Pricing Derivatives and Financial Options in … files/Modeling Financial Options... · Web viewPricing Derivatives and Financial Options in Globalized Markets Jamila Awad Pricing

Pricing Derivatives and Financial Options in Globalized Markets Jamila Awad

Pricing Derivatives and Financial Options in Globalized Markets

AuthorJamila Awad

Rights Reserved

JAW Group

DateJune 17, 2014

Paper: “Pricing Derivatives and Financial Options in Globalized Markets” (2014) Rights Reserved: JAW GroupAuthor: Jamila Awad JAW Group, 3440 Durocher # 1109Date: June 17, 2014 Montreal, Quebec, H2X 2E2, Canada Mobile: (1) 514 799-4565 E-mail: [email protected]


Executive Summary

The commencement of an unprecedented era of prudential financial risk management in globalized markets fosters the necessity to cement a widespread derivative pricing manual. The disquisition strives to map a frame of reference in pricing various derivative instruments and delivers executed examples that heighten the prescript: American option on a basket of stocks or ETFs, Asian option on futures, Asian option on correlated derivatives, Asian Call option on commodities, and lastly, barrier option on stocks. The discourse is partitioned in four sections. The introduction articulates the problematic incurred in the derivative universe and shadows the concepts inferred in the discourse. The second section umbrages the mathematical, theoretical and logic inferred in the prescribed code. The third section presents the results of the conducted derivative pricing models with true market data. The conclusion summarizes the main findings. In brief, market operators reinforce the soundness and robustness of globalized financial markets by transacting accurately priced derivatives.

2Paper: “Pricing Derivatives and Financial Options in Globalized Markets” (2014) Rights Reserved: JAW GroupAuthor: Jamila Awad JAW Group, 3440 Durocher # 1109Date: June 17, 2014 Montreal, Quebec, H2X 2E2, Canada Mobile: (1) 514 799-4565


1. Introduction

The burgeoning era of derivatives enhances the necessity to cement a pricing model framework for financial options that umbrages sound pricing techniques in line with risk management systems. Sound derivative instruments revolutionized globalized frontiers whereas market operators as well as investors grasp tools to hedge risk thus thwart gambling and manipulation practices. Furthermore, derivatives represent mediums used in trading transactions for tax purposes. Financial markets in the continuous epoch of globalization are referred as neither perfectly integrated nor entirely segmented in an environment with asymmetric information. The research paper maps a manual of reference that dictates the instructions to price various derivatives in line with market constraints such as automated trading platforms in asymmetric information environments.

The research paper is partitioned in four sections. The introduction articulates the problematic incurred in the derivative universe and shadows the concepts inferred in the narration. The second section umbrages the mathematical, theoretical and logic inferred in the prescribed code. Precisely, the proposed financial instruments are: American option on a basket of equity assets, Asian option on futures, Asian option on correlated derivatives, Asian Call option on commodities, and lastly, barrier option on stocks. The third section presents the results of the conducted derivative pricing models with true market data. The conclusion summarizes the findings.

The pioneers of derivative option pricing methodologies are mathematical engineers named Black, Scholes and Merton (1973) who revolutionized the universe of arbitrage, hedging, and lastly, automated trading in financial markets. Precisely, Black and Scholes (1973) delivered a closed form solution to value European options. Merton (1973) cemented an analytical framework for down and out barrier Call options and his publications became a reference for extended barrier options (Reiner and Rubinstein, 1991).

Derivatives depict financial instruments or contracts whose values are derived from the appraisal of underlying financial assets. The payoff feature of derivatives entails the modeling of various cash-flow patterns. Furthermore, contingent claims render investors flexible features such as redemption at specific dates, early exercise, and lastly, tailored payments for customized contracts.

Financial options depict contracts between buyers and sellers whereas the owner of an option is provided the choice to exercise the underlying and, on the other side, the underwriter is confined to pursue an action, buy or sell, in reaction to the counterparty’s decision. The two types of options are stated as Call or Put.

American options portray financial instruments that enable the holder of the option the choice to exercise the underlying at a specified time and with established features. In addition, American options are more flexible than European options whereas the holders of American options are entitled to exercise the underlying anytime between the inception date and expiry date. The



technical aspect of pricing options plays a dominant role that becomes beneficial for investors. Precisely, option detainers desiring to exercise at-the-money options are bound to weigh the payoff value in order to capture the highest financial redemption. On the other hand, option underwriters are also rewarded when options are adequately modeled with enhanced pricing precision and reduced level of uncertainty.

Exotic options portray derivatives whose values derive on underlying asset path simulations. In addition, exotic options offer sophisticated investors malleable financial instruments with various payoff opportunities while mitigating risk.

Barrier options depict exotic derivatives that offer a cheaper alternative compared to plain vanilla options. In fact, barrier options are activated or expired when the underlying asset price either hits or not a specified barrier before maturity. In precise terms, the barrier option is exercised when it hits a determined level.

Future contracts represent derivatives that set predetermined investments with specified features. They also portray popular mediums to hedge risk and heighten profits between two entangled parties.

Asian options depict instruments that protect against price manipulation whereas the pricing of Asian options depend on the average underlying assets path simulations (Kemna and Vorst, 1990).

The primitive methods for modeling American option prices are the lattice tree techniques or the finite differential equations. These standardized option pricing frameworks are bound to assumptions that map an exaggerated simplification of real financial market environments. Binomial methods examine the dynamics of an option’s theoretical value for discrete time intervals over the option’s duration to estimate the value of the option at tree nods. Finite differential equations implement the Taylor series expansion theorem to approximate partial derivatives and thus value the price of options at potential exercise points.

The pricing of American options in higher dimensions is adequately conducted with the Least Squares Monte Carlo technique (Longstaff and Schwartz, 2001) that solutions conditional expectations via linear combinations of basic functions.

The Monte Carlo simulation approach depicts an efficient and docile solution to simulate asset paths. Furthermore, this technique also delivers a flexible backward induction algorithm that simulates bundled financial instruments and accurate pricing arrangements for options with early exercise features (Hull and White, 1990).

The implementation of Monte Carlo simulations evolved in option pricing models with examples such as plain vanilla options (Boyle, 1977). In fact, antithetic control variates portray complementary tools that enhance variance minimization in confidence intervals when pricing options with Monte Carlo simulations.



In conclusion, the accurate pricing of derivatives enhances liquidity in globalized markets and enable participants to grasp investment opportunities with various payoff features.



2.The Derivative Instruments Framework and Market Environment

The second section umbrages the mathematical, theoretical and logic implied in the presented derivatives. The selected financial instruments are: American option on a basket of stocks or ETFs, Asian option on futures, Asian option on correlated derivatives, Asian Call option on commodities, and lastly, barrier option on stocks.

2.1 The American Option

American Put options are in some occasions exercised prior to maturity which is not the case for American Call options.

An American Put option on a unitary asset with non-dividend paying stock whose price is determined by a geometric Brownian motion is expressed with the following mathematical formula:

St = S0exp[σWt + (r – σ2/2)t] (1)

Variable nomenclature:

St: The price of the asset at exercise timeS0: The initial price of the assetr: The constant risk-free rate of interestσ: The constant volatility of the assetWt: A standard Brownian motion

A stochastic process W follows a Brownian motion movement in line with the axioms:

The increments (Wt – Ws) represent a normal distribution with a mean of zero and a variance equal to (t – s).

The normal increments are said independent (without memory of past behavior). The paths of Wt depict continuous functions of t.

The basis function linked to the American Put option is determined by Laguerre polynomials whereas X depicts the value of the asset underlying the option that follows a Markov process:

L3(X) = (e-X/2)(eX/3!)(d3/dX3)(X3e-X) (2)

The discounted realized payoffs are then regressed from the above stated formula. The value of the early exercise option equals the difference between the American and the European Put values.



In the occurrence of an American Put on a basket of assets, the payoff of the option represents a function of the maximum of the bundled assets.

The Brownian motion for a pool of assets is given by:

Si(t)= Si(0)exp[σiWi(t) + (r – σi2/2)t] whereas i = 1,…,n (3)


Si(t): The price of the asset at exercise timeSi(0): The initial price of the assetr: The constant risk-free rate of interestσ: The constant volatility of the assetWt: A standard Brownian motion

The payoff value for an American Put on a basket of assets is evaluated by:

V(Si,…Sn) = Max[0, K – Max (Si,…,Sn)] whereas whereas i = 1,…,n (4)


V(Si,…Sn): The value of the American Put option at exercise time (Si,…,Sn): The price of the underlying assetsK: The strike price

Options are modeled under a risk-neutral valuation environment whereas an asset pricing path (S(t)) follows a geometric Brownian motion process complying with the stochastic differential equation:

dS = μSdt + σSdX (5)


dS: The asset price stochastic differentialμ: The average growth rate of the asset pricedt: The stochastic differential σ: The volatility of the assetS:The asset pricedX: The stochastic differential of the standard Brownian motion.

The stochastic differential equation is also defined on a probability space (Ω, F, P). The Girsanov theorem is implemented to evaluate the probability Q in liaison with the standard Brownian motion.



The option payoff function is represented by u(S,t). A simple non-path dependent option is described by:

u(S,t)=u(S(t),t) (6)

Otherwise, an option that relies on the entire path is illustrated by {S(t):0<t’<t}. The early exercise (τ) of an American option prior to time T is expressed with the following payout formula:

F(τ) = (S(τ),τ) (7)

The Black-Scholes-Merton risk-neutral valuation equation is formalized with:

F(S,t)=Max E[e-r(τ-t)μ(S(τ),τ|S(t)=S)] (8)


E[.]: The risk-neutral expectationμ: The growth rateτ: The time when the option is exercisedμ(S(τ),τ|S(t)=S):The payoff of the option at the time it is exercised conditional to an asset price

The Monte Carlo simulation procedure therefore applies the previously elaborated mathematical frameworks.

The Black-Scholes PDE values options (V(S,t) whereas V depicts the value of the entire portfolio combining different options) with the Ito lemma stochastic differential formula represented by:

df = σS(∂f/∂S)dx + [μS(∂f/∂S) + 0.5σ2S2(∂2f/∂S2) + (∂f/∂t)]dt (9)

The mathematical expression considers the random walk followed by V:

dV = σS(∂V/∂S)dx + [μS(∂V/∂S) + 0.5σ2S2(∂2V/∂S2) + (∂V/∂t)]dt (10)

The value of a portfolio combining one option and an undetermined number of underlying assets, Δ, is calculated with the frame:

Π = V – ΔS (11)

The jump process in one-time step of the value of the above illustrated portfolio is characterized with the equation:



dΠ = dV – ΔdS (12)

Therefore, the Ito lemma stochastic differential formula is reorganized whereas dΠ follows a random walk:

dΠ = σS(∂V/∂S - Δ)dx + [μS(∂V/∂S) + 0.5σ2S2(∂2V/∂S2) + (∂V/∂t) - μΔS]dt (13)

The deterministic equation follows:

dΠ = [0.5σ2S2(∂2V/∂S2) + (∂V/∂t)]dt (14)The return on Π sum invested in riskless asset is equal to rΠdt. The risk-neutral world simplifies the trading universe by supposing that transactions occur in the absence of transaction costs. Arbitrageurs gain a riskless profit greater than the cost of borrowing when the deterministic equation of the value of a portfolio diverges from the value dΠ. The arbitrage opportunity context is also applicable when the deterministic equation of the value of a portfolio is inferior to rΠdt.

The deterministic component of the formula is rearranged:

rΠdt = [0.5σ2S2(∂2V/∂S2) + (∂V/∂t)]dt (15)

Lastly, the Black-Scholes partial differential equation is obtained by considering previously explained formulas that are divided throughout by dt:

0.5σ2S2(∂2V/∂S2) + (∂V/∂t) + rS(∂V/∂S) – rV = 0 (16)

The PDE enables to value an option in terms of the function V(S,t) and removes the drift parameter μ.

2.1.1 The Monte Carlo Technique and The Least Squares Monte Carlo Method

The Monte Carlo technique depicts a flexible framework that models high-dimensional integrals. The methodology values the price of securities in cascade steps:

1. Paths of the underlying state variables in a risk-neutral frame are simulated over the time horizon.

2. The discounted cash-flows of the security on each sample path are evaluated.3. The average discounted cash-flows of the security over all traced sample paths are

quantified.

The Monte Carlo procedure is also relevant to price financial instruments with complex structures such as assets with correlated features. The technical arrangement configures the



option price, μ, as an integral that characterizes the mathematical expectation of the discounted payoff under a risk-neutral probability measure.

The expectation is then schematized as an integral over an s-dimensional unit hypercube with the expression:

μ = μ(f) =∫(0,1)

❑

f (u ) du = E[f(U)] whereas (0,1) = {u: 0 <uj <1 for all j} (17)

The Monte Carlo simulation is to some extent restrictive for American options due to the early exercise feature. Ergo, the Least Squares Monte Carlo (LSM) procedure represents an interesting alternative to determine the early exercise boundary. The LSM technique values American options with the least square algorithm of Carriere that enables to compare payoffs from early exercise against the payout until option maturity. The expected payoff for option continuation rather than early exercise is therefore derived from the cross-sectional information used in the least squares simulation. Hence, the LSM framework delivers a flexible arrangement to estimate path-dependent and multifactor simulation environments. The conditional expectation function is obtained by regressing the realized payouts from option continuation on a set of basis functions for each simulated asset path.

The LSM technique is performed with the following cascade steps:

1. The method executes a valuation of N random paths defined as Skn,tn (for 1 ≤ k ≤ N and tn

= ndt) by rolling-back on these paths.2. The current asset value is then determined for points (Sk

n,tn) in the set X= Skn with the

equation Y=e-rdtF(Skn+1,tn+1) whereas Y portrays the value of the deferred exercise and

F(Skn+1,tn+1) depicts the value for the simulated path.

3. A regression of Y is analyzed as a function of polynomials X,…,Xm whereas m represents the value in the basic function.

4. The least squares fit of the previously described polynomials in X then approximates Yk to compare the payoff of early exercise against the option payout at maturity.

The number of asset paths grow exponentially with the number of polynomials basis functions, therefore the convergence rate of the algorithm is compelling to value the early exercise of American Put options (Glasserman, 2004).

2.2 The Asian Option

An Asian option portrays a derivative instrument whose value depends on the average price of the underlying assets over a period of time. This exotic financial tool therefore induces a lower volatility and is less expensive compared to plain vanilla options. A panoply of Asian options exist in financial markets such as continuous arithmetic average Asian options as well as discreet geometric forms.

The assumptions relevant to the Asian option modeling implemented in this redaction are:



The price of assets follow a log-normal distribution continuous in time The product of log-normal distributed random variable is log-normal as well The sum of log-normal distributed random factors is not log-normal The presence of a risk-neutral environment

The pricing of a geometric average Asian option derives from the Black-Scholes framework.

The payout for discreet geometric average Asian options is illustrated with the mathematical expression:

In the occurrence of a Call: Φ(S) =[ ∏i=0

m

S ( ¿m )1/(m+1) - K]+ whereas i = 1,…,m (18)

And

S(t1)/S0 = exp[(r - σ2/2)T/m + σ(T/mXm)1/2] (19)

that follows a Brownian motion with independent distributed Xm ~ N(0,1) random variables .

The payoff Call option formula is then rearranged:

Φ(S) =log[ ∏i=0

m

S (t i )1/(m+1) /S0] = [(r – σ2/2)T]/2 + [σ (T/m)1/2∑i=1

m

iXi]/(m+1) (20)

The portion of the payout Call option frame is restructured to integrate the additive mean and the variance property of independent normal random variables:

[σ(T/m)1/2∑i=1

m

iXi]/(m+1) = σ [((2m+1)T)/(6(m+1))]1/2Z whereas Z ~ N(0,1) (21)

Finally, the Call option value in a risk-neutral environment is illustrated by the formula:

Φ(S) = [ρ – σ2Z/2]T + σZ(T)1/2Z whereas ρ = [(r - σ2/2) + σ2

Z]/2 (22)

And

σZ = σ [(2m+1)/(6(m+1))]1/2 (23)

In brief, the payoff values for Call and Put geometric average Asian options in a risk-neutral environment are:

Call: CK,g(S0,T) = exp [(ρ – r)T]~CK(S0,T) (24)

Put: PK,g(S0,T) = exp [(ρ – r)T]~P K(S0,T) (25)




CK,g(S0,T): The payout value of a geometric (g) average Asian Call option PK,g(S0,T): The payout value of a geometric (g) average Asian Put option ~CK(S0,T): The price of a European Call option with risk-free interest rate (r) and volatility σZ~PK(S0,T): The price of a European Put option with risk-free interest rate (r) and volatility σZ

The arithmetic average Asian option pricing is however more complex to model and cannot be simplified in line with the Black-Scholes framework. The proposition introduces the following inequality that explains why an arithmetic average Call Asian option is cheaper and more appealing compared to a European Call option:

CK,a(t,S(t)) ≤ CK(t,S(t)) (26)

The inequality is rewritten in proof format with the following parameters:

EQt,S(t)[(1/1+m) + (m/1+m)Q – K’]+ ≤ EQ

t,S(t) [Q – K’]+ (27)


Q:The risk-neutral factor measured by Q=[(α1+ α1α2…+ α1…αm)/m] whereas S(T)=α0* α1*…* αm

K’: The strike price given by K’= K/α0

Thus, the appealing cheaper cost feature of arithmetic average Asian options compared to traditional European options.

2.2.1 The Monte Carlo technique for the arithmetic average Asian option

The Monte Carlo technique projects a flexible framework to simulate arithmetic average Asian option paths in the pricing architecture.

The Monte Carlo method approximates an estimator entitled θ with the following formula:

Θ = E[g(X)] (28)

Whereas the parameter g(X) depicts an arbitrary random function that induces E[|g(X)|] < ∞ to generate n independent arbitrary observations X1,…,Xn from the probability function f(X).

The estimator is quantified as:

θ̂= (1/n)∑i=1

n

g (Xi) (29)

Therefore, the theorem of large numbers is applied such:

(1/n)∑i=1

n

g (Xi)a . s .→

E[g(X)] whereas θ̂→∞ when n→∞ (30)



The variance (s2) of a sample of the Monte Carlo simulation is characterized by:

s2 = (1/n-1)∑i=1

n

¿¿ θ̂)2 (31)

The central limit theorem implies the following proposition:

[n1/2(θ̂ – θ ¿/ s¿ d→ N(0,1) when n→∞ (32)

The parameter (θ̂ – θ) is said to be an approximation of a standard normal variable scaled by (s/n1/2). Hence, the solution is summarized with the expression as n increases:

P[θ̂ – zα/2s/n1/2 < θ < θ̂ + zα/2s/n1/2] ≈ 1 – α (33)

The convergence rate in the Monte Carlo technique enables to estimate the variance of a population whereas s ≈ σ for large n numbers.

The Monte Carlo technical framework to price arithmetic average Asian Call options generates

S(T/m),S(2T/m),…,S(T) and evaluates the Max [0, ((∑i=0

m

S ( ¿m

))/m+1) - K]. (34)

However, the procedure is combined to a variance reduction methodology with a control variate to enhance the convergence rate and to minimize the variance.

The variance minimization modus operandi focuses on reducing the size of n thus to narrow the confidence interval. The control variate arrangement generates an unbiased estimator of θ (θ:E(Y) with Y = g(X)) by inducing another random variable entitled control variate, denoted Z, with a known mean E(Z) such:

θ̂ = Y to attain the target estimator θ̂c = Y + c(Z – E(Z)) whereas c represents a real number E(θ̂c) = E(Y) + c[E(Z) – E(Z)] = E(Y) = θ

The variance of the control variate is measured by:

Var(θ̂c) = Var(Y) + c2Var(Z) + 2cCov(Y,Z) (35)

However, the procedure minimizes the real number c with the solution:

Cmin = Cov(Y,Z)/Var(Z) (36)

The rearranged formula for the control variate estimator variance is calculated by:

Var(θ̂Cmin) = Var(θ̂) - Cov(Y,Z)2/Var(Z) with Cov(Y,Z) ≠ 0 (37)

Precisely, the selected control variable is the geometric average Asian option.



In light with the articulated propositions, the arithmetic average Asian option represents an appealing derivative compared to a standard European option.

In conclusion, the Monte Carlo technique amalgamated to the geometric average Asian option as an antithetic variate delivers a flexible and sound frame of reference to price options with precise estimations.

2.3 The Barrier Option

A Barrier option portrays an exotic derivative that offers a cost wisely alternative compared to a plain vanilla option. Precisely, a barrier option is activated or expired when the underlying asset price either hits or not a specified level, denoted Sb, before maturity.

The two main classes of barrier financial instruments are stated as knock-in or knock-out options. In light, knock-in refers to a barrier option that is activated if Sb is crossed. On the other hand, knock-out describes a canceled contract if Sb is crossed throughout the entire existence of the option. Lastly, an exotic contract is said to be an up barrier option if Sb is higher than the current asset price, S0, otherwise the derivative is stated as a down barrier option when Sb < S0.

Therefore, a down-and-out Put option is less costly than a plain vanilla derivative due to the exotic instrument’s pre-expiration feature if the barrier level is crossed.

However, a mathematical relationship exists between the sum of two types of Put barrier options and a plain vanilla Put option. Precisely, the replication of a plain vanilla Put option is performed by holding a down-and-out as well as a down-and-in Put option. The parity formula is expressed as:

PP.V. = PD&I + PD&O (38)


PP.V.: The price of a plain vanilla Put optionPD&I: The price of a down-and-in Put Barrier optionPD&O: The price of a down-and-out Put Barrier option (Sb < S0 and Sb < K)

Therefore, augmenting the absolute difference between the barrier level and the initial asset spot price reduces the option price in the case of the knock-in option. In addition, enhanced volatility heightens the value of a knock-in option. Conversely, elevating the absolute difference between the barrier wall and the initial asset spot price accrues the value of the knock-out option. Lastly, risk increase measured by sigma plunges the value of a knock-out option.

Barrier options are modeled with continuous or discreet techniques. In light, the following equation depicts the analytical solution to price a Down-and-Out Put barrier option in the occurrence of a continuous framework:

P = Ke-rT{N(d4) – N(d2) - a[N(d7) – N(d5)]}- S0{N(d3) – N(d1) - b[N(d8) – N(d6)]} (39)



Whereas

a = (Sb/S0)-1+2r/2σ ;b = (Sb/S0)1+2r/2σ ; d1 = (log(S0/K) + (r + 1/2*σ2)*(T))/(σ*T1/2);d2 = (log(S0/ K) + (r - 1/2*σ2)*T)/(σ*T1/2);d3 = (log(S0/ Sb) + (r + 1/2*σ2)*T)/(σ*T1/2);d4 = (log(S0/ Sb) + (r - 1/2*σ2)*T)/(σ*T1/2);d5 = (log(S0/ Sb) - (r - 1/2*σ2)*T)/(σ*T1/2);d6 = (log(S0/ Sb) - (r + 1/2*σ2)*T)/(σ*T1/2);d7 = (log((S0*K)/ Sb

2) - (r - 1/2*σ2)*T)/(σ*T1/2);d8 = (log((S0*K)/ Sb

2) - (r + 1/2*σ2)*T)/(σ*T1/2);


P: Price of a down-and-out Put Barrier optionK: The strike priceSb: The barrier levelS0: The initial asset priceT: The option expiration timeσ: The volatilityr: The risk-free rate

The continuous option pricing arrangement can be harmonized with the discreet form by increasing the frequency of the continuous pricing process. The shift frequency is measured by considering the following parameters: the monitoring frequency (δt), the asset volatility (σ), and lastly, a constant β (Broadie and Glasserman, 1997).

The relationship between the continuously monitored Barrier option and the discreet price of a knock-in or knock-out down Call or up Put Barrier option is expressed as:

PBM(Sb) = P(Sbe±βσ(δt)1/2) + σ(1/m1/2) (40)

And

β = -0.5ζ/(2π)1/2 ≈ 0.5826 (41)


PBM(Sb): The value of a monitored barrier optionSb: The corrected barrier level equal to Sbe±0.5826σ(δt)1/2 (The ± depends of the option type: Call or Put) to use the continuous value as a discreet price estimate σ: The volatilityζ: The Riemann zeta functionβ: A constant



Finally, the value of a discreet monitored down-and-out option is higher than its continuous form due to the lower occurrence of a discreet version barrier option of crossing the barrier level.

The technical aspect of pricing barrier options follows the Monte Carlo simulation arrangement and the variance reduction concept described in the previous section. In light, the Monte Carlo procedure is complemented with an antithetic variate.

12.4 The Asian Option on Derivatives

Options on derivatives represent an enhanced investment vehicle that shield against price fluctuations in assets, interest rates, and lastly, foreign exchange. Furthermore, option of future contracts generally demand less investment compared to options directly linked to physical goods.

An option on a futures contract is summarized by an option with expiration date T1 and an underlying, the futures contract, with an ending date T2 therefore T1 ≤ T2. Hence, movements in the value of the futures contract impact the option price. The strike price depicts the specified futures price at which the future contract is traded at time T1 if the option is exercised.

The payoff for options on future contracts is expressed as:

In the occurrence of a Call: PC(T1) = [G(F(t,T2)) - K]+ (42)

In the event of a Put: PP(T1) = [K - G(F(t,T2))]+ (43)

Whereas G(F(t,T2)) defines a function of the futures price at time t.

The discounted payout for options on futures conditional to the risk-neutral measure (F(ti)) and the growth rate (er(T1-ti)) is illustrated by:

In the occurrence of a Call: PC(T1) = e-r(T-t)E[G(F(t,T2)) – K| F(ti)]+ (44)

In the event of a Put: PP(T1) = e-r(T-t)E[K - G(F(t,T2))| F(ti)]+ (45)

Asian options on a bundle of future contracts depict a flexible hedging alternative for investors seeking to combine future contracts of commodities in various geographical locations. Standard plain vanilla options are biased to hedge the risk of a basket of correlation future contracts whereas the underlying does not comply with the lognormal distribution assumption.

An Asian option depicts an exotic derivative example and its value is estimated by the average underlying price over a pre-determined time period. Future contracts enable to lock in future asset prices therefore the entangled parties benefit from such instruments. However, future contracts requisite that parties honor margin requirements but option purchases only necessitate the payment of a premium.



The proposed framework to price arithmetic discreet Asian options on a basket of futures entails to consider the martingale stochastic expression, the geometric Brownian motion, and finally, the Ito lemma drift-diffusion process, described in previous sections.

The simulation of asset future prices is performed with the log likelihood function and a partial differentiation. The future asset return (Ri) for future prices (F0,…,Fi whereas i = 0,…, n) is measured by: Ri = (Fi – Fi-1)/Fi-1. (46)

The future asset returns (R1,…,Rn) become normally distributed with mean μ and variance σ2dt when a time step ∆t is fixed. The future price dynamics is then implemented via the maximum likelihood estimation technique and an estimator for σ obtained from n data points.

Suppose a sample X1,…,Xn from a density pθ (whereas θ = θ1,.., θn and n depicts the number of variables dependent on the density). The maximum likelihood function for the maximizer θ is

expressed as: Lx(θ) = ∏i=1

n

❑pθ(Xi). (47)

In addition, the log likelihood function is illustrated by: log Lx(θ) = ∑i=1

n

❑log(pθ(Xi)). (48)

In light, pθ is partially differentiable in regards to the vector θ: ∂logLx(θ)/ ∂θj = 0 whereas j = 1,…,n.

The variable X is then transformed to Z ~ N(μ,σ2) whereas f(X) = Z and θ = [μ,σ2]. The distribution function of X is measured by: P(X ≤ x) = Φ0,1[(f(x) – μ)/σ]. (49)

The density ƒx is expressed as: ƒx = dΦ/d[(f(x) – μ)/σ] and Φ(.) depicts the cumulative standard normal distribution.

Therefore, the log likelihood function for a sample distribution of X is given by:

Log Lx(μ,σ) = nlog(2π)1/2 - nlogσ – (1/2σ2)∑i=1

n

( f (Xi) – μ)2 + ∑i=1

n

logdf (Xi )/dx (50)

The differentiation process in regards to μ and σ generates:

dlogLx/dμ = (1/σ2)∑i=1

n

( f (Xi) – μ) (51)

And

dlogLx/dσ2 = -(n/2σ2) + (1/2σ4)∑i=1

n

( f (Xi) – μ)2 (52)

Hence, the parameters that maximize the log likelihood function are:



μ = (1/n)∑i=1

n

( f (Xi)) (53)

And

σ2 = (1/n)∑i=1

n

( f (Xi) – μ)2 (54)

In terms of future price dynamics:

μ = (1/n)∑i=1

n

(log( Xi)) (55)

And

σ2 = (1/n)∑i=1

n

(log( Xi)– μ)2 (56)

Lastly, the unbiased volatility measure is illustrated by:

σ2 = (1/n - 1)∑i=1

n

(log( Xi)– μ)2 (57)

Options on a basket of assets enhance portfolio diversification and decrease volatility.

The following equation represents a geometric Brownian motion on the future price of assets for an option on bundled future contracts:

dFi(t)/Fi(t) = σidWi(t) whereas i = 1,..,p (58)


Fi(t): The futures price at time t of asset iσi: The constant volatilityp: The number of assets in the basket

The correlation between the future prices is given by: Cov(Wi(t), Wj(t)) = ρ(i,j)t. (59)

Therefore, the option payoff on a bundle of futures at expiration time T and with a number of averaging dates (n) with a strike (K) is illustrated by:

P(T) = [A(T) - K]+ whereas A(T) = (1/n)∑j=1

n

❑ ∑i=1

p

Fi (tj) (60)



The multidimensional frame in modeling an option on a bundle of futures is changed from dependent to independent by implementing the Cholesky theorem.

Suppose a vector-value process X(t) described by:

dX(t) = μXdt + σX(t)d ˇW ¿¿t) (61)


ˇW ¿¿t): The vector of independent Brownian motionσ:The volatility

The Cholesky decomposition states that every symmetric positive definite matrix, denoted C, has a unique factorization therefore: C = LLT whereas L depicts a lower triangular matrix with positive diagonal entries and LT portrays the upper triangular matrix.

To solve the expressions in the occurrence of correlated Brownian motions such as a basket of futures, by setting W(t) = [W1(t), W2(t)]T for a portfolio containing two correlated assets denoted S1(t) and S2(t):

dS1 = μ1S1(t)dt + σ1(t)S1(t)dW1(t) (62)

dS2 = μ2S2(t)dt + σ2(t)S2(t)dW2(t) (63)

The Cholesky decomposition for matrix C is illustrated by: C =[ 1 ρρ 1 ] (64)

In brief, the dependent Brownian motions W1 and W2 are transformed to be independent by dW(t) = Ld ˇW ¿¿t) with:

dW1(t) = dW̌ 1(t) (65)

dW2(t) = ρdW̌ 1(t) + (1 – ρ2)1/2dW̌ 2(t) (66)

The Monte Carlo technique depicts a flexible framework to price options on a bundle of futures whereas underlying prices follow geometric Brownian motions.

The discreet equation format to value changes in future prices is illustrated by:

F(tj+1) = F(tj)exp[-1/2σ2∆t + σ(W(tj+1) – W(tj))] (67)


F(tj): The futures price at time tj (tj = j∆t with j = 0,…,n and ∆t = T/n whereas T depicts the maturity date of the future contract and n equals the number of dates used to average the futures price)



σ: The futures annual price volatility

The Monte Carlo simulation is initiated by simulating a sample path of future prices {F0,F1,…,Fn}. The simulation is then performed M times to capture M*n futures prices. Lastly, the option payoff is estimated by discounting the average of all simulation paths payoff functions.

Therefore, the Monte Carlo simulation on an Asian option on a unitary future contract is approximated by simulating the price path using:

F(tj+1) = F(ti) + σF(ti)Zi+1(∆t)1/2 (68)

Whereas the parameter Zi+1 represents a normally distributed random variable and ∆t depicts the time step.

The framework for valuing an Asian option on two correlated future contracts is performed by firstly executing a Cholesky decomposition that generates the following pricing paths:

F1(ti+1) = F1(ti) + σ1F1(ti)[W1(ti+1) – W1(ti)] (69)

F2(ti+1) = F2(ti) + σ2F2(ti)[ρ1,2(W1(ti+1) – W1(ti))] + (1 – ρ21,2)1/2 [W2(ti+1) – W2(ti)] (70)

Therefore, the dynamics of the pricing paths for the futures are expressed as:

F1(ti+1) = F1(ti)exp[-1/2 σ21∆t + σ1(W1(ti+1) – W1(ti))] (71)

F2(ti+1) = F2(ti)exp[-1/2 σ22∆t + σ2(ρ1,2(W1(ti+1) – W1(ti)) + (1 – ρ2

1,2)1/2 (W2(ti+1) – W2(ti))] (72)

The approximation method amalgamated to the Monte Carlo simulation is used to model Asian options on a bundle of future contracts. Precisely, the approximation technique estimates the sum of futures with a geometric average and a lognormally distributed variable.

The lognormal financial expression to describe the averaged sum of future prices is illustrated by:

A(T) = Mexp[-1/2σ2T + σW(T)] (73)

Whereas the approximation process requires to estimate the parameters M and σ. The expectation of A(T) enables to approximate M and the variance of the geometric average is used to quantify σ.

Therefore, the approximation method generates A1(T) = Mexp[-1/2σ2T + σW(T)] (74)

but the accurate value is A2(T) = (1/n)∑j=1

n

❑ ∑i=1

p

Fi (tj). (75)

Hence, the parameter M is obtained by setting E[A1(T)] = E[A2(T)]. (76)

The solution to estimate E[A1(T)] is illustrated by: 20

Paper: “Pricing Derivatives and Financial Options in Globalized Markets” (2014) Rights Reserved: JAW GroupAuthor: Jamila Awad JAW Group, 3440 Durocher # 1109Date: June 17, 2014 Montreal, Quebec, H2X 2E2, Canada Mobile: (1) 514 799-4565


[A1(T)] = E[Mexp(-1/2σ2T + σW(T))] (77)

= Mexp(-1/2 σ2)E[exp(σW(T)]

= Mexp(-1/2 σ2)exp(1/2 σ2)

= M

The analytical expression to value E[A2(T)] is described by:

[A2(T)] = E[(1/n)∑j=1

n

❑ ∑i=1

p

Fi (tj)] (78)

= (1/n)∑j=1

n

❑ ∑i=1

p

E (Fi(tj)¿)¿

= (1/n)∑j=1

n

❑ ∑i=1

p

Fi (0¿)¿

= ∑i=1

p

Fi (0¿)¿

= A(0)

The geometric average is obtained by: G(T) = ∏j=1

n

❑∏i=1

p

[( 1nai

)Fi( tj)]ai (79)

Whereas ai= [(1/n)Fi(0)]/∏i=1

p

Fi (0) (80)

And

∏j=1

n

❑∏i=1

p

ai= 1 (81)

Therefore, G(T) = A(0)exp[∑j=1

n

❑∑i=1

p

❑(-1/2aiσ2itj + aiσiWi(tj)] (82)

Hence, the variance of the geometric average is used as the volatility of the arithmetic mean. Suppose Y(T) = log(G(T)). (83)

The variance is estimated by the mathematical formula:

Var(Y) = Var [logG(T)] (84)

= E[log(G(T))2] - [E(log(G(T))]2



The volatility of Y(T) is quantified by measuring the first and second moments:

E[Y(T)] = log(A(0)) + ∑j=1

n

❑∑i=1

p

❑¿1/2aiσitj) = y0 (85)

E[Y2(T)] = y20 + 2y0.0 + E[(∑

j=1

n

❑∑i=1

p

❑aiσiWi(tj)))2] (86)

In brief, the expression Y(T) is quantified by:

Var (Y) = y20 + ∑

i 1 ,i 2=1

p

❑ ai1ai2σi1σi2. ∑j 1 , j 2=1

n

❑ ρ(i1,i2)min(tj1,tj2) - y20 (87)

The variance of the geometric average is expressed as:

σ2T= ∑j 1 , j 2=1

n

❑ ∑i 1 , i2=1

p

❑ai1ai2σi1σi2ρ(i1,i2)min(tj1,tj2) (88)

Lastly, the Black formula is used to evaluate the parameters M and σ for options.



3.Implementation of the described derivatives with true market dataThe third section presents the results following the execution of the various derivatives articulated in the previous chapter. The examples strive to deliver a frame of reference whereas true asset market data are matched with the best suited financial instrument. The data source is Bloomberg. The asset tickers refer to the Bloomberg ticker nomenclature.The daily data window is from June 1, 2009 to June 1, 2014. The average standard type is obtained with the following mathematical expression: Dividing the daily average asset closing price for the selected window by the average standard-type for the selected window. The average dividend yield is simply the window arithmetic mean of the dividend yield for the asset.

3.1 Examples for American Put Option

The calculated American Put option is a variant of the described formula in the second section. Precisely, the used formula for the illustrated example estimates the option value with the following equation:

V(Si,…Sn) = Mean(Max[0, (K1,…,Kn) – Max (S1,…,Sn), ] whereas i = 1,…,n (89) The basket 1 contains the stocks: Pepsico Inc (PEP), Goldman Sachs Group Inc. (GS), Ford Motor Company (F), and lastly, Pfizer Inc (PFE).

The basket 2 combines the stocks: J.P. Morgan (JPM), Wal-Mart Stores (WMT), Novartis AG (NVS), and lastly, Owens-Illinois (OI).

The basket 3 contains the stocks: Chevron Corporation (CVS), Amazon.com (AMZN), Oracle Corporation (ORCL), and lastly, Owens-Illinois (OI).

The basket 4 combines the Exchange-Traded-Funds (ETFs): SPDR Gold Shares (GLD), Aggregate Bond (AGG), Vanguard REIT (VNQ), and finally, Health care (XLV).

The basket 5 contains the Exchange-Traded-Funds (ETFs): iShares Gold (IAU), Vanguard Total Bond (BND), Dow Jones REIT (RWR), and lastly, Ultra Basic Material (UYM).

The basket 6 combines the Exchange-Traded-Funds (ETFs): Swiss Gold Shares (SGOL), Power Shares Build America Bond (BAB), Global Wind Energy Portfolio (PWND), and lastly, Performance S&P 500 (SPY).



Table I: The American Put option price for basket 1 constituted of stocks

Stock tickerin basket 1

Initial stockPrice

(U.S. $)

Stock strikePrice

(U.S. $)

StockAverageDividend

yield

Stockaveragestandarddeviation

American Put

Option Price*(U.S. $)

American PutOption Price

95% Confidence Interval*(U.S. $)

PEPGSFPFE

87.73160.7416.5429.60

901652030

2.88691%1.12404%0.80227%4.01401%

12.1227%17.6631%23.6908%24.0713%

6.4093 [6.3906;6.4280]

PEPGSFPFE

87.73160.7416.5429.60

951702535

2.88691%1.12404%0.80227%4.01401%

12.1227%17.6631%23.6908%24.0713%

10.1005 [10.0804;10.1207]

PEPGSFPFE

87.73160.7416.5429.60

901652535

2.88691%1.12404%0.80227%4.01401%

12.1227%17.6631%23.6908%24.0713%

8.5152 [8.4965;8.5339]

PEPGSFPFE

87.73160.7416.5429.60

951702030

2.88691%1.12404%0.80227%4.01401%

12.1227%17.6631%23.6908%24.0713%

8.0312 [8.0112;8.0512]

*With interest rate equal to 2% for all stocks with 100 000 simulations and T=1.



Table II: The American Put option price for basket 2 constituted of stocks


Initial stockPrice

(U.S. $)

Stock strikePrice

(U.S. $)


yield

Stockaverage

Standarddeviation

American Put




JPMWMTNVSOI

55.7275.9890.1633.24

60809535

1.85685%2.23683%

0%0%

16.8273%17.0107%18.2236%20.2747%

6.3985 [6.3807; 6.4164]

JPMWMTNVSOI

55.7275.9890.1633.24

658510040

1.85685%2.23683%

0%0%

16.8273%17.0107%18.2236%20.2747%

9.9978 [9.9787; 10.0168]

JPMWMTNVSOI

55.7275.9890.1633.24

608010040

1.85685%2.23683%

0%0%

16.8273%17.0107%18.2236%20.2747%

8.1550 [8.1375; 8.1725]

JPMWMTNVSOI

55.7275.9890.1633.24

65859535

1.85685%2.23683%

0%0%

16.8273%17.0107%18.2236%20.2747%

8.1951 [8.1768; 8.2134]




Table III: The American Put option price for basket 3 constituted of stocks


Initial stockPrice

(U.S. $)

Stock strikePrice

(U.S. $)


yield

Stockaverage

Standarddeviation

American Put




CVXAMZNORCLKO

122.32313.7842.2040.66

1253154545

3.19753%0%

0.70097%2.79541%

18.3925%37.9277%17.1581%15.3489%

16.3265 [16.2759; 16.3772]

CVXAMZNORCLKO

122.32313.7842.2040.66

1253154545

3.19753%0%

0.70097%2.79541%

18.3925%37.9277%17.1581%15.3489%

19.8241 [19.7717; 19.8764]

CVXAMZNORCLKO

122.32313.7842.2040.66

1253155050

3.19753%0%

0.70097%2.79541%

18.3925%37.9277%17.1581%15.3489%

18.3069 [18.2566; 18.3573]

CVXAMZNORCLKO

122.32313.7842.2040.66

1253155050

3.19753%0%

0.70097%2.79541%

18.3925%37.9277%17.1581%15.3489%

17.6683 [17.6163; 17.7203]




Table IV: The American Put option price for basket 4 constituted of ETFs


Initial stockPrice

(U.S. $)

Stock strikePrice

(U.S. $)

StockAverageDividen

dyield

Stockaverage

Standarddeviation

American Put




GLDAGGVNQXLV

120.94109.5774.3559.63

1251107560

0%0%0%0%

16.8616%2.5647%18.2805%24.3017%

5.0976 [5.0812; 5.1140]

GLDAGGVNQXLV

120.94109.5774.3559.63

1301158065

0%0%0%0%

16.8616%2.5647%18.2805%24.3017%

8.5733 [8.5547; 8.5920]

GLDAGGVNQXLV

120.94109.5774.3559.63

1251108065

0%0%0%0%

16.8616%2.5647%18.2805%24.3017%

6.5825 [6.5645; 6.6005]

GLDAGGVNQXLV

120.94109.5774.3559.63

1301157560

0%0%0%0%

16.8616%2.5647%18.2805%24.3017%

7.0373 [7.0210; 7.0535]




Table V: The American Put option price for basket 5 constituted of ETFs


Initial stockPrice

(U.S. $)

Stock strikePrice

(U.S. $)


yield

Stockaverage

Standarddeviation

American Put




IAUBNDRWRUYM

12.1782.4182.5154.27

15858555

0%0%0%0%

16.9154%2.4881%18.4913%23.2479%

4.2421 [4.2324; 4.2518]

IAUBNDRWRUYM

12.1782.4182.5154.27

20909060

0%0%0%0%

16.9154%2.4881%18.4913%23.2479%

8.3019 [8.2909; 8.3129]

IAUBNDRWRUYM

12.1782.4182.5154.27

15859060

0%0%0%0%

16.9154%2.4881%18.4913%23.2479%

4.2305 [4.2211; 4.2400]

IAUBNDRWRUYM

12.1782.4182.5154.27

20908555

0%0%0%0%

16.9154%2.4881%18.4913%23.2479%

6.7298 [6.7204; 6.7393]




Table VI: The American Put option price for basket 6 constituted of ETFs


Initial stockPrice

(U.S. $)

Stock strikePrice

(U.S. $)


yield

Stockaverage

Standarddeviation

American Put




SGOLBABPWNDSPY

123.3029.526.20

192.37

1253010195

0%0%0%0%

15.1147%6.9565%35.9671%18.7559%

6.5020 [6.4828; 6.5212]

SGOLBABPWNDSPY

123.3029.526.20

192.37

1303515200

0%0%0%0%

15.1147%6.9565%35.9671%18.7559%

10.2915 [10.2719; 10.3110]

SGOLBABPWNDSPY

123.3029.526.20

192.37

1253015200

0%0%0%0%

15.1147%6.9565%35.9671%18.7559%

8.4194 [8.3995; 8.4392]

SGOLBABPWNDSPY

123.3029.526.20

192.37

1303510195

0%0%0%0%

15.1147%6.9565%35.9671%18.7559%

8.3527 [8.3341; 8.3713]




3.2 Examples for Asian option on derivatives

The Asian option on future contracts is illustrated with the commodities: Crude Oil (CL1), Silver (SIN4), Copper (HGN4), and lastly, Gold (GCQ4).

Table VII: The Asian option price on future contracts

FutureContractticker

Initial Price

(U.S. $)

StrikePrice

(U.S. $)

Commodityaverage

Standarddeviation

Number of fixeddates

Asian Option Price*

(U.S. $)

Asian Option Price


CL1 103.58 103.58 12.7108% 1 5.1467 [5.1307; 5.1626]SIN4 19.014 19.014 27.0509% 1 2.0037 [1.9969; 2.0105]

HGN4 314.45 314.45 9.1630% 1 11.2517 [11.2176; 11.2857]GCQ4 1257.10 1257.10 12.8505% 1 63.0559 [62.8609; 63.2510]CL1 103.58 103.58 12.7108% 10 3.1988 [3.1891;3.2084]SIN4 19.014 19.014 27.0509% 10 1.2448 [1.2408;1.2488]

HGN4 314.45 314.45 9.1630% 10 7.0068 [6.9860;7.0277]GCQ4 1257.10 1257.10 12.8505% 10 39.1222 [39.0036;39.2407]

*With interest rate equal to 2% for all futures with 1 000 000 simulations and T=1.



3.3 Examples for Asian option on correlated futures

The Asian option on correlated future contracts is performed with Crude Oil (CL1 and CLN4).

Table VIII: The Asian option price on correlated future contracts

Basket of2 futureContracts**

Initial Price

(U.S. $)

BasketstrikePrice

(U.S. $)

Commodityaverage

Standarddeviation

Number of fixeddates

Asian Option Price*

(U.S. $)

Asian Option Price


CL1CLN4

103.58103.58

207.16 12.7108%5.9689%

1 7.3820 [7.3595 ; 7.4046]

CL1CLN4

103.58103.58

207.16 12.7108%5.9689%

10 4.5793 [4.5656; 4.5931]

CL1CLN4

103.58103.58

207.16 12.7108%5.9689%

50 4.3540 [4.3128; 4.3951]

CL1CLN4

103.58103.58

207.16 12.7108%5.9689%

100 4.3230 [4.2821; 4.3640]

*For 1 and 10 fixed dates: Interest rate equal to 2% for all futures with 1 000 000 simulations. For 50 and 100 fixed dates: Interest rate equal to 2% for all futures with 100 000 simulations.**The correlation between the future contracts CL1 and CLN4 is equal to 0.86456 calculated with the covariance between the two commodities and the daily average standard deviation for each commodity.



3.4 Examples for Asian Call option on commodities

The Asian Call option is illustrated with the commodities: Crude Oil (CL1 and CLN4), Silver (SIN4), and finally, Copper (HGN4).

Table IX: The Asian Call option price with antithetic variate on commodities

Assetticker

Initial Price

(U.S. $)

StrikePrice

(U.S. $)

Commodityaverage

Standarddeviation

Asian Calloption price

with antitheticvariate*(U.S. $)

Asian Option Price


CL1 103.58 100 12.7108% 5.6774 [5.6734; 5.6814]CLN4 103.58 100 5.9689% 4.6708 [4.6699; 4.6718]HGN4 314.45 310 9.1630% 10.7623 [10.7532; 10.7715]SIN4 19.014 17 27.0509% 2.4763 [2.4746; 2.4780]CL1 103.58 95 12.7108% 9.7086 [9.7062; 9.7111]

CLN4 103.58 95 5.9689% 9.4341 [9.4339; 9.4344]HGN4 314.45 305 9.1630% 14.3011 [14.2934; 14.3087]SIN4 19.014 15 27.0509% 4.1689 [4.1678;4.1700]

*With interest rate equal to 2% for all assets with 1 000 000 simulations and T=1.



3.5 Examples for Down-&-Out Put Barrier option on a stock

The Down-&-Out Put Barrier option is performed with the Home Depot Inc (HD) stock.

Table X: The Down-&-OutPut Barrier option price on a unitary stock

Stockticker

Initial stockPrice(U.S.

$)

StockStrikePrice(U.S.

$)

BarrierCross (Sb)

Price(U.S. $)

Stockaverage

Standarddeviation

Down-&-OutPut Barrier

option price*(U.S. $)

Down-&-Out Put Barrier option price


HD 55.88 55 45 13.1817% 1.2455 [1.2424; 1.2486]HD 55.88 55 40 13.1817% 1.8949 [1.8906; 1.8991]HD 55.88 55 35 13.1817% 2.0028 [1.9982; 2.0073]HD 55.88 55 30 13.1817% 2.0101 [2.0055; 2.0146]

*With interest rate equal to 2% for the stock with 1 000 000 simulations, T=1 and 100 steps.



ConclusionThe LSM technique and its implementation to value the early exercise of American Put options enables to attain a high-dimensional flexible framework compared to traditional option pricing procedures. The proposed American Put option pricing model is well fitted to estimate bundled equity assets. The barrier option pricing arrangement is well performed with the Monte Carlo simulation process complemented to an antithetic variate. The Barrier option is well suited to value stocks. The Asian option portrays a derivative that is well adapted to estimate the energy sector and commodities due its averaging feature and decreased risk compared to American as well as European options. The examples with true market data umbrage the usefulness of adequately matching accurate derivative pricing techniques with asset classes or categories.

In conclusion, modeling derivatives in a globalized financial environment whereas market frontiers are neither perfectly integrated nor completely segmented is performed with robust and sound mathematical frameworks.



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