: n =, n = Strike Pries: K = 60, 70, 80, 90, 00, 0, 0, 30, 40. Risk-eutral Probabilities Option Probabilities Strike Put/Call Pries Risk-neutraormal Diffs P P P P P P P P P P C C C C C C C C C C C risk-neutral probabilities Risk-neutral ormal differene

15 Two-Asset Case() Steps: 0 Differenes to the results from -D Binomial Tree Differenes to the results from -D Binomial Tree Two-Asset Case() Steps: 0 Differenes to the results from -D Binomial Tree Differenes to the results from -D Binomial Tree (0000) (0000) Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree Binomial Tree Implied Tree Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree Binomial Tree Implied Tree Binomial Tree Implied Tree

16 Two-Asset Case() Steps: 50 (0000) (0000) Differenes to the results from -D Binomial Tree Differenes to the results from -D Binomial Tree Two-Asset Case() Steps: 00 Differenes to the results from -D Binomial Tree Differenes to the results from -D Binomial Tree (0000) Binomial Tree Implied Tree (0000) Binomial Tree Implied Tree Binomial Tree Implied Tree Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree Binomial Tree Implied Tree Binomial Tree Implied Tree steps are applied to both simulation and the Least Squares.

17 Two-asset options (Case ) Time to Expiration: T =.0 year Interest Rate: r = 5% Dividend Yield: δ = 5%, δ = 5% Volatilities: σ = 0%, σ = 90% Correlation: ρ = -0.9 Spot Pries: S,0 = 50, S,0 = 50 # of shares: n =, n = Strike Pries: K = 60, 70, 80, 90, 00, 0, 0, 30, 40. Risk-eutral Probabilities Option Probabilities Strike Put/Call Pries Risk-neutraormal Diffs.884 P P P P P P P P P P C C C C C C C C C C C risk-neutral probabilities Risk-neutral ormal 0.0 differene

18 Two-Asset Case() Steps: 0 Differenes to the results from -D Binomial Tree Differenes to the results from -D Binomial Tree Two-Asset Case() Steps: 0 Differenes to the results from -D Binomial Tree Differenes to the results from -D Binomial Tree (0000) (0000) Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree Binomial Tree Implied Tree Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree Binomial Tree Implied Tree Binomial Tree Implied Tree

19 Two-Asset Case() Steps: 50 Differenes to the results from -D Binomial Tree Differenes to the results from -D Binomial Tree Two-Asset Case() Steps: 3 00 Differenes to the results from -D Binomial Tree Differenes to the results from -D Binomial Tree (0000) (0000) Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree Binomial Tree Implied Tree Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree Binomial Tree Implied Tree Binomial Tree Implied Tree steps are applied to both simulation and the Least Squares.

20 Three-asset options (Case ) Time to Expiration: T =.0 year Interest Rate: r = 5% Dividend Yield: δ = 5%, δ = 5%, δ 3 = 5% Volatilities: σ = 0%, σ = 0%, σ = 0% Correlation: Spot Pries: S,0 = 33.33, S,0 = 33.33, S 3,0 = # of shares: n =, n =, n 3 = Strike Pries: K = 60, 70, 80, 90, 00, 0, 0, 30, 40. Risk-eutral Probabilities Option Probabilities Strike Put/Call Pries Risk-neutraormal Diffs 47.5 P P P P P P P P P P C C C C C C C C C C C risk-neutral probabilities Risk-neutral ormal differene Three-Asset Case() Steps: 0 Differenes to the results from 3-D Binomial Tree Differenes to the results from 3-D Binomial Tree (0000) (0000) Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree Binomial Tree Implied Tree Binomial Tree Implied Tree