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

## 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

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

By

For vectors

We introduce the following convenient notations. For a

Also, for a differentiable function

The

We analyze European basket options whose payoff at maturity

where

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

where

We define two sequences of independent and identically distributed

The process

where

The processes

For the sake of concreteness, we assume Gaussian jumps, i.e., we assume for any

Let

First, we write the price of the basket contract in terms of its conditional price when the number of jumps and

Notice that, for any

We also introduce the vector

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

*In addition assume* *and* *for any* *, where* *is a* *diagonal matrix with components* *and* *is also a* *matrix with components*

*Then, we have*

*where for any*

*with* *a standard normal random variable independent of* *and*

*Also*

*Here* *is the* *component of the matrix:*

**Remark 2.** *Notice that when* *is nonnegative,* *is the well-known Black-Scholes price of a call option with maturity at* *, volatility* *, spot price* *, and strike price* *. A sufficient condition for* *to be positive is* *while* *. It is the case of spreads and crack spreads. When* *is negative, it does not have the meaning of a strike price anymore.*

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

For any fixed

Approximations based on the three expansions are discussed below.

(i) An order

with

Notice the existence of the derivatives of any order in the functions

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

In a region

where the sums are taken over the sets

Here

Notice that, by the orthogonality of the polynomials, the coefficients in the expansion are

For convenience, we write the Chebyshev polynomials in terms of powers of their variables, where

In particular, for a rectangular region

Hence, for

See, for example, [11] for specific expressions of

(iii) Approximation by cubic splines.

On a rectangular region

where

The local coefficients

In the case of

In order to approximate the prices, we replace the function

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

In particular when

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

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

**Theorem 4.** *Let* *be the price of a European basket contract with maturity* *, strike price* *, and payoff* *under a model given by* Eqs. (1) *and* (2)*. In addition assume* *and* *for any* *, where* *is a* *diagonal matrix with components* *. Let* *be a* *matrix with components*

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

*for some truncation vector*

*The n-th-order Chebyshev approximation on a region* *is*

*where* *and*

*The* *-th-order approximation by cubic splines on the region* *is given by*

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

## 3. Approximating the price of spread contracts

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

Hence for

where

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

where

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

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:

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

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

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

where

and

Hence

where

Higher derivatives can be calculated recursively.

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

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

Moreover

Hence

where

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

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

Similarly for a cubic spline approximation, we specialize Eq. (13) with

where

## 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

Volatilities corresponding to the diffusion part of both assets are

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

MC | Taylor (f.o) | Taylor (s.o.) | Spl. | |
---|---|---|---|---|

Price | 14.7784 | 10.2980 | 14.29068 | 14.8842 |

Interval | — | — | — | |

Run time | 624.312 | 1.68806 | 1.68806 | 54.1720 |

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

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

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

Truncation values for the number of jumps, denoted in the paper by

where

until

In Figure 3 we show the probability distribution

## 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

## A.2 Proof of Theorem 1

From Eq. (2) written in its integral form

it is easy to see that

From the expression above, in the case of

Similarly, for

Then, conditionally on

Hence, the price is expressed as

where

On the other hand, conditioning on

where

Taking into account Eq. (19), again conditioning on the events

Hence, we can write, on the set

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

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

## A.3 Proof of Theorem 4

In Eq. (6) we replace the function

Then, the Taylor approximation of

Eq. (11) follows after replacing

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

Eq. (12) easily follows.

Finally, by similar arguments,

from which (13) follows.