Prices obtained using the benchmark parameter set and Monte Carlo, first- and second-order Taylor, and cubic spline approximations.
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.
- Taylor approximations
- Chebyshev polynomials
- cubic splines
- basket options
- spread options
- jump-diffusion model
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 , 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 out of the total number of assets, 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  for a vanilla European option and in [3, 4] for spread contracts under a bivariate Black-Scholes model. See also . A Chebyshev expansion has been recently considered in . 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 . See also  for expansions in terms of Fourier series and  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 be a filtered probability space. We define the filtration as the -algebra generated by the random variables completed in the usual way. Denote by an equivalent martingale measure (EMM), respectively, by , , and the expectation, characteristic, and moment-generating functions of a random variable under . The function is its probability density function.
By we denote the (constant) interest rate, is the componentwise product between matrices and , and represents the transpose of matrix , while is a vector with components . The symbol is the usual Kronecker’s number. The vector is created from the vector after eliminating the first component. For a function with domain in and a vector with , the symbol represents the mixed partial derivative of the function differentiated times w.r.t. the -th variable.
For vectors and , we set and .
We introduce the following convenient notations. For a vector , , and
Also, for a differentiable function , we set the vector .
The -dimensional process of spot prices is denoted by , while is the corresponding log-price process. They are related by
We analyze European basket options whose payoff at maturity , for a strike price , are given by
where are some deterministic weights and .
Furthermore, for the log-prices, we assume a multidimensional jump-diffusion dynamics under given by
where is a multivariate Brownian motion with independent components and . The matrix is symmetric, positive definite, while is such that . The value is the compensator of a compound Poisson process .
We define two sequences of independent and identically distributed -dimensional random vectors and . The components of the random vectors in the first sequence are independent.
The process is a d-variate compound Poisson process, independent of such that
where is a vector of independent Poisson processes with respective intensities .
The processes and correspond, respectively, to idiosyncratic and common jumps of the -th underlying asset on the interval . Their jump sizes are and .
For the sake of concreteness, we assume Gaussian jumps, i.e., we assume for any that , where is a diagonal matrix with components and , with a matrix of components . The compensator across each dimension takes the form
First, we write the price of the basket contract in terms of its conditional price when the number of jumps and underlying assets are fixed. Results are given in Theorem 1 below.
Notice that, for any
We also introduce the vector with components
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 .
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 , we approximate the conditional price on the variable by 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 Taylor approximation of around is described by
with , where the second sum is taken on the set
Notice the existence of the derivatives of any order in the functions and .
(ii) An approximation based on Chebyshev polynomials is given as follows:
In a region , we consider an expansion of order of the function as
where the sums are taken over the sets
Here is a family of -dimensional Chebyshev polynomials with degrees defined in the region , while the quantities are suitable approximations of the corresponding Chebyshev coefficients , computed using the trapezoidal rule.
Notice that, by the orthogonality of the polynomials, the coefficients in the expansion are , where is the scalar product of functions and , conveniently weighted by a function . See, for example,  for a general account on Chebyshev polynomials.
For convenience, we write the Chebyshev polynomials in terms of powers of their variables, where are the coefficients of this expansion.
In particular, for a rectangular region and valued vectors and , we write
See, for example,  for specific expressions of in one dimension.
(iii) Approximation by cubic splines.
On a rectangular region , we consider an approximation based on cubic splines given by
where is some point on a (d − 1)-dimensional grid with points in .
The local coefficients are determined by imposing the conditions . The family of sets is a partition of . Notice that the coefficients depend on the particular rectangle in the grid. See  for a general account on multivariate splines.
In the case of , splines used to approximate the conditional price become one-dimensional polynomials. Additional conditions on the derivatives to smoothen these curves are imposed, namely, , , where and are, respectively, the derivatives from the left and the right of the function at point . Moreover, for end points in the grid, .
In order to approximate the prices, we replace the function by its respective expansions. The conditional prices on the event are 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 , , and .
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 and a Borel set , we define
In particular when , we write .
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 are related to the partial derivatives of the corresponding moment-generating function Indeed, for , we have
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 , conditionally on , the log-prices of the first asset are normally distributed, i.e., , with
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 on the interval is needed. To this end we have
where is the -th moment of a standard normal random variable constrained to the interval and is a vector with components .
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 with respect to .
First, notice that, from the Black-Scholes pricing formula:
and is the cumulative distribution function of a standard normal distribution.
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 and constrained to the interval . To this end we denote
Notice that, taking into account Eq. (14),
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 . Therefore, we have
where is defined as in Eq. (16) but replacing by .
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 , maturity year, spot prices , , and a fix interest rate of 3%.
Volatilities corresponding to the diffusion part of both assets are and , while the correlation coefficient between the two Brownian noises is . Regarding the jump part, we consider an average intensity of the common jumps equal to jumps per year and idiosyncratic intensities jumps per year for the respective assets, while jump sizes have means equal to zero; volatilities of common jump sizes are , with a linear correlation . Volatilities of the idiosyncratic jumps are taken as and .
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 repetitions 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.
|MC||Taylor (f.o)||Taylor (s.o.)||Spl.|
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 .
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 , evaluations in the order of are needed, where is the maximum truncation level in the number of jumps. In a Chebyshev approach of the same order about , evaluations of the conditional price should be performed, when a grid of points is used in a trapezoidal approximation of the corresponding integrals. In a cubic splines approximation . Here is also the number of points in the grid where the polynomial coefficients are adjusted.
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 and is shown. The remaining three curves represent the first-order (green), second-order (red), and third-order (magenta) Taylor polynomials around the average value . 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 is continuous in for any value of ; 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 2 shows the differences between the conditional price and the cubic spline for different values of the underlying price. Truncation values were selected as and . 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.
Truncation values for the number of jumps, denoted in the paper by and , should cover most of the jump probability distribution . 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 ’s is attained, namely,
where represents the maximum of the integer part of and zero, then adding expression (18) for points over the set
until , where is a predetermined value close to one.
In Figure 3 we show the probability distribution , for varying and . We observe probabilities become negligible after certain values of with a peak around the center of the distribution. For the benchmark parameter set truncation values capture 99.67% of the probability mass.
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.
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 and the corresponding derivatives of the moment-generating function of , we can compute Taylor approximations up to third order around the point as
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 , we have
Then, conditionally on , we have
Hence, the price is expressed as
On the other hand, conditioning on :
Taking into account Eq. (19), again conditioning on the events and , it is well known that has a univariate normal distribution with mean and variance given, respectively, by and . See, for example, .
Hence, we can write, on the set :
Then, replacing the expression above in Eq. (21), we have
A.3 Proof of Theorem 4
Then, the Taylor approximation of is
Eq. (12) easily follows.
Finally, by similar arguments,
from which (13) follows.