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Exotic option and interest rate derivative pricing protocolAuthorJamila AwadRights ReservedJAW GroupDateMarch 2014Executive SummaryFinancial engineering innovation such as the development of new derivative instruments solicits investors to participate in the globalization of financial markets by endorsing various types of payoffs. The disquisition strives to model an exotic option and interest rate derivative pricing prototype in compliance with non-multinormal distributions. The redaction is partitioned in three sections. The introduction umbrages the concepts and elusive foundations in standard derivative pricing techniques. The second section articulates the mathematical, the theoretical and the reconstitution systems integrated to construct the financial instrument pricing exemplary. The final section describes the cascade steps to implement the derivative pricing archetype. In brief, the introduction of transparent derivative instruments shall reinforce sound transactions in capital markets. 1. IntroductionMarket participants in capital markets quest transparent derivative instruments that render various payoff opportunities and embrace prudential financial risk management. The pricing of sophisticated multiasset derivatives is complex however requisites a guideline to safeguard unequivocal and accurate trading transactions in capital markets. Individual assets expose stochastic volatility and jumps. Standard derivative pricing techniques such as geometric Brownian motion (Johnson, 1987) rely on the assumption that co-movements of underlying assets are captured by the linear correlation matrix. The stated hypothesis leads to an erroneous pricing foundation because the entire dependence structure is not encapsulated in the linear correlation matrix thus as observed in leptokurtic and abnormal distributions of asset returns. The redaction aspires to cement a pricing protocol to engineer exotic options and interest rate derivatives in compliance with accurate assumptions of asset returns with non-multinormal distributions. The choice dependence structure contains rich information relevant to evaluate the payoff of financial instruments such as options and interest rate derivatives particularly in volatile market conditions. Equity options portray financial derivatives whose redemption is bonded to the performance of a bundle of underlying stocks. The financial compensation for interest rate instruments is determined from the price fluctuations of the underlying interest rate basket derivatives. The narration is partitioned in three sections. The introduction articulates the concepts and elusive foundations in standard derivative pricing techniques. The second section umbrages the mathematical, the theoretical and the reconstitution systems integrated to construct the financial instrument pricing exemplary. The final section details the cascade steps to implement the derivative pricing archetype.A copula depicts a multivariate distribution function whereas each marginal is uniform on the unit interval. Sklar (1959) introduced the theorem that multivariate continuous distribution functions are uniquely factored into its marginals and a copula. Therefore, the dependency between multiple assets is adequately captured by the copula (Bennett and Kennedy, 2004). The technical advantage of a copula-based modeling approach relies on the foundation that appropriate marginal distributions for the components of a multivariate system are selected by various flexible methods, and then bonded to a retained copula to expose the dependency prevailing between the constituents (Genest and Werker, 2005). The delivered derivative pricing prototype strives as well to introduce a flexible dynamic density specification approach to jointly model the underlying assets and apprehend correlated jump processes. The procedure consists of firstly calibrating market prices of univariate equity options or interest rate derivatives, and secondly assembling the correlation matrix to characterize the dependence structure amongst the underlying assets (Rosenberg, 2003). The joint behavior of financial asset returns is best modeled by a copulae framework that underpins the junction in the asymmetric tail movements (Mashal and Zeevi, 2002). Model risk represents a dominating factor in valuating exotic products (Hull and Suo, 2002). The sensitivity of derivative prices to different copula specifications varies according to the composition of the underlying asset bundle, the payoff structure and lastly to economic conditions. Derivative instruments are simulated via Monte Carlo techniques. Options also portray excellent tools to hedge the risk of multiple assets that usually take the form of Calls and Puts thus render the right to buy or sell a number of underlying assets (Rubinstein, 1979). The valuation of exotic options heavily relies on correlation coefficients between the bundled assets. The protocol presents a pricing methodology for best-of and worst-of options to expose the enhanced sensitivity of the payoff structures to the copula dependency fundament. The Monte Carlo simulation technique enables to simulate multiasset option prices for each combination of derivative payoff and underlying basket. In more volatile market periods, the distribution of individual logarithmic returns becomes more leptokurtic and elucidates flexible distributional assumptions (Boyer et al., 1999). In contrast, in less turbulent market scenarios, a misspecification for the margins induces a negligible effect on option prices. The fair value of every option is calculated through a Monte Carlo 100,000 paths simulation of monthly correlated stock or interest rate returns under the selected copula family for each pair of marginals. A crucial component of pricing options is the correlation among the different underlying assets. The sensitivity of option prices to the choice of copula function is enhanced when the constituents of the underlying bundle of assets are more correlated. Precisely, the basket value of an option would increase if the correlation among the bundled assets increases. A sound derivative pricing prescript is primordial to safeguard an exact mark-to-market evaluation of exotic products and to exhibit a correct assessment of the financial instrument’s risk profile. The correlation among asset returns as well as the joint occurrences of large losses, measured in tail dependence, significantly inflate in times of distress such as during the financial crisis of 2008. Plain vanilla options are priced through a Black-Scholes model despite divergence in the distribution assumptions of the model. Ergo, the implementation of copulas for evaluating exotic option prices is sound because the procedure exposes a transparent picture about the estimated parameters. Furthermore, the copulae technique enhances high dimensional multiasset derivative pricing procedures by enabling copula calibration for every pair of marginals examined. In essence, the dependence structure under the objective measure is the same as under the risk-neutral measure (Cherubini and Luciano, 2002). The convenience of transacting basket options is their cost efficient advantage for portfolio insurance compared to plain vanilla options on individual assets. The valuation of multivariate financial derivatives requisites the analysis of the joint distribution of underlying variables in a risk-neutral world. The standard option pricing theory based on Black-Scholes (1973) framework introduces significant errors in the pricing of basket options because it assumes log-normal distributional characteristics of individual underlying assets and a simple dependency correlation (Carmona and Durrleman, 2006). In addition, the existence of volatility smiles or skew bears weakens the Black-Scholes option pricing methodology (Merton, 1974). Therefore, capital markets quest for pricing techniques that model multiasset returns in non-multinormal distributions. The tail dependence cannot be omitted in option pricing whereas the impact of smiles has reinforced the necessity to introduce accurate derivative valuation methods. 2. The Exotic Option and Interest Rate Derivative Pricing PrototypeThe second chapter articulates the logic inferred to address the exotic option and interest rate derivative pricing prototype with a copula approach and delivers the cascade steps to implement the protocol.2.1 The Copulae PrescriptThe quest for sage financial risk management as well as the bourgeoning era of sophisticated derivative instruments requisite practitioners to enforce option pricing foundations that illustrate non-multinormal return distributions, and thus adequately capture and hedge risks. The copulae framework underpins the joint behavior of marginal pairs such as bundled stock assets and interest rate derivatives traded in portfolio management. Furthermore, the copula function is best suited to value multidimensional arrangements and to measure extreme events. The theorem of Sklar (1959) decapsulates an n-dimensional joint distribution into its n-marginal distributions and a copula that characterizes the dependency between the n-variables. The copula framework sides the Gaussian assumption to permit financial modeling with a multifaceted approach. The copula function is induced from a random vector with a cumulative distribution function.The following mathematical expression describes a copula function, noted as (X1,….,XN) ? RN, presented as a random vector with a cumulative distribution function (F) and marginal functions (Fn(xn)):F (X1,….,XN) = P ( X1 ≤ x1,…..,XN ≤ xN) and (1)Fn(xn) = P (Xn ≤ xn), 1 ≤ n ≤ N (2)In addition, a copula function C of vector F is described as a cumulative distribution function of a probability measure bounded to [0,1]N such:Cn(un) = C (1,….,1,un,1,….,1) = un whereas ≤ n ≤ N, 0 ≤ un ≤ 1 and (3)F(x1,….,xN) = C (F1(x1),….,Fn(xN)) for the interval(x1,….,xN) and whereas xN is a continuity point of Fn (1 ≤ n ≤ N).The copula function is schematized with maxims:Every distribution function F holds at least one copula function that is uniquely defined in the dimension (F1(x1),….,FN(xN)) of the points (x1,….,xN). Hence, all numbers included in the interval ≤ n ≤ N become a continuity point of Fn. The copula function of F is unique when all the marginal functions are continuous.Every copula C is continuous and complies to the following inequality: ( ≤ n ≤ N, 0 ≤ un, vn ≤1): | C(u1,….,uN) – C( v1,….,vN) | | un – vn | The ensemble C of all copula functions is convex and follows one set of convergence: punctual, uniform or weak. All copula functions of (X1,….,XN) are also copula functions of (h1(X1),….,hN(XN)) when (h1,….,hN) are monotone and non-decreasing mappings of R.If {F(m), m ≥1}is a sequence of probability cumulative distribution function in RN, the convergence of F(m) to a distribution function F with continuous margins Fn when m→∞ is equivalent to the following states:Fn(m) → Fn for all figures included in the interval ≤ n ≤ N.C(m) →C when C is a unique copula function bonded to F and when C(m) illustrates a copula function associated to F(m). The theory of the conditional copula states that the conditional distribution of (X,Y) given W, be denoted by the parameter H. Also, the conditional marginal distributions of X|W and Y|W are denoted F and G respectively. All previously stated variables are supposed continuous. The cumulative distribution functions of a random parameter as well as the corresponding probability density function are then schematized.The copula approach alleviates the analysis of two random variables by separating the dependency between arbitrary variables in a general form. The copula assembles information from the joint distribution that is not enclosed in the marginal distributions. Hence, the transformation of X and Y to u and v coordinates filters out information incorporated in the marginal distributions. In precise terms, the copula C contains all of the data about the dependency between X and Y, but disregards the univariate individual characteristics of X and Y. A two-dimensional conditional copula portrays a function C that conforms with the following axiom: [0,1]* [0,1]*Z→[0,1] whereas Z ? Rk and k is a finite integer in line with the properties:C(u, 0|z) = C(0,v|z) = 0, and C(u,1|z) = u and C(1,v|z) = v, for every u,v in [0,1] and all z ? Z.Vc([u1,u2] x [v1,v2]|z) ≡ C(u2, v2|z) – C(u1, v2|z) – C(u2, v1|z) + C(u1, v1|z) ≥ 0 for all u1, v1, u2, v2 ? [0,1] such that u1 ≤ u2 and v1 ≤ v2, and all z ? Z. The Sklar theorem for continuous conditional distributions is illustrated with the parameters: F is the conditional distribution of X|Z, G is the conditional distribution of Y |Z, and finally H is the joint conditional distribution of (X,Y)|Z. If parameters F and G are continuous in x and y, then a unique conditional copula C is generated: H(x,y|Z) = ((C(F(x|Z),G(y|Z)|Z)).Conversely, if the parameter F describes a conditional distribution of X|Z, G characterizes a conditional distribution of Y|Z and C depicts a conditional copula, then the function H becomes a conditional bivariate distribution function encapsulating the conditional marginal distributions F and G.2.2The Archimedean CopulasThe Archimedean class of copula encompasses many families due to its achievement in reducing dimensionality. The Archimedean copula family contains the following copula names: Ali-Mikhail-Haq (AMH), AP, Clayton, Frank, Gumbel and Joe. The mathematical properties of the Archimedean class are captured by an additive generator function φ: II → [ 0,∞] which is continuous, convex and a decreasing function (φ’(t) < 0 ). Moreover, the additive generator function may also be indexed by the association parameter θ permitting an entire family of copulas to be Archimedean. Therefore, any function φ that satisfies the above stated conditions can be utilized to generate a valid bivariate cumulative distribution function (cdf). The rotated flexibility of the Archimedean family of copulas delivers an appealing technique to investigate marginal distributions by surmounting standard dilemmas arising from the biased assumption that returns are normally distributed. In addition, financial engineering frameworks with Archimedean copula functions also enable to underpin fat-tail features of the underlying and dependence structures. The combination of survival and Archimedean copulas strengthens the model without imposing symmetric dependence on variables. Tail dependence apprehends the behavior of random variables during extreme events. Precisely, it measures the probability of observing an extremely large positive or negative realization of one variable, given that the other variable also took on an extremely large positive or negative value. Therefore, the assembled copula design enables upper and lower tail dependency to range anywhere from zero to one. The following paragraph presents properties related to other Archimedean families. The normal copula is a copula bonded to the bivariate normal distribution and thus depicts the dependence function implicitly assumed whenever the bivariate normal distribution is schematized. Plackett's copula is symmetric, similar to the normal copula, however exhibits less dependence in the bivariate tails of the distribution. Clayton's copula represents an asymmetric copula, exhibiting greater dependence in the negative tail than in the positive tail. The generator of the Clayton copulae is determined by ΦC(u) = (u-α – 1) and hence Φ-1(t) = ((t+1) -1/ α). The Clayton copula is suitable for describing dependencies in the left tail and during financial market meltdowns (Longin and Solnik, 2001). The generator of the Gumbel copulae is provided by ΦG(u) = ((-ln(u))α and therefore ΦG(u)-1(t) = exp(-t1/ α). The Gumbel copulas schematize multivariate extreme value distributions and they are often used to model extreme distributions. They depict asymmetric Archimedean copulas, exposing greater dependence in the positive tail than in the negative tail. From the practical point of view, the Archimedean copulas depict a highly malleable copulae family due to their capabilities to generate a number of copulas from interpolating between certain copulas, and thus capture various dependency structures. The Archimedean copulas are constructed by using a continuous, decreasing or convex generator function Φ: I →R+ such that Φ(0) = 1. The generator function is called strict whenever Φ(0) = + ∞.The pseudo inverse of the generator function, subtitled Φ-1, is continuous, does not increase in boundaries [0, ∞] and strictly decreases in interval [0, Φ(0)]. Given a generator and its inverse, an Archimedean copula entitled CArchimedean is generated according to the Kimberling theorem such: C(u1,u2) = Φ-1 (Φ(u1) + Φ(u2)) in interval [0,1]2 → [0,1] (4)for quantiles of two random variables X and Y.2.3 The Survival Copulae TheoremThe survival copulae theorem derives from Sklar’s copula theorem and illustrates the generation of a survival copula that arises from a mirrored version of the original examined copula. It elicits the interchange of upper and lower tail properties.Consider a pair (X,Y) of random variables with joint distribution function S, the joint survival function is provided by ?(x,y) = P[X > x,Y > y]. (5)The margins of the function ? are the functions ?(x,-∞) and ?(-∞,y), which are the univariate survival functions F(x) = P[X > x] = 1 – F(x) and G(y) = P[Y > y] = 1 - G(y), respectively. (6)The relationship between the univariate and joint survival functions is illustrated by: ?(x,y) = 1 – F(x) – G(y) + S(x,y) (7) ?(x,y) = F(x) + G(y) – 1 + C(1 – F(x), 1 - G(y)) (8) Suppose the random variables X ~ F and Y ~ G for a copula subtitled C and the joint survival function for the vector (F(X),G(Y)) of the uniform random variables, thus the joint probability that (F(X),G(Y)) is greater than (u,v) when evaluated in the marginals (u,v) which reflect the original copula is the survival copula ?. The survival copula is resumed with the equation: ? (u,v) = 1 – u – v + C(u,v) (9) 2.4 The Maximum Likelihood Estimate OptimizationThe copula values for a predetermined array of copula families are accomplished with the maximum likelihood estimation. Parametric estimation with the maximum likelihood portrays a sound methodology to prescribe the optimal copula choice by fitting the data into a copula and afterwards analyzing the parameters to pursue further analysis.The maximum likelihood technique is obtained from the following logic. The density of the joint distribution F for the copula C and the margins Fn is illustrated:f(x1,…,xn,…,xN) = c(F1(x1),…,Fn(xn),…,FN(xN))*ΠNfn(xn) (10)whereas fn is the density of the margin Fn and c is the density of the copula induced by:c(u1,…,un,…,uN) = [? C (u1,…,un,…,uN)] ÷ [?u1,…,?un,…,?uN] (11)The logarithm likelihood (l(θ)) estimator is illustrated with the equation:l(θ) = lnPr{(X1,…,XN) = (xt1,…,xtN)} (12)Sklar's theorem decomposes a bivariate distribution, Ht, into three components: the two marginal distributions, Ft and Gt, and finally the copula, Ct. The maximum likelihood analysis is initiated with the differentiation of Ft and Gt as well as the double differentiation of Ct: ht(x,y|z) ≡ ft(x|z)*gt(y|z)*ct(u,v|z) where u ≡ Ft(x|z) and v ≡ Gt(y|z) (13)Reformatting the equation on both sides with the logarithm results: LXY = LX + LY + LC (14) The joint log-likelihood is equal to the sum of the marginal log-likelihoods and thecopula log-likelihood. For the purposes of multivariate density design, the copula representation enhances greater flexibility in the specification by designing individual variables using whichever marginal distributions with best fitting properties. Furthermore, the maximum likelihood estimator portrays the most efficient estimator by reaching the minimum asymptotic variance bound.2.5 Correlation MeasuresThe commonly used measure of dependence structure is the linear correlation however the joint distribution of real world financial global markets is not suitable for the linear correlation (Embrechts et al., 2002). In precise terms, financial instruments depict non-linear products that are skewed and heavy tailed which invalidates the hypothesis of a normal distribution. Ergo, the linear correlation misleads the level of real dependence between different risks and defaults to adequately capture the degree of dependence in the tail of the underlying distribution where extreme events remain highly dependent.The correlations among securities as well as the joint occurrence of large losses increase significantly in times of market distress. Consequently, the choice of dependence structure plays a predominant role in periods of high volatility. The correlation matrix portrays a schematization of the entire dependence structure. In finance, the correlation depicts a widespread computing measure for dependence whereas high-order co-movements for copulas are evaluated with the following correlation coefficients: the Spearman's rho and the Kendall's tau. The Spearman's rho and the Kendall's tau satisfy the maxims A1 to A4 for continuous marginals:The axiom A1 relates to the symmetry principle whereas δ(X,Y) = δ(Y,X). The property A2 illustrates the boundedness theory: -1 ≤ δ(X,Y) ≤ 1.The maxim A3 describes the comonotonic and counter-comonotonic principles: δ(X,Y) = 1 ? X,Y is considered comonotonic and δ(X,Y) = -1 ? X,Y is stated counter-comonotonic.An n-dimensional random vector X = (X1,…,Xn) is comonotonic if there is a random variable Z and increasing functions hi to satisfy (X1,…,Xn) = d(h1(Z),…,hn(Z)). On the other hand, the random vector X = (X1,X2)' is counter-comonotonic if a random variable Z holds (X1,X2) = d(h1(Z),h2(Z)) where h1 presents an increasing function and h2 describes a decreasing function.The dictum A4 defines the strictly monotonic function on the range of X for a variable h: δ (h(X),Y) is equivalent to δ(X,Y) when h is increasing and equal to δ(X,Y) when h is decreasing. Variable nomenclature:δ: The dependence measure.h: The parameter to define a bijection. X: The variable to illustrate a distribution (X1,…,Xn) as an n-dimensional continuous random vector.Y: The variable that portrays an independent copy of X. 2.5.1 The Spearman Correlation MeasureNevertheless, for random variables Y1 and Y2 with respective marginal functions F1 and F2, the Spearman correlation (ρs) statistic is defined as ρs = (Y1, Y2) ρ(F1(Y1),F2(Y2)) (15)Therefore, the Spearman’s rho for continuous random variables (X1,X2) with copula C is computed for (X1,X2)': ρs(X1,X2) = 12 ∫∫[0,1]uvdC(u,v) – 3 (16)2.5.2 The Kendall Tau Correlation MeasureThe Kendall Tau (ρτ) portrays a measure of concordance for bivariate random vectors. The concordance statistic is described with two distinct points (x1,y1) and (x2,y2). The points are pronounced concordant when (x1 – x2)( y1 – y2) > 0 and discordant when (x1 – x2)( y1 – y2) < 0. A random vector is defined with the two parameters (Y1, Y2) whereas (?1, ?2) portrays an independent copy of the stated random vector. The Kendall’s tau is then computed with the equation: = E (sign ((Y1 - ?1) (Y2 - ?2))) (17)whereas E denotes the expectation operator. Suppose (X1,X2) are continuous random variables with copula C, then Kendall's tau (ρτ) for (X1,X2)' is provided by: ρτ(X1,X2) = 4 ∫∫[0,1]C(u,c)dC(u,v) – 1 (18)The strengths of implementing copula based measures over standard correlation coefficients are summarized:An array of copula families can be used to construct a suitable measure depending on the modeled random variables of the multivariate data.Copulas are invariant under strictly increasing transformations of random variables. Copula based measures permit the parameterization of dependence structure. The creation of multivariate distribution functions by joining the marginal distributions enable to adequately measure correlation.Copulas extract the dependency structure from the joint distribution function, and afterwards partition the dependence and marginal behavior.2.6 The Black-Scholes ModelThe Black-Scholes frame of reference in option pricing illustrates the process of two underlying assets, entitled S1 and S2, under a risk-neutral world described by:dS1 = r S1dt + σ1S1dW1 (19)dS2 = r S2dt + σ2S2dW2 (20)dW1dW2 = ρdt (21)dB = rBdt (22)Variable nomenclature:ρ: The linear correlation coefficient between the two underlying assets.σ1,σ2:The volatility parameters of asset 1 and asset 2 respectively.r:The risk-free constant interest rate.W1,W2: Two correlated Wiener processes.B: The deterministic process. The Black-Scholes technique assumes that the marginal distribution of the underlying assets follows a lognormal distribution and that the joint terminals as well as the bonded transition distribution functions comply with a bivariate normal distribution.The marginal density function under the Black-Scholes model considers a normal distribution for lnS1(tj+1) – lnS1(tj) between time step j to j+1 whereas j = 0,1,…,n-1, tn = T and Δt = tj+1 – tj, with mean ((r – σ2/2)Δt) and variance (σ2Δt) is provided by:lnS1(tj+1) – lnS1(tj) ~ ф[(r – σ2/2)Δt, σ(Δt)1/2] (23)Variable nomenclature:S1(tj+1): The stock price for asset 1 at a future period tj+1.S1(tj): The stock price at time tj.Φ(m,s): A normal distribution with mean m and standard deviation s.Furthermore, the joint transition density of lnS1 and lnS2 conforms to a bivariate normal distribution.The multidimensional Black-Scholes technique enables to value the payoff of multiasset baskets. The option price is the discounted risk-neutral expectation of its payoff. Suppose one risk-free asset, B(t), and n risky assets, entitled S1(t),…,Sn(t). The differential stochastic equation that defines the dynamics of the stated assets is described by:dSit = ?iSit(t)dt + SitσidWit whereas i = 1,…,n and t ? [0,T] (24)ρijdt = dWit dWjt (25)Variable nomenclature:?i: The parameter that represents the drifts equal to (r-di) under a risk-neutral probabilityσi: The constant volatility parameterdWit,dWjt: The correlated increments of a Wiener processρij: The correlation of normally distributed assets combined in the basketThe geometric Brownian motion when σi is stated independent of time is provided by:SiT = Sitexp{(?i - σi2/2)(T – t) + σi(dWiT - dWit)} (26)The geometric Brownian motion is however reformulated to calculate the price of an option containing a bundle of assets. 2.7 The Monte Carlo TechniqueThe price of an option containing multiassets depicts the discounted risk-neutral expectation of its payoff. The Monte Carlo principle derives from the central limit theorem that dictates the behavior of a normal distribution for a large sum of numbers that are independent and identically distributed variables. The Monte Carlo technique involves simulating paths of the underlying assets and retaining the discounted mean of the simulated payoffs. Assume an environment whereas the risk-free rate and the volatility are constant, the Monte Carlo simulation method to price options follows the cascade steps:Simulates n trajectories for SiT under a risk-neutral putes the payoff of each option for every simulation.Solutions the mean of simulated payoffs to measure an estimate of the risk-neutral expectation.Discounts the estimated expectation with the risk-free interest rate.The integration of the Monte Carlo technique into the option price for an asset follows the equation:Sitj = Sitj-1exp{(?i - σi2/2)dt+ σi(dt)1/2Zij} (27)Whereas Zij equivalents the correlated standard normal distributed random variable that considers the correlation between the underlying assets.Hence, the option price combining multiassets is given by: LΠMc(t,S) = e-r(T-t)(1/L)ΣΦi(t,S) (28) i=1Variable nomenclature?:ΠMc: The option price estimator.L: The number of Monte Carlo simulationsΦi: The ith payoff computed with the ith simulationThe Monte Carlo technique is flexible and renders the opportunity to value complex options for path dependent financial instruments as well as for derivatives relying on the value of underlying assets at maturity. 2.8 Exotic OptionsAn arbitrage-free technique is implemented to perform the pricing of derivatives in financial markets. The risk-free rate, the volatility, the strike price and the maturity represent parameters that impact the pricing of options. The implied surface volatility is characterized by the smile linked to the strike and the decay factor until maturity. The calibration of the copula is performed with log returns of stocks and interest rate derivatives. A competitive advantage of implementing a copula approach to pricing a bundled multiasset options is to value the multivariate pricing kernel as a function of univariate pricing functions. Ergo, the technique leads to a sensitivity analysis in regards to the dependence structure of the underlying assets in a distinct step from univariate prices. The protocol evaluates the pricing of best-of and worst-of call as well as put options for multistock baskets or for interest rate derivatives.The formulas to calculate the retained exotic options follow: Best-of-call option = Max (Max (S1(T)/S1(0),S2(T)/S2(0),…,Sn(T)/Sn(0)) – K,0.0) (29)Best-of-put option = Max (0.0, K - Max (S1(T)/S1(0),S2(T)/S2(0),…,Sn(T)/Sn(0)) (30)Worst-of-call option = Max (Min(S1(T)/S1(0),S2(T)/S2(0),…,Sn(T)/Sn(0)) – K,0.0) (31)Worst-of-put option = Max (0.0, K - Min(S1(T)/S1(0),S2(T)/S2(0),…,Sn(T)/Sn(0)) (32)In conclusion, the copula function stands an attractive loop computing tool in financial risk management due to its flexibility to divulgate information relating to the structure of dependence. 2.9 The Cascade Steps to Execute the Exotic Option and Interest Rate Derivative Pricing PrototypeThe cascade steps of the architected exotic pricing process aspires to map a dimensional frame of reference that will enable market participants to scissor the protocol and engineer sound derivative instruments in conformity with described environmental, mathematical, and theoretical regimes. The prototype has been designed with docility to accurately integrate the exhaustive steps and to render market participants a new spectre in financial engineering innovation. The exotic option and interest rate derivative pricing prototype is segmented in nine cascade steps:Gather data about the underlying assets combined in the basket. Then, pursue with statistical calculations: the log returns adjusted for dividends (for stocks), the standard error and the correlation matrix.Collect information about the market yield on U.S. Treasury securities or LIBOR at constant maturity, quoted on investment basis, to calculate the risk-free rate benchmark.Transform log returns of combined assets in the basket into a matrix with [0,1] pute correlations between marginal pairs. Integrate the first and second coordinate of marginal pairs to fit the best suited copula family via the maximum likelihood estimate methodology, that proceeds with the Akaike criteria at 95% confidence intervals, restricted to Archimedean and Survival copula families. Infuse retained copula families into a maximum likelihood estimate script for every pair of marginals.Engineer a Monte Carlo simulation script customized to integrate maginal pairs for all underlying assets combined in the basket.Integrate the following parameters in the engineered Monte Carlo script and launch 100,000 paths simulation: individual asset spot prices, basket strike price, marginal pairs with respective weights, mean risk-free rate, maturity in years, and lastly, option pricing mean payoff formula.Repeat step eight by changing basket strike price or other relevant parameters. 3. The Implementation of the Pioneered Financial Instrument Pricing Protocol with examples.The data used are monthly log returns on each stock or swap rate between May 31, 2008 and May 31, 2013. The retained time period encapsulates turbulences in capital markets following the worldwide financial crisis of 2008 as well as the liquidity crisis in the European continent. Data source: Computstat and the Board of Federal Reserve of Governors website. All stock asset quotes are denominated in USD currency whereas returns on Canadian stocks were converted to be uniformed in USD currency with American and global stocks. Spot prices integrated in the step 8 for Monte Carlo simulations are spot prices in USD currency as of May 31, 2013. The mean risk-free rate measured for the selected period from the market yield on U.S. Treasury securities at 5-year constant maturity, quoted on investment basis, is 1.67%.The maturity considered for all baskets and used in the payoff formula integrated in the Monte Carlo simulation script is 5 years. The formulas integrated in the Monte Carlo technique to value the prices of options are:Best-of-call option = Mean[Max(Max (S1(T)/S1(0),S2(T)/S2(0),…,Sn(T)/Sn(0)) – K,0.0)] Best-of-put option = Mean[Max(0.0, K - Max (S1(T)/S1(0),S2(T)/S2(0),…,Sn(T)/Sn(0))] Worst-of-call option = Mean[Max(Min(S1(T)/S1(0),S2(T)/S2(0),…,Sn(T)/Sn(0)) – K,0.0)] Worst-of-put option = Mean[Max(0.0, K - Min(S1(T)/S1(0),S2(T)/S2(0),…,Sn(T)/Sn(0))] All examples follow the cascade steps articulated in the previous sub-section.The following equally weighed assets compose each evaluated baskets:Basket 1:WMT: Wal-Mart Stores Inc with spot price $75.31.DOW: Dow Chemical with spot price $34.46.PEP: Pepsico Inc. with spot price $80.77.CM: Canadian Imperial Bank with spot price $76.84.TM: Toyota Motor Corp. with spot price $117.55.Basket 2:SNC: SNC-Lavalin Group with spot price $40.27.SLB: Schlumberger Ltd. with spot price $73.03.XOM: Exxon Mobil Corp. with spot price $91.10.ORCL: Oracle Corp. with spot price $33.78.SNY: Sanofi with spot price $54.88.Basket 3:ABT: Abbott Laboratories with spot price $36.67.MMM: 3M Co. with spot price $110.91.YUM: Yum Brands Inc. with spot price $67.75.ABX: Barrick Gold Corp. with spot price $21.56.CAN: Accenture Plc. with spot price $82.11.Basket 4: EOG: EOG Resources Inc. with spot price $129.10.TIF: Tiffany & Co. with spot price $77.78.KO: Coca-Cola Co. with spot price $39.99.TOT: Total SA with spot price $49.85.BNE: Bonterra Energy Corp. with spot price $50.77.Basket 5:GE: General Electric Co. with spot price $23.32.HD: Home Depot with spot price $78.66.PX: Praxair Inc. with spot price $114.33.UN: Unilever NV with spot price $41.13.PPL: Pembina Pipeline Corp. with spot price $31.93Basket 6:PG: Proctor & Gamble Co. with spot price $76.76.BMY: Bristol-Myers Squibb Co. with spot price $46.01.GS: Goldman Sachs Group with spot price $162.58.PBR: Petrobras Brasileiro SA with spot price $17.77.OCX: Onex Corp. with spot price $48.09.Basket 7:HPQ: Hewlett-Packard Co. with spot price $24.42.BA: Boeing Co. with spot price $99.51.NOV: National Oilwell Varco Inc. with spot price $70.30.CS: Credit Suisse Group with spot price $29.55.NMC: Newmont Mining Canada with spot price $34.88Basket 8: CVX: Chevron Corp. with spot price $123.75.WFM: Whole Foods Market Inc. with spot price $51.86.MCD: McDonald’s Corp. with spot price $97.34.STO: Statoil ASA with spot price $23.67.RCI.B: Rogers Communications with spot price $46.09Basket 9:Federal Reserve Interest Rate SWAP 1-year with spot rate 0.31%.Federal Reserve Interest Rate SWAP 10-years with spot rate 2.06%.Basket 10:Federal Reserve Interest Rate SWAP 1-year with spot rate 0.31%.Federal Reserve Interest Rate SWAP 30-years with spot rate 3.03%.Marginal pairs, entitled X1 to X10, are optimized with the maximum likelihood estimate methodology that selects the best fitted copula family with Akaike criteria at 95% confidence intervals. Precisely, marginal X1 to X10 represent: X1→pair asset 1 and asset 2; X2→pair asset 1 and asset 3; X3→pair asset 1 and asset 4;X4→pair asset 1 and asset 5; X5→pair asset 2 and asset 3; X6→pair asset 2 and asset 4; X7→pair asset 2 and asset 5; X8→pair asset 3 and asset 4; X9→pair asset 3 and asset 5;X10→pair asset 4 and asset 5.Table I:The selected copula family and the parameter Tau for optimized marginal pairs in basket 1 and in basket 2.Marginal PairBASKET 1BASKET 2CopulaFamilyTauParameterCopulaFamilyTauParameterX1Frank0.971009Survival Clayton 180°0.9867231X2Frank0.9876272Joe0.9896994X3Frank0.9855336Joe0.9883898X4Frank0.9817389Joe0.9893169X5Frank0.9709756Joe0.9889404X6Frank0.9747924Joe0.9883416X7Frank0.9730001Joe0.9870683X8Frank0.98745Gumbel0.9826846X9Frank0.9835121Frank0.9842805X10Frank0.9850065Frank0.9851588 Table II:The selected copula family and the parameter Tau for optimized marginal pairs in basket 3 and in basket 4.Marginal PairBASKET 3BASKET 4CopulaFamilyTauParameterCopulaFamilyTauParameterX1Frank0.9820586Joe0.9840988X2Frank0.981884Survival Clayton 180°0.9866521X3Frank0.971166Joe0.9892843X4Frank0.98176Survival Clayton 180°0.9885479X5Frank0.9856966Joe0.9857635X6Frank0.9693666Frank0.9759523X7Frank0.9849491Frank0.9748056X8Survival Clayton 180°0.9845783Frank0.9832539X9Gumbel0.9817971Survival Clayton 180°0.9895742X10Frank0.9692675Frank0.98226Table III:The selected copula family and the parameter Tau for optimized marginal pairs in basket 5 and in basket 6.Marginal PairBASKET 5BASKET 6CopulaFamilyTauParameterCopulaFamilyTauParameterX1Joe0.9894812Survival Joe 180°0.9342103X2Joe0.9898493Survival Clayton 180°0.9867991X3Survival Clayton 180°0.9885842Survival Clayton 180°0.9828045X4Frank0.9739416Frank0.9853354X5Frank0.9842391Joe0.9850402X6Frank0.9837455Survival Clayton 180°0.981574X7Frank0.9833997Joe0.9895653X8Frank0.9858827Survival Clayton 180°0.9874485X9Frank0.9845042Frank0.9780245X10Frank0.9851027Survival Clayton 180°0.9849014Table IV:The selected copula family and the parameter Tau for optimized marginal pairs in basket 7 and in basket 8.Marginal PairBASKET 7BASKET 8CopulaFamilyTauParameterCopulaFamilyTauParameterX1Survival Clayton 180°0.9869989Joe0.9855499X2Survival Clayton 180°0.9838207Frank0.9875328X3Gumbel0.9702724Frank0.9872144X4Frank0.9667889Clayton0.9310049X5Survival Clayton 180°0.9835559Frank0.9707417X6Survival Clayton 180°0.9863088Joe0.9843347X7Frank0.9682567Frank0.9692495X8Survival Clayton 180°0.9828627Joe0.99X9Survival Clayton 180°0.9809191Frank0.9889584X10Frank0.9644411Survival Clayton 180°0.989465Table V:The selected copula family and the parameter Tau for optimized marginal pairs in basket 9 and in basket 10.Marginal PairBASKET 9BASKET 10CopulaFamilyTauParameterCopulaFamilyTauParameterX1Joe0.9843163Joe0.984316Table VI:The prices of best-of and worst-of call and put options with mean payoff for basket 1 with different strikes at maturity 5 years.Options for basket 1Price with strike =0.5Price with strike =0.25Price with strike =0.75Price with strike =1Price with strike =2Price with strike =4worst-of-call-option0.13142480.34004070.02305810.00168107900worst-of-put-option0.021645510.00049940430.14327360.35168191.2689583.109798best-of-call-option1.3990291.6270121.1669390.93710490.17089590.00828372best-of-put-option0006.425093e-070.15409191.833652Table VII:The prices of best-of and worst-of call and put options with mean payoff for basket 2 with different strikes at maturity 5 years.Options for basket 2Price with strike =0.25Price with strike =0.5Price with strike =0.75Price with strike =1Price with strike =2Price with strike =4worst-of-call-option0.33222130.13025970.029863820.00398527900worst-of-put-option0.00027772550.028031940.15886360.36251711.2776223.118569best-of-call-option1.549341.3194141.0890160.86174930.14091810.0003594303best-of-put-option007.699336e-075.610919e-050.20179131.899853Table VIII:The prices of best-of and worst-of call and put options with mean payoff for basket 3 with different strikes at maturity 5 years.Options for basket 3Price with strike =0.25Price with strike =0. 5Price with strike =0.75Price with strike =1Price with strike =2Price with strike =4worst-of-call-option0.2376540.069290890.012889430.00189133400worst-of-put-option0.0018251230.063996510.2364650.45516391.3740913.213626best-of-call-option1.9715261.7440341.5157321.281030.40816760.003139898best-of-put-option00000.04483991.480802Table IX:The prices of best-of and worst-of call and put options with mean payoff for basket 4 with different strikes at maturity 5 years.Option for basket 4Price with strike =0.25Price with strike =0.5Price with strike =0.75Price with strike =1Price with strike =2Price with strike =4worst-of-call-option0.39578250.17622480.043298670.00638666900worst-of-put-option2.45412e-050.010802360.10740730.30119561.2144463.054868best-of-call-option1.9011671.6659521.4399861.2066630.39495940.02294544best-of-put-option0004.838732e-060.10692361.573567Table X:The prices of best-of and worst-of call and put options with mean payoff for basket 5 with different strikes at maturity 5 years.Options for basket 5Price with strike =0.25Price with strike =0.5Price with strike =0.75Price with strike =1Price with strike =2Price with strike =4worst-of-call-option0.55698820.33247270.15842130.05543951.946427e-050worst-of-put-option5.521747e-050.0085299870.063844140.19047171.0533792.89385best-of-call-option2.9082842.6802022.4467742.2155061.302130.1163408best-of-put-option00000.00017678540.65995Table XI:The prices of best-of and worst-of call and put options with mean payoff for basket 6 with different strikes at maturity 5 years.Options for basket 6Price with strike =0.25Price with strike =0.5Price with strike =0.75Price with strike =1Price with strike =2Price with strike =4worst-of-call-option0.067301880.011645410.0018369030.000283888300worst-of-put-option0.036758490.21023820.43051240.65939751.5793883.418675best-of-call-option2.1366361.907761.6791071.4482280.56339540.009777349best-of-put-option0007.363487e-080.034433851.320408Table XII:The prices of best-of and worst-of call and put options with mean payoff for basket 7 with different strikes at maturity 5 years.Options for basket 7Price with strike =0.25Price with strike =0.5Price with strike =0.75Price with strike =1Price with strike =2Price with strike =4worst-of-call-option0.16718430.028299560.002494390.000155894800worst-of-put-option0.0029984830.094332360.29765180.52528251.4459913.285466best-of-call-option1.146410.91879610.68817290.47447360.055924980.0008175573best-of-put-option04.75677e-060.0012056640.01619170.51888742.301448Table XIII:The prices of best-of and worst-of call and put options with mean payoff for basket 8 with different strikes at maturity 5 years.Options for basket 8Price with strike =0.25Price with strike =0.5Price with strike =0.75Price with strike =1Price with strike =2Price with strike =4worst-of-call-option0.4304510.21079850.068112380.014191981.269653e-070worst-of-put-option1.797017e-050.011109340.098509990.27415871.1805133.020869best-of-call-option3.2316853.0059942.7829492.5452111.6537160.5833539best-of-put-option00000.020246860.7905438Table XIV:The prices of best-of and worst-of call and put options with mean payoff for basket 9 with different strikes at maturity 5 years.Options for basket 9Price with strike =0Price with strike =0.1Price with strike =0.2Price with strike =0.5Price with strike =1Price with strike =2Price with strike =4worst-of-call-option0.097929570.039843960.020158660.0041373020.0006088713.725236e-059.996421e-07worst-of-put-option00.034130750.10597250.36655760.82257461.7420443.581905best-of-call-option2.9941632.9021512.8101282.5341392.0742521.1543421.209372e-05best-of-put-option0000000.6854948Table XV:The prices of best-of and worst-of call and put options with mean payoff for basket 10 with different strikes at maturity 5 years.Options for basket 10Price with strike =0Price with strike =0.1Price with strike =0.2Price with strike =0.5Price with strike =1Price with strike =2Price with strike =4worst-of-call-option0.097660230.040169290.019986850.0043380180.00071667343.724379e-051.948093e-05worst-of-put-option00.03405690.10632890.36642710.8223861.7417353.581962best-of-call-option2.9940872.9021612.8101722.5341422.0741951.1543052.495265e-05best-of-put-option0000000.68551064- ConclusionIn conclusion, the superiority of the copulae approach over standard correlation measurement techniques and the 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