Open access peer-reviewed chapter

Pricing Basket Options by Polynomial Approximations

By Pablo Olivares

Submitted: July 24th 2018Reviewed: November 4th 2018Published: January 21st 2019

DOI: 10.5772/intechopen.82383

Downloaded: 176

Abstract

In this paper, we use polynomial approximations in terms of Taylor, Chebyshev, and cubic splines to compute the price of basket options. The paper extends the use of a similar pricing technique applied under a multivariate Black-Scholes model to a framework where the dynamic of the underlying assets is described by dependent exponential Levy processes generated by a combination of Brownian motions and compound Poisson processes. This model captures some empirical features of the asset dynamics such as common and idiosyncratic random jumps. The approach is implemented in the context of spread options and a multivariate Merton model, i.e., a jump diffusion with Gaussian jumps. Our findings show that, within the range of parameters analyzed, polynomial approximations are comparable in accuracy to a standard Monte Carlo approach with a considerable reduction in computational effort. Among the three expansions, cubic splines show the best performance.

Keywords

  • Taylor approximations
  • Chebyshev polynomials
  • cubic splines
  • basket options
  • spread options
  • jump-diffusion model

1. Introduction

We study the pricing of basket contracts under a multivariate jump-diffusion process. The paper extends the use of a similar pricing technique applied under a multivariate Black-Scholes model, see [1], to a framework where the dynamic of the underlying assets is described by dependent exponential Levy processes generated by a combination of Brownian motions and compound Poisson processes. This model captures some empirical features of the asset dynamics such as common and idiosyncratic random jumps. The dependence between assets is reflected in both the covariance structure of the Brownian motion and the joint probability law of the common jump sizes.

For such class of models, no pricing closed-form formula is available. In single-asset contracts, well-established numerical methods have proven to be effective, but their extensions to several dimensions reveal important instabilities and a costly computational effort. Our paper introduces a novel approach based on polynomial approximations of the conditional price. It is, in the framework considered, less time demanding than a standard Monte Carlo approach to achieve similar results. Moreover, the use of Chebyshev polynomials and cubic splines improves the convergence over previous attempts based on Taylor expansions.

We consider a pricing methodology consisting in a two-step procedure. First, conditioning on d1out of the total number of dassets, we find the price of a payoff based on a single asset with a more complex conditional distribution.

Secondly, we consider some expansions of the conditional price, given either in terms of Taylor, Chebyshev, or cubic spline polynomials, allowing to write the corresponding price as a linear combination of mixed exponential-power moments.

This approach is implemented in the context of spread options and a multivariate Merton model, that is, a jump diffusion with Gaussian jumps. Our findings show that, within the range of parameters analyzed, polynomial approximations are comparable in accuracy to a standard Monte Carlo approach with a considerable reduction in computational effort. Among the three expansions, cubic splines show the best performance.

The use of a Taylor expansion to pricing has been considered in the pioneering work of [2] for a vanilla European option and in [3, 4] for spread contracts under a bivariate Black-Scholes model. See also [5]. A Chebyshev expansion has been recently considered in [6]. Applications under a multivariate jump-diffusion model have been less explored. Our paper intends to fill this gap.

Although a comparison with alternative approaches is beyond the scope of this paper, it is worth noticing the existence of pricing methods based on Fourier or Hilbert transforms. For example, for spread contracts under a different class of Levy processes, a Fast Fourier transform method can be found in [7]. See also [8] for expansions in terms of Fourier series and [9] for Hilbert transforms.

The organization of the paper is as follows: in Section 2, we introduce the model and obtain the pricing expressions for basket contracts under the approximations. In Section 3, we specialize the three expansions in the case of spreads contracts. In Section 4, we discuss the implementation of the methods and present our numerical findings. Finally in Section 5, we present conclusions. Proofs are deferred to the appendix.

2. Pricing under jump-diffusion models

Let ΩAFtt0Pbe a filtered probability space. We define the filtration FXtσXs0stas the σ-algebra generated by the random variables Xs0stcompleted in the usual way. Denote by Qan equivalent martingale measure (EMM), respectively, by EQ, φX, and MXthe expectation, characteristic, and moment-generating functions of a random variable Xunder Q. The function fXis its probability density function.

By rwe denote the (constant) interest rate, ABis the componentwise product between matrices Aand B, and Arepresents the transpose of matrix A=aij1i,jd, while diagAis a vector with components aii1id. The symbol δijis the usual Kronecker’s number. The vector Y˜is created from the vector Yafter eliminating the first component. For a function fwith domain in Rdand a vector L=l1l2ldwith lkN, the symbol DLfrepresents the mixed partial derivative of the function fdifferentiated lktimes w.r.t. the k-th variable.

For vectors v=v1v2vdand n=n1n2nd, we set v!=k=1dvkand νn=k=1dvknk.

We introduce the following convenient notations. For a 1×n+1vector Va, bR, and nN

binnVab=m=0nnmVambnmPVa=1Va1Vann

Also, for a differentiable function f, we set the vector DVf=fD1fDnf.

The d-dimensional process of spot prices is denoted by Stt0, while Ytt0is the corresponding log-price process. They are related by

Stj=S0jexpYtj,j=1,2,,dE1

We analyze European basket options whose payoff at maturity T, for a strike price K, are given by

hST=j=1dwjSTjK+

where wj1jdare some deterministic weights and x+=maxx0.

Furthermore, for the log-prices, we assume a multidimensional jump-diffusion dynamics under Qgiven by

dYt=μdt+Σ12dBt+dZtE2

where Btt0is a multivariate Brownian motion with independent components and μ=r12diagΣm. The matrix Σ=σjlj,lis symmetric, positive definite, while Σ12is such that Σ12Σ12=Σ. The value mis the compensator of a compound Poisson process m=logφZ1i.

We define two sequences of independent and identically distributed 1×d-dimensional random vectors XkkNand X0,kkN. The components of the random vectors in the first sequence are independent.

The process Ztt0is a d-variate compound Poisson process, independent of Btt0such that

Ztj=k=1NtjXkj+k=1Nt0X0,kj,j=1,,d

where Ntt0=Nt0Nt1Ntdt0is a vector of independent Poisson processes with respective intensities λj.

The processes Ntjt0and Nt0t0correspond, respectively, to idiosyncratic and common jumps of the j-th underlying asset on the interval 0t. Their jump sizes are Xkjand X0,kj.

For the sake of concreteness, we assume Gaussian jumps, i.e., we assume for any kNthat XkNμJDJ, where DJis a diagonal matrix with components DJjl=δjlσJj2and X0,kNμ0,JΣ0,J, with Σ0,Ja matrix of components Σ0,Jjl=σ0j,l. The compensator across each dimension takes the form

mj=λjexpμJj+12σJj21+λ0expμ0,Jj+12σ0jj21,j=1,2,d

Let CJDdenote the price of a European basket option with payoff hSTunder the model given by Eqs. (1) and (2).

First, we write the price of the basket contract in terms of its conditional price when the number of jumps and d1underlying assets are fixed. Results are given in Theorem 1 below.

Notice that, for any kNd+1

pk=PNT=k=expj=0dλjTj=0dλjkjTj=0dkjk!E3

We also introduce the vector μ¯kwith components

μ¯jk=μjT+kjμJj+k0μ0,Jjj=1,2,,d.

Theorem 1. Let CJDbe the price of a European basket contract with maturity T, strike price K, and payoff hYT, under a model given by Eqs. (1) and (2). See proof in Appendix A.2.

In addition assume XkNμJDJand X0,kNμ0,JΣ0,Jfor any kN, where DJis a d×ddiagonal matrix with components DJjl=δjlσJj2and ΣJ0is also a d×dmatrix with components Σ0,Jjl=σ0j,l.

Then, we have

CJD=kNd+1CkpkE4

where for any kNd+1

Ck:=w1exp12σ2kTEQexpμY˜TNTC(Y˜TNT)/NT=kE5
Cyk=erTEQ[S01exp(r12σ2NTT+ σNTTZ))K(Y˜TNT)+/NT=k,Y˜T=y]E6

with Za standard normal random variable independent of NTand Y˜T.

Also

Kyk=expr12σ2kTμykKw1j=2dwjw1S0jexpyj,foryRd1μyk=μ¯1k+Σ1Y˜kΣY˜1kyμ˜kσk=1Tσ11kΣ1Y˜kΣY˜1kΣ1Y˜k

Here σjlkis the jlcomponent of the matrix:

ΣYk=ΣT+DJDN+k0Σ0,JandDNjl=δjlNTjΣ1Y˜k=σ12kσ13kσ1,d1k

Remark 2. Notice that when Kykis nonnegative, Cykis the well-known Black-Scholes price of a call option with maturity at T>0, volatility σk, spot price S01, and strike price Kyk. A sufficient condition for Kykto be positive is w10while wj0,2jd. It is the case of spreads and crack spreads. When Kykis negative, it does not have the meaning of a strike price anymore.

Remark 3. The values μykand σkare, respectively, the mean and variance of the first asset after conditioning on a value yof the remaining assets and the certain number of jumps k.

For any fixed kNd+1, we approximate the conditional price Cykon the variable yby a suitable polynomial. In particular we consider Taylor, Chebyshev polynomials and cubic splines.

Approximations based on the three expansions are discussed below.

(i) An order nTaylor approximation of Cykaround yRd1is described by

CTyykn=l=0nLRlDLCykL!yyLE7

with L=l1l2ld1, where the second sum is taken on the set

Rl=LNd1/l1+l2++ld1=l0ljl.

Notice the existence of the derivatives of any order in the functions Kyand Cyk.

(ii) An approximation based on Chebyshev polynomials is given as follows:

In a region DRd1, we consider an expansion of order n=n1n2nd1of the function Cykas

CChykn=12ĉ0k1Dy+lBnĉlkTlDy1Dy=12ĉ0k1Dy+lBnmClĉlkbm,lyl2m1DyE8

where the sums are taken over the sets

Bn=lNd1/0lnjj=12d1.Cl=mNd1/0mjlj2j=12d1

Here TlDlBnis a family of d1-dimensional Chebyshev polynomials with degrees lBndefined in the region D, while the quantities ĉlkare suitable approximations of the corresponding Chebyshev coefficients clk, computed using the trapezoidal rule.

Notice that, by the orthogonality of the polynomials, the coefficients in the expansion are clk=<C,TlD>W, where <f,g>Wis the scalar product of functions fand g, conveniently weighted by a function W. See, for example, [10] for a general account on Chebyshev polynomials.

For convenience, we write the Chebyshev polynomials in terms of powers of their variables, where bm,lare the coefficients of this expansion.

In particular, for a rectangular region D=abd1and valued vectors a=a1a2ad1and b=b1b2bd1, we write

TDyTla,by=Tl1,11+2yaba

Hence, for d=2

CChyk=12ĉ0k1Dy+l=1n m=0l2ba2mlĉlkbm,l2ya+bl2m1DyE9

See, for example, [11] for specific expressions of bm,lin one dimension.

(iii) Approximation by cubic splines.

On a rectangular region D=abd1, we consider an approximation based on cubic splines given by

Csplyk=j=1NlB3αj,lkybj1l1DjyE10

where bjis some point on a (d − 1)-dimensional grid b0b1bNwith N+1points in D.

The local coefficients αj,lkare determined by imposing the conditions Cyjk=zjk,j,k=1,,N+1. The family of sets Djj=01Nis a partition of D. Notice that the coefficients αj,lkdepend on the particular rectangle in the grid. See [12] for a general account on multivariate splines.

In the case of d=2, splines used to approximate the conditional price become one-dimensional polynomials. Additional conditions on the derivatives to smoothen these curves are imposed, namely, DlCyjk=D+lCyjk,j=1,2,,N, l=1,2, where DlCyjkand D+lCyjkare, respectively, the derivatives from the left and the right of the function Cykat point y=yj. Moreover, for end points in the grid, D2y0k=D2yNk=0.

In order to approximate the prices, we replace the function Cykby its respective expansions. The conditional prices on the event NT=kare estimated by approximating the corresponding conditional expected values. Substituting the approximations of conditional prices into Eq. (4), we obtain, after truncation, estimates of the price of the basket contract, under the jump-diffusion model described by Eqs. (1) and (2). They are denoted, respectively, by CJDTy, CJDCh, and CJDspl.

Notice that these estimates depend on the mixing exponential-power moments of the log-prices. The latter can be computed from its conditional moment-generating function under the selected EMM. Hence, for a vector Xand a Borel set D, we define

MXuk=EQexpuX/NT=kMXukD=EQexpuX1DX/NT=k

In particular when D=abd1, we write MXukD=MXukab.

Concrete expressions of these approximations under a two-dimensional Gaussian model are shown in Theorem 4.

As it is well known, the conditional mixed exponential-power moments of a random vector Xare related to the partial derivatives of the corresponding moment-generating function Indeed, for νNd1, we have

DνMXukD=EQexpuXXν1DX/NT=k,uRd1

In order to simplify notations, we introduce the following quantities:

A1k=12σ2kT+μ¯1kΣ1Y˜kΣY˜1kμ˜kA2k=A1k+Σ1Y˜kΣY˜1kyA3k=A1k+12a+bΣ1Y˜kΣY˜1k
and the set
NMd+1=k=k0k1kd/kj=01Mjj=01d

Theorem 4. Let CJDbe the price of a European basket contract with maturity T, strike price K, and payoff hYTunder a model given by Eqs. (1) and (2). In addition assume XkNμJDJand X0,kNμ0,JΣ0,Jfor any kN, where DJis a d×ddiagonal matrix with components DJjl=δjlσJj2. Let ΣJ0be a d×dmatrix with components Σ0,Jjl=σ0j,l.

Then, its n-th-order approximation around yRd1in terms of Taylor polynomials is given by

CJDTy=w1kNMd+1 l=0n LRlexpA2kDLCykL!DLMY˜TyΣ1Y˜kΣY˜1kkpkE11

for some truncation vector MNd+1.

The n-th-order Chebyshev approximation on a region D=abd1is

CJDCh=w12kNMd+1ĉ0kK1(abk)+ w1kNMd+1 lBn mClexpA3kĉlkbm,lba2mlDl2mMV˜T(12Σ1Y˜kΣY˜1kkbaba)pkE12

where V˜T=2Y˜Ta+band K1abn=expA1kMY˜TΣ1Y˜kΣY˜1kkab.

The n-th-order approximation by cubic splines on the region D=abd1is given by

CJDspl=w1kNMd+1exp12σ2kTj=1NlB3expΣ1Y˜kΣY˜1kbj1αj,lkDmMY˜bj1Σ1Y˜kΣY˜1kkDjpkE13

Remark 5. The point yaround which the Taylor expansion is taken, in general, depends on k.

3. Approximating the price of spread contracts

Spread contracts are the most common basket derivatives. In this case the payoff is written as hST=ST1ST2K+.

Hence for d=2, conditionally on YT2=yNT=k, the log-prices of the first asset are normally distributed, i.e., YT1Nμykσ2k, with

μyk=μ¯1k+σ11kσ22kρ¯kyμ¯2kσ2k=1Tσ11kσ122kσ22k=1T1ρ¯k2σ11k

where

ρ¯k=σ12kσ11kσ22k

is the conditional correlation coefficient between the two assets.

A result about the derivatives of the moment-generating function of a constrained standard normal random variable Zon the interval bis needed. To this end we have

DmMZσ11kρ¯kkb=exp12σ11kρ¯k2binmμVbσ11kρ¯kσ11kρ¯kE14

where μmab=μmbμma=abzmfZzdzis the m-th moment of a standard normal random variable constrained to the interval aband μVabis a vector with components μjab,j=0,1,,m.

By integration by parts, the later can be calculated recursively as

μ0ab=NbNaμ1ab=fZafZbμmab=m1μm2ab+am1fZabm1fZb),m2

For a Taylor expansion, derivatives of the moment-generating function and constrained moment-generating function for the second component of the log-prices are computed as follows:

DlMYT2yσ11kσ22kρ¯kk=expσ11kσ22kρ¯kμ¯2kybinlPVσ22k1DVMZσ11kρ¯kμ¯2ky

Now, combining the expressions above with Eq. (11), we have

CJDTy=w1kNM3exp12σkT+μ¯1kl=0nlmDlCykl!μ¯2kylmbinlPVσ22k1DVMZσ11kρ¯kμ¯2kypkE15

Next, we obtain the Taylor approximations up to third order. By elementary calculation we can compute the derivatives of the function Cykwith respect to y.

First, notice that, from the Black-Scholes pricing formula:

Cyk=S01N(d1KykKykerTN(d2Kyk

where

d1Kyk=logS01Kyk+r+σk2TσkTd2Kyk=d1KykσkT

and N.is the cumulative distribution function of a standard normal distribution.

Hence

D1Cyk=1σkTT1ykAyk

where

T1yk=D1KykKykAyk=S01fZd1Kyk+σkTerTKykNd2KykerTKykfZd2Kyk

Higher derivatives can be calculated recursively.

DnCyk=1σkTl=0n1n1lDlT1ykDnl1Ayk

Concrete expressions for second- and third-order derivatives are shown in the appendix.

Regarding the approximation based on Chebyshev polynomials, we first compute the moment-generating function of the random variables YT2and VT2constrained to the interval ab. To this end we denote

b˜=bμ¯2kσ22k,a˜=aμ¯2kσ22kE16

Notice that, taking into account Eq. (14),

DmMZσ11ρ¯ka˜b˜=exp12σ11kρ¯k2binmμm1a˜σ11kρ¯kb˜σ11kρ¯kσ11kρ¯kE17

Moreover

MYT2σ11kσ22kρ¯kkb=expσ11kσ22kρ¯kμ¯2k+12σ11kρ¯k2Nb˜σ11kρ¯k

Hence

DνMVT212σ11kσ22kρ¯kkbaba=exp12σ11kσ22kρ¯k2μ¯2kabGνk

where

Gνk=binνMZσ11kρ¯kka˜b˜PV2σ2212k12μ¯2kab

Then, combining Eq. (12) with the results above, we get

CChkn=w12ĉ0kK1abk+ w1exp12σkT+μ¯1kl=1nm=0l2ĉlkbm,lKablmGl2mk

Finally, the n-th-order Chebyshev approximation is given by

CJDCh=kNM3CChknpk

Similarly for a cubic spline approximation, we specialize Eq. (13) with D=ab,Dj=bj1bj,b0=a,bN+1=b. Therefore, we have

CJDspl=w1kNMd+1exp12σkT+μ¯1kj=1Nl=03αj,lkσ22kl2bin(lDVMZ(σ11kρ¯kb˜j1b˜j)b˜j1)pkE18

where b˜jis defined as b˜in Eq. (16) but replacing bby bj.

4. Numerical results

We implement the results from the previous section to price spread contracts and show that the approximations considered above produce accurate price values when compared with a standard Monte Carlo approach, at a lesser computational effort.

To this end we consider the following benchmark set of parameters:

The contract specifications consist a strike price of K=$1, maturity T=1year, spot prices S01=$100, S02=$96, and a fix interest rate of 3%.

Volatilities corresponding to the diffusion part of both assets are σ1=10%and σ2=30%, while the correlation coefficient between the two Brownian noises is ρ=0.3. Regarding the jump part, we consider an average intensity of the common jumps equal to λ0=3jumps per year and idiosyncratic intensities λ1=λ2=2jumps per year for the respective assets, while jump sizes have means equal to zero; volatilities of common jump sizes are σ0,1=1%,σ0,2=5%, with a linear correlation ρJ=0.5. Volatilities of the idiosyncratic jumps are taken as σJ,1=10%and σJ,2=20%.

Although these values are somehow arbitrary, they have been selected to produce reasonable asset prices in connection with contracts based on crude oil prices. It is worth noting that there is not a general agreement about the range of the parameters in a jump-diffusion model. Indeed they may depend on the market into consideration.

In Table 1 prices of spread contracts under different methods are shown. Prices are obtained using Taylor and cubic splines approximations and contrasted with a Monte Carlo approach. For the latter we carry 107repetitions to achieve stable results, with a relative average error of 0.1%. In addition, 95% Monte Carlo confidence intervals and running times are provided. Implementation is done on a Surface Pro 4 i7 computer, using MATLAB language.

MCTaylor (f.o)Taylor (s.o.)Spl.
Price14.778410.298014.2906814.8842
Interval14.7683,14.7885
Run time624.3121.688061.6880654.1720

Table 1.

Prices obtained using the benchmark parameter set and Monte Carlo, first- and second-order Taylor, and cubic spline approximations.

In row three the average computer time (in seconds) for different pricing methods is shown.

The efficiency of the Monte Carlo method can be improved by considering only the simulation of a single asset with the corresponding conditional probability and then computing the discounted average of the conditional Black-Scholes price. It reduces the computational time by half, still considerably higher than those based on polynomial expansions. Chebyshev polynomial approximation is discussed in [1].

The expansions also require repetitive evaluations of conditional prices, which turn out to be given by simple Black-Scholes closed formulas.

For a Taylor approach of order n, evaluations in the order of nM3are needed, where Mis the maximum truncation level in the number of jumps. In a Chebyshev approach of the same order about n2NM3, evaluations of the conditional price should be performed, when a grid of Npoints is used in a trapezoidal approximation of the corresponding integrals. In a cubic splines approximation 3NM3. Here Nis also the number of points in the grid where the polynomial coefficients are adjusted.

For a theoretical analysis of the error using Taylor and Chebyshev expansions, although in different contexts, see [13] for Taylor and [6] for Chebyshev cases.

In Figure 1a, a graph of conditional prices in function of log-price values of the first asset (blue line) with average number of jumps equal to k0=3and k1=k2=2is shown. The remaining three curves represent the first-order (green), second-order (red), and third-order (magenta) Taylor polynomials around the average value y=EQYT2. In Figure 1b, conditional prices and its cubic spline approximation are shown. At this scale both are indistinguishable. Notice that, although the Taylor approximation is excellent in a neighborhood of the expansion point, there are significant deviations for values far from the mean. These deviations, under the assumption of normality of the jump sizes, result to be infrequent; therefore, they do not impact the global error, but might be significant when other probability distributions, in particular heavy-tailed ones, or even normal jumps with higher volatilities, are taken into account. Instead of local approximations, as the case of Taylor polynomial expansion, uniform approximations on a given interval may reduce the error. Expansions based on orthogonal basis, e.g., Chebyshev or varying coefficients as in the case of cubic splines, are suggested. Notice that the function Cykis continuous in yfor any value of k; therefore, Weierstrass’ theorem of uniform convergence applies. Curiously, the convergence of Bernstein polynomials, applied in the original proof of the theorem, is remarkably slow.

Figure 1.

(a) Conditional price (blue curve) as function of log-price values and its Taylor approximations up to third order around the average. (b) Conditional price vs. its cubic spline approximation.

Figure 2 shows the differences between the conditional price and the cubic spline for different values of the underlying price. Truncation values were selected as a=1and b=1. Generally speaking the choice of these values depends on the probability distribution of the underlying asset. In practice it requires an exploratory study of the available data. On the other hand, the larger the interval, the more accurate is the approximation but also is the computational effort. Moreover, we have found that the results are sensible to this choice, though rather robust to the number of splines and the truncation values.

Figure 2.

Curve representing the difference between conditional price and cubic spline approximation for the benchmark parameters.

Truncation values for the number of jumps, denoted in the paper by M0,M1and M2, should cover most of the jump probability distribution pkkN3. An efficient way of choosing these values consists in starting to evaluate the sum at a point close to where the maximum value of the pk’s is attained, namely,

k=k0k1k2=λ0T1+λ1T1+λ2T1+

where x+represents the maximum of the integer part of xand zero, then adding expression (18) for points j=j0j1j2over the set

NMk=k0+j0k1+j1k2+j2N3/klMl2+jlkl+Ml2l=0,1,2

until kpkδ, where δis a predetermined value close to one.

In Figure 3 we show the probability distribution pkkN3, for k2=5varying k0and k1. We observe probabilities become negligible after certain values of k0k1with a peak around the center of the distribution. For the benchmark parameter set truncation values M0=15,M1=10,M2=10capture 99.67% of the probability mass.

Figure 3.

Probabilities p k to observe k = k 0 k 1 jumps when k 2 = 5 . Truncation values M 0 = 15 , M 1 = 10 , M 2 = 10 capture 99.67% of the probability distribution in the number of jumps.

5. Conclusions and future developments

The paper establishes a methodology over the use of polynomial approximations based on Taylor, Chebyshev, and cubic splines to the price of basket contracts. This approach produces accurate results at a lesser computational effort than a standard Monte Carlo technique. The claim is supported by numerical evidence in the case of spread options, under a bivariate jump-diffusion model with a complex Gaussian jump structure that allows to capture the dependence between assets.

The study needs to be extended to different parameter values to corroborate the results in a wider scope. Moreover, optimal choices in the numerical implementation, for example, the order of the polynomials, the number of points in the grid, and truncation levels, require a further study.

Sensitivities with respect to the parameters in the model and the contract, i.e., maturity, strike, interest rate, correlation, etc., can be easily calculated with a straightforward adaptation of the current method. It is enough to approximate the corresponding derivatives instead.

A natural question is how to adapt our method when a non-Gaussian joint distribution of the jump sizes is considered. In this setting, the conditional probability distribution is generally unknown; nonetheless, the use of a copula approach to capture the dependence may provide some insight.

Acknowledgments

This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.

A.1 Taylor implementation up to third order

After computing the second and third derivatives of Cykand the corresponding derivatives of the moment-generating function of Z, we can compute Taylor approximations up to third order around the point yas

CJDTy1=w1kNM3exp12σkT+μ¯1k+12σ11kρ¯k2Cyk+D1C(yk)μ¯2ky+σ11kσ22kρ¯kpkCJDTy2=CJDTy1+w1kNM3exp12σkT+μ¯1k+12σ11kρ¯k2D2Cyk12μ¯2ky2+μ¯2kyσ11kσ22kρ¯k+ 12σ22k1+σ11kρ¯k2pkCJDTy3=CJDTy2+ w1kNM3exp12σkT+μ¯1k+12σ11kρ¯k2D3Cyk16μ¯2ky3+12μ¯2ky2σ11kσ22kρ¯k+12μ¯2kyσ22k1+σ11kρ¯k2+16σ22k32σ11kρ¯kσ11kρ¯k2+3pk

A.2 Proof of Theorem 1

From Eq. (2) written in its integral form

YT=μT+Σ12BT+ZT

it is easy to see that

EQYT/NT=μT+N˜TμJ+NT0μ0,J=μ¯jN˜TΣYNT:=VarYT/NT=Σ12VarBTΣ12+VarZT/NT=ΣT+VarZT/NT

From the expression above, in the case of jl, we have

covZTjZTl/NT=EQk=1NT0k=1NT0X0,kjEQX0,kjX˜klEQX0,kl/NT=NT0covX0,kjX0,kl/NT=NT0σ0j,l

Similarly, for j=l

covZTjZTl/NT=NTjσJj2+NT0σ0j,j

Then, conditionally on NT, we have

YTNμT+NTμJ+NT0μ0,JΣYNTE19

Hence, the price is expressed as

CJD=erTkNd+1EQhST/NT=kpk=kNd+1CkpkE20

where Ck=erTEQhST/NT=k.

On the other hand, conditioning on NT=kY˜T:

CkerTEQhST/NT=k=erTEQEQhST/NT=kY˜T/NT=k=w1erTEQEQS01expYT1Kw1j=2dwjw1S0jexpYTj+/NT=kY˜T/NT=k=w1erTEQEQS01expYT1K1Y˜T+/NT=kY˜T/NT=kE21

where K1y=Kw1j=2dwjw1S0jeyj.

Taking into account Eq. (19), again conditioning on the events Y˜T=yand NT=k, it is well known that YT1has a univariate normal distribution with mean and variance given, respectively, by μykand σ2kT. See, for example, [14].

Hence, we can write, on the set Y˜=yNT=k:

YT1=μyk+σkTZ

Then, replacing the expression above in Eq. (21), we have

Ck=w1erTEQEQS01expμY˜TNT+σNTTZK1Y˜T+/Y˜TNT=k/NT=k=w1erTEQexp(r12σkT+μ(Y˜Tk))EQS01expr12σ2NTT+σNTTZexp(r12σ2NTTμ(Y˜TNT))K1Y˜T+/Y˜TNT/NT=k=w1exp12σ2kTEQexpμY˜TNTC(Y˜TNT)/NT=kE22

Eq. (4) easily follows after replacing Eq. (22) into Eq. (20).

A.3 Proof of Theorem 4

In Eq. (6) we replace the function Cykby its Taylor expansion given in Eq. (7).

Then, the Taylor approximation of Ckis

CTyk=w1exp12σ2kTEQexpμY˜TNTCT(Y˜TyNT)/NT=k=w1l=0nLRlDLCykL!exp12σ2kT+μ¯1kΣ1Y˜kΣY˜1kμ˜kEQexpΣ1Y˜NTΣY˜1NTY˜TY˜TyL/NT=k=w1expA2kl=0nRlDLCykL!DLMY˜TyΣ1Y˜kΣY˜1kk

Eq. (11) follows after replacing Ckin Eq. (20) by the expression above and truncating at point M.

After replacing Eq. (9) into Eq. (22), we have

CChk=w12ĉ0kexpA1kEQexpΣ1Y˜NTΣY˜1NTY˜T1DY˜T/NT=k+w1expA1klBnĉlkEQexpΣ1Y˜NTΣY˜1NTY˜TTlDY˜T/NT=k=w12ĉ0kK1kab+w1expA1klBnmClĉlkbm,lEQexpΣ1Y˜NTΣY˜1NTY˜T1+2Y˜Tabal2m1DY˜T

Eq. (12) easily follows.

Finally, by similar arguments,

CJDsplk=w1exp12σ2kTj=1NlB3αj,lkEQexpΣ1Y˜NTΣY˜1NTY˜TY˜Tcjl1DjY˜T/NT=k
=w1exp12σ2kTj=1NlB3expΣ1Y˜kΣY˜1kcjαj,lkDlMY˜cjΣ1Y˜kΣY˜1kkDj

from which (13) follows.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Pablo Olivares (January 21st 2019). Pricing Basket Options by Polynomial Approximations, Polynomials - Theory and Application, Cheon Seoung Ryoo, IntechOpen, DOI: 10.5772/intechopen.82383. Available from:

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