Pricing Americanstyle Basket Options By Implied Binomial Tree


 Ζένα Αλεξιάδης
 11 μήνες πριν
 Προβολές:
Transcript
1 Priing Amerianstyle Basket Options By Implied Binomial Tree Henry Wan * Draft: Marh 00 Abstrat It is known that the most diffiult problem of priing and hedging multiasset 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 multiasset 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, Amerianstyle 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 multivariate 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, finitedifferene 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 Europeanstyle options. It is however known that the most diffiult problems of priing and hedging multiasset 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 multidimensional Amerianstyle options. They an be ategorized by either lattiebased or simulationbased approahes. Lattiebased 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 onedimension 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 lattiebased 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 lattiebased 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 Europeanstyle 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 multiasset 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 primaldual simulation algorithm by Andersen and Broadie (00) belong to this ategory. In the simulated tree method, eah single node generates its own nonombining 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 onedimension Amerian options. Sine then artiles were published on the improvement of this basi onept. Longstaff and Shwartz (00) used leastsquares regression on polynomials to approximate the holding value and optimum to exerise. Barraquand and Martineau (995) applied a state aggregation tehnique and redued a highdimensional problem into a single dimension one sine the exerise deision only depends on the single variable, the intrinsi value. The resulting onedimension problem an be solved quikly by standard bakward dynami programming and provides signifiant omputation saving for highdimensional 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 multidimensional 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 highdimensional 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 Europeanstyle standard options to infer the riskneutral 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 Europeanstyle standard all and put options with different strikes, whose underlying is the multiasset basket, by simulations. Seond, we infer the riskneutral probabilities of the basket from the set of Europeanstyle options on the basket, the assoiated market prie of the basket, and the riskless interest rate bond. Third, we reover the fully speified stohasti proess of the basket, in the form of an implied binomial tree, from its riskneutral 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 Europeanstyle standard basket options The first step is valuing Europeanstyle 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 lattiebased method to value highdimensional Europenstyle basket options. Another set of approahes to value multiasset basket options is by simulations. Sine the value of a Europeanstyle option is the riskneutral expetation of its disounted payoff at the time of maturity, simulations obtain an estimate of suh a riskneutral 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 Europeanstyle multiasset 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 Amerianstyle 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 riskneutral distribution of the underlying basket. The approah involves three steps in applying simulations : () Simulate the ending asset pries in a basket under the riskneutral measure. A way of simulating multidimensional 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 riskneutral 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 riskneutral 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 Europeanstyle standard all and put options based on a set of equally logspaed 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 outofmoney 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 timeonsuming 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 Riskeutral Probabilities The seond step of the approah is to infer the implied ending riskneutral probabilities of the univariate basket from the set of Europeanstyle alls and puts on the basket. Consider a state seurity, or ArrowDebreu 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 Europeanstyle all and put options on the basket through the simulation. We an use them to imply the state pries and hene the riskneutral 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 zerooupon riskfree 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 riskfree 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 m3 ) K m3 K m K m3 K m K m3 K m K m3 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 noarbitrage onditions for the forward prie of underlying asset and the prie of riskfree 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 riskneutral 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 riskneutral 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 Amerianstyle or other exoti options should follow the same proess. Here, the standard Europeanstyle 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 riskneutral 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 upthendown and downthenup 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 nth 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 bellshaped node probabilities. With the assumption of ombining binomial tree, there are nhoosej 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 riskneutral 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 nth 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 upmove 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 downmove 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 riskneutral 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 riskneutral 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[(rd)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 Amerianstyle 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 twostep 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 onehalf for the asset prie to either move up or move down. ½ ½ 6.93 ½ ½ 88. ½ ½ Exhibit : Standard Twostep Binomial Tree This tree an be used to prie both European and Amerianstyle 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 Europeanstyle options. It is believed that binomial tree is the disretization of the ontinuoustime model. ow, we want to show that we an reover the above binomial tree by using a set of Europeanstyle standard all and puts, the urrent asset prie and the riskfree 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 logreturns 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 Europeanstyle 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 twostep 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 riskfree bond, we an ompute the riskneutral probabilities of the three states. The payoff table for the three seurities under three different states and the alulated ending riskneutral 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 Riskneutral probabilities By using the ending riskneutral 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 (OneAsset) 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 (00step trees)
11 Twoasset 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 twodimensional binomial tree, or the so alled the "binomial pyramids." As a simple example, we start with a twostep twodimensional 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 3by3 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 TwoDimension Tree As we see, the twostep bivariate binomial model effetively reates nine different states for the basket, and basket options are nothing but options on the singlevariate basekt. We will show that a singlevariate implied tree reovered by the riskneutral 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 3by3 possible ending values of the basket in order to build a twosteps implied tree. This is done by the equal logspaed sampling method desribed in the previous example of singleasset options. We start with alulating the mean and standard deviation of the basket's logreturns. Knowing the 3by3 possible ending values of the basket and the orresponding nodal probabilities, we an ompute the mean and standard deviation of logreturns on the basket. In this ase, they are and 0.733, respetively. We sample three states out for onstruting a twosteps 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 riskneutral probabilities of the three states. They are obtained by reating a noarbitrage payoff table using the pries of standard European options, urrent prie of the basket and the riskfree bond. The proess is similar to that of in the single asset example. Exhibit 8 shows the "binomial pyramids" in priing a twoasset all option struk at Its prie is 7.3. The resultant payoff table aross states as well as the riskneutral probabilities is in Exhibit Exhibit 8: TwoDimension 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 Riskneutral Probabilities Our third step is using the riskneutral probabilities to onstrut implied binomial tree. The singlevariate 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 (TwoAsset 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 multidimension 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 riskneutral probability, () simulation for European options, and (3) the Leastsquares simulation for Amerian options. In most ases, we keep all five methods with the same steps exept some of the three and fourasset ases where neither the multidimensional binomial trees nor simulations are able to handle 00 steps or more. Conlusion This artile ontributes the ongoing hallenge of priing Amerianstyle multiasset 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 Twoasset 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. Riskeutral Probabilities Option Probabilities Strike Put/Call Pries Riskneutraormal Diffs P P P P P P P P P P C C C C C C C C C C C riskneutral probabilities Riskneutral ormal differene
15 TwoAsset Case() Steps: 0 Differenes to the results from D Binomial Tree Differenes to the results from D Binomial Tree TwoAsset 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 TwoAsset Case() Steps: 50 (0000) (0000) Differenes to the results from D Binomial Tree Differenes to the results from D Binomial Tree TwoAsset 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 Twoasset 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. Riskeutral Probabilities Option Probabilities Strike Put/Call Pries Riskneutraormal Diffs.884 P P P P P P P P P P C C C C C C C C C C C riskneutral probabilities Riskneutral ormal 0.0 differene
18 TwoAsset Case() Steps: 0 Differenes to the results from D Binomial Tree Differenes to the results from D Binomial Tree TwoAsset 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 TwoAsset Case() Steps: 50 Differenes to the results from D Binomial Tree Differenes to the results from D Binomial Tree TwoAsset 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 Threeasset 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. Riskeutral Probabilities Option Probabilities Strike Put/Call Pries Riskneutraormal Diffs 47.5 P P P P P P P P P P C C C C C C C C C C C riskneutral probabilities Riskneutral ormal differene ThreeAsset Case() Steps: 0 Differenes to the results from 3D Binomial Tree Differenes to the results from 3D Binomial Tree (0000) (0000) Binomial Tree Implied Tree (0000) (0000) Binomial Tree Implied Tree Binomial Tree Implied Tree Binomial Tree Implied Tree
The Canadian Pensioners Mortality Table
The Canadian Pensioners Mortality Table Historical Trends in Mortality Improvement and a Proposed Projection Model Based on CPP/QPP Data as at December 31, 2007 Louis Adam, FSA, FCIA Université Laval,
Διαβάστε περισσότεραFull Cryptanalysis of LPS and Morgenstern Hash Functions
Full Cryptanalysis of LPS and Morgenstern Hash Functions Christophe Petit 1, Kristin Lauter 2 and JeanJacques Quisquater 1 1 UCL Crypto Group, 2 Microsoft Research. emails: christophe.petit@uclouvain.be,klauter@microsoft.com,jjq@uclouvain.be
Διαβάστε περισσότεραΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΙΕΠΙΣΤΗΜΟΝΙΚΟ  ΙΑΤΜΗΜΑΤΙΚΟ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥ ΩΝ ΒΕΛΤΙΣΤΟΣ ΣΧΕ ΙΑΣΜΟΣ ΣΥΣΤΗΜΑΤΩΝ
ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΙΕΠΙΣΤΗΜΟΝΙΚΟ  ΙΑΤΜΗΜΑΤΙΚΟ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥ ΩΝ «ΕΠΙΣΤΗΜΗ ΚΑΙ ΤΕΧΝΟΛΟΓΙΑ Υ ΑΤΙΚΩΝ ΠΟΡΩΝ» ΒΕΛΤΙΣΤΟΣ ΣΧΕ ΙΑΣΜΟΣ ΣΥΣΤΗΜΑΤΩΝ ΑΠΟΚΑΤΑΣΤΑΣΗΣ ΥΠΟΓΕΙΩΝ Υ ΡΟΦΟΡΕΩΝ ΣΕ ΠΕΡΙΒΑΛΛΟΝ
Διαβάστε περισσότεραWAVELETBASED OUTLIER DETECTION AND DENOISING OF AIRBORNE LASER SCANNING DATA
WAVELETBASED OUTLIER DETECTION AND DENOISING OF AIRBORNE LASER SCANNING DATA A THESIS SUBMITTED TO GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY TOLGA AKYAY IN
Διαβάστε περισσότεραarxiv:0807.0514v1 [hepph] 3 Jul 2008
Preprint typeset in JHEP style  HYPER VERSION arxiv:yymm.nnnn [hepph] CP ZUTH / Analytic integration of realvirtual counterterms in NNLO jet cross sections I arxiv:7.v [hepph] Jul Ugo Aglietti Dipartimento
Διαβάστε περισσότεραDo Media Data Help to Predict German Industrial Production?
No 149 Do Media Data Help to Predict German Industrial Production? Konstantin A. Kholodilin, Tobias Thomas, Dirk Ulbricht July 2014 IMPRINT DICE DISCUSSION PAPER Published by düsseldorf university press
Διαβάστε περισσότεραAre Short Term Stock Asset Returns Predictable? An Extended Empirical Analysis
Are Short Term Stock Asset Returns Predictable? An Extended Empirical Analysis Thomas Mazzoni Diskussionsbeitrag Nr. 449 März 2010 Diskussionsbeiträge der Fakultät für Wirtschaftswissenschaft der FernUniversität
Διαβάστε περισσότεραINSTITUTE OF THEORETICAL PHYSICS AND ASTRONOMY, VILNIUS UNIVERSITY. Justas Zdanavičius
INSTITUTE OF THEORETICAL PHYSICS AND ASTRONOMY, VILNIUS UNIVERSITY Justas Zdanavičius INTERSTELLAR EXTINCTION IN THE DIRECTION OF THE CAMELOPARDALIS DARK CLOUDS Doctoral dissertation Physical sciences,
Διαβάστε περισσότεραTop down vs. bottom up parsing. Top down vs. bottom up parsing
CMSC 331 notes (9/17/2004) 1 Top down vs. bottom up parsing A grammar describes the strings of tokens that are syntactically legal in a PL A recogniser simply accepts or rejects strings. A generator produces
Διαβάστε περισσότεραΠανεπιστήμιο Κρήτης, Τμήμα Επιστήμης Υπολογιστών. Business Process Management and Knowledge Toolkit
Business Process Management and Knowledge Toolkit Last update: 13/3/15 1 ADONIS 2 Business Process Management The goals of Business Process Management are the optimization of both the processes of an enterprise,
Διαβάστε περισσότεραEconomic Analysis Papers
UNIVERSITY OF CYPRUS ECONOMICS RESEARCH CENTRE Economic Analysis Papers A small macroeconomic model of the Cyprus economy Louis Christofides Andros Kourtellos Department of Economics & Economics Research
Διαβάστε περισσότεραL A TEXbabelÏ. ǫφτ. ë16 19 Οκτώβριος/October 2007. Σ αὐτὸτὸτεῦχος/in this issue. iii. iv Οἱ σημειώσεις τοῦ τυπογράφου
eutypon1619 2007/10/3 12:47 page 1 #1 Ì ØÙÔÓÒ Ì ÙØÝÔÓÒ ÁËËÆ½½¼ ¹ ½ ¼ ë16 19 Οκτώβριος/October 2007 Σ αὐτὸτὸτεῦχος/in this issue iii ΕνTEXνα καὶ ἄtexνα iv Οἱ σημειώσεις τοῦ τυπογράφου 1 Erik Meijer Citations,
Διαβάστε περισσότεραMultivariate Models, PCA, ICA and ICS
Multivariate Models, PCA, ICA and ICS Klaus Nordhausen 1 Hannu Oja 1 Esa Ollila 2 David E. Tyler 3 1 Tampere School of Public Health University of Tampere 2 Department of Mathematical Science University
Διαβάστε περισσότεραTsenka Lazarova Stoyanova
ΜΟΝΤΕΛΟΠΟΙΗΣΗ ΜΕΤΑΔΟΣΗΣ ΡΑΔΙΟΣΗΜΑΤΩΝ ΚΑΙ ΣΤΡΑΤΗΓΙΚΗ ΑΝΑΠΤΥΞΗΣ ΑΣΥΡΜΑΤΩΝ ΔΙΚΤΥΩΝ ΑΙΣΘΗΤΗΡΩΝ ΕΞΩΤΕΡΙΚΟΥ ΧΩΡΟΥ ΜΕ ΣΤΟΧΟ ΤΟΝ ΕΝΤΟΠΙΣΜΟ ΚΑΙ ΙΧΝΗΛΑΤΗΣΗ ΜΕΣΩ ΤΟΥ ΛΑΜΒΑΝΟΜΕΝΟΥ ΡΑΔΙΟΣΗΜΑΤΟΣ ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ
Διαβάστε περισσότεραIndividual vs. aggregate models of land use changes: Using spatial econometrics to improve predictive accuracy?
Individual vs. aggregate models of land use changes: Using spatial econometrics to improve predictive accuracy? JeanSauveur AY, Raja CHAKIR Julie LE GALLO March 28, 2014 Abstract It is a widespread practice
Διαβάστε περισσότεραPARALLEL SECTION 1 Economics and Finance Session Chair: Eleutherios Thalassinos, University of Piraeus Room: Conference Room Time: 18:4520:00
PARALLEL SECTION 1 Economics and Finance Session Chair: Eleutherios Thalassinos, University of Piraeus Room: Conference Room Time: 18:4520:00 Eleutherios Thalassinos, Department of Maritime Studies University
Διαβάστε περισσότεραDeliverable D5.1. Text Summarisation Tools. Pilot B
Deliverable D5.1 Deliverable D5.1 Text Summarisation Tools Grant Agreement number: 250467 Project acronym: ATLAS Project type: Pilot B Deliverable D5.1 Text Summarisation Tools 31.08.2012 Project coordinator
Διαβάστε περισσότεραD14+D15: RECOMMENDATIONS
ura Contract no. 030145 Coordinating Regional Primary Sector Policies for Boosting Innovation CoRIn Coordination Action Support for the coherent development of policies Regions of Knowledge 2 D14+D15:
Διαβάστε περισσότεραΕΝΤΥΠΑ ΥΠΟΒΟΛΗΣ ΕΝΔΙΑΜΕΣΗΣ ΕΚΘΕΣΗΣ ΠΡΟΟΔΟΥ ΕΡΕΥΝΗΤΙΚΟΥ ΕΡΓΟΥ ΤΗΣ ΔΕΣΜΗΣ 2008
1 ΕΝΤΥΠΑ ΥΠΟΒΟΛΗΣ ΕΝΔΙΑΜΕΣΗΣ ΕΚΘΕΣΗΣ ΠΡΟΟΔΟΥ ΕΡΕΥΝΗΤΙΚΟΥ ΕΡΓΟΥ ΤΗΣ ΔΕΣΜΗΣ 2008 Η Ενδιάμεση Έκθεση, υποβάλλεται σε δύο αντίγραφα, μέχρι και δύο μήνες μετά τη συμπλήρωση της μισής διάρκειας του χρόνου υλοποίησης
Διαβάστε περισσότεραEstimating a dynamic labour demand equation using small, unbalanced panels: An application to Italian manufacturing sectors
Estimating a dynamic labour demand equation using small, unbalanced panels: An application to Italian manufacturing sectors Giovanni S.F. Bruno +, Anna M. Falzoni + and Rodolfo Helg *+ + Università Bocconi,
Διαβάστε περισσότεραInternship Report JulyDecember 2012. Katerina Tzortzi
Internship Report JulyDecember 2012 Katerina Tzortzi Techniques for the enrichment of Ekfrasis, the conceptually organized lexicon of the Institute for Language and Speech Processing/R.C. Athena : the
Διαβάστε περισσότεραCYPRUS UNIVERSITY OF TECHNOLOGY FACULTY OF MANAGEMENT AND ECONOMICS. Dissertation ECONOMIES OF SCALE AND OPTIMAL SIZE SHIP IN LNG CARRIERS
CYPRUS UNIVERSITY OF TECHNOLOGY FACULTY OF MANAGEMENT AND ECONOMICS Dissertation ECONOMIES OF SCALE AND OPTIMAL SIZE SHIP IN LNG CARRIERS Prokopis Appios Limassol 2013 CYPRUS UNIVERSITY OF TECHNOLOGY
Διαβάστε περισσότεραSession 1: Fund Performance and Performance Measurement
Session 1: Fund Performance and Performance Measurement Dikaios Tserkezos, Department of Economics,University of Crete Eleni Thanou, Graduate Program on Banking Hellenic Open University Non linear diachronic
Διαβάστε περισσότεραFactors Influencing Cloud ERP Adoption
Factors Influencing Cloud ERP Adoption A Comparison Between SMEs and Large Companies Master Thesis 15 ECTS, INFM03 June 2013 Authors: Amar Alajbegovic Vasileios Alexopoulos Achilles Desalermos Supervisor:
Διαβάστε περισσότεραPointtopoint digital link design
1516η Διάλεξη Pointtopoint digital link design Γ. Έλληνας, Διάλεξη 1516, σελ. 1 Outline Link Design RiseTime Intro to WDM WDMLANs Γ. Έλληνας, Διάλεξη 1516, σελ. 2 Page 1 Pointtopoint digital link
Διαβάστε περισσότεραVirtual Music School Usercentred plan
Virtual Music School Usercentred plan USER CENTER PLAN USER CENTER PLAN VEMUS Project IST 27952 Usercentred plan Project Information Project No. IST27952 Project acronym VEMUS Project title Virtual
Διαβάστε περισσότεραUniversity of the Aegean. Business School. Department of Financial Management and Engineering
Business School Fleet Sizing Decisions for Prioritized PickUp and Delivery Operations using the Branch and Price Method DimitrisGeorgios Baklagis Supervisor Professor Ioannis Minis Chios 2013 Acknowledgments
Διαβάστε περισσότεραCollege Readiness Systems Longitudinal Evaluation: EXCELerator Program Impact, Year 2 Report
College Readiness Systems Longitudinal Evaluation: EXCELerator Program Impact, Year 2 Report August 2011 Deborah J. Holtzman Frances Stancavage 2800 Campus Drive, Suite 200 San Mateo, CA 94403 8003562735
Διαβάστε περισσότεραSESSION 1 ENERGY MARKETS
SESSION 1 ENERGY MARKETS Alexandridis, A. Zapranis, S. Livanis, Department of Accounting and Finance, University of Macedonia of Economics and Social Studies Analyzing Crude Oil Prices and Returns Using
Διαβάστε περισσότεραPaper No. 33 Determinants of Regional Entry and Exit in Industrial Sectors Kristina Nyström
CESIS Electronic Working Paper Series Paper No. 33 Determinants of Regional Entry and Exit in Industrial Sectors Kristina Nyström (JIBS, CESIS) May 2005 The Royal Institute of technology Centre of Excellence
Διαβάστε περισσότερα