In this paper we present a new multi-asset pricing model, which is built upon newly developed families of solvable multi-parameter single-asset diffusions with a nonlinear smile-shaped volatility and an affine drift. Our multi-asset pricing model arises by employing copula methods. In particular, all discounted single-asset price processes are modeled as martingale diffusions under a risk-neutral measure. The price processes are so-called UOU diffusions and they are each generated by combining a variable (Ito) transformation with a measure change performed on an underlying Ornstein-Uhlenbeck (Gaussian) process. Consequently, we exploit the use of a normal bridge copula for coupling the single-asset dynamics while reducing the distribution of the multi-asset price process to a multivariate normal distribution. Such an approach allows us to simulate multidimensional price paths in a precise and fast manner and hence to price path-dependent financial derivatives such as Asian-style and Bermudan options using the Monte Carlo method. We also demonstrate how to successfully calibrate our multi-asset pricing model by fitting respective equity option and asset market prices to the single-asset models and their return correlations (i.e. the copula function) using the least-square and maximum-likelihood estimation methods.
Many quantitative finance applications require a multi-asset pricing model with dependencies between the single-asset price components. Compared to the variety of univariate asset price models, the pool of multi-asset pricing models is not so extensive. Most of multivariate models are based on multidimensional geometric Brownian motion with the possible inclusion of a jump process. In this paper we develop and explore a new multi-asset arbitrage-free pricing model based on a special family of nonlinear diffusions. The development of efficient computational methods for pricing multi-asset equity derivatives under such a model and the calibration of the multi-asset model to both standard equity option data as well as historical equity prices are the objectives of the current paper.
Here, we specialize on option pricing applications under so-called UOU diffusion models, which are obtained by transforming an underlying Ornstein-Uhlenbeck diffusion process via the use of a diffusion canonical transformation method (see [2,3,5,8] and references therein). For all choices of model parameters, all discounted (single-asset) price processes UOU are conservative martingales under a riskneutral measure. Since the univariate diffusions are solvable, the single-asset risk-neutral transition probability density function is given in analytically closed form. Moreover, implied volatility surfaces for this highly nonlinear asset price model exhibit a wide range of pronounced smiles and skews of the type observed in the option markets. The main relevant features of the univariate UOU model are summarized in Section 1.
To construct a multivariate probability distribution, one can use a copula function that allows us to couple univariate distribution functions. Sampling from the obtained joint multivariate distribution function thereby reduces to sampling from the copula function and from the univariate distributions. Therefore, the copula method allows us to construct the joint distribution and density functions as well as to obtain an exact path sampling algorithm.
The main computational disadvantage of such an approach is the calculation of inverses of the distribution functions. This operation can be a rather time-consuming computational problem for a complicated multi-parameter distribution. Nevertheless, such a drawback can be significantly improved if the copula function and univariate distributions have a similar structure. As is shown in Subsection 1.3, the bridge probability density function (conditional on the values of the process at the endpoints of a time interval) of a UOU diffusion is reduced to a normal density. Hence it is natural to couple univariate UOU bridges using a Gaussian copula. Based on this idea, in Section 2, we construct a two-step algorithm for the exact path simulation of the multidimensional (nonlinear) UOU process in the risk-neutral measure. Firstly, we apply a usual copula method for sampling the multi-asset process at the terminal (maturity) time. Secondly, we use a bridge sampling along with a multivariate normal distribution to model the process at any intermediate time.
In Section 3, we demonstrate the calibration of the univariate and multivariate models to historical asset and equity option prices. The calibration process has two stages. First, we calibrate all univariate (marginal) asset price models independently of each other. Using the least-square method the models can be fitted to standard European option prices. Alternatively, the maximum likelihood estimation (MLE) allows the models to be fitted to historical asset prices. Second, we fit the copula function to historical observations. Since, our model assumes a normal copula, we need to find a best-fitted normal correlation matrix.
In Section 4, we give computational applications of the model to pricing multi-asset path-dependent Asian-style and Bermudan options. In pricing Bermudan options we use a regression-based Monte Carlo method.
In summary, the main results of our paper include: the construction a new family of multivariate models for which marginal processes are local volatility smile diffusions; the development of calibration schemes for the single-asset and multi-asset pricing UOU diffusion models based on the least-square and MLE methods; the construction of an exact multivariate path simulation method that can be used for Monte Carlo pricing of generally path-dependent European-style and American-style options.
1 Ornstein-Uhlenbeck Family of Univariate State-Dependent Volatility Diffusion Models
The diffusion canonical transformation method, first presented in [2] and then further developed in [15,5,8], leads to various families of solvable one-dimensional diffusions with a nonlinear diffusion coefficient function and an affine drift. In this paper, we consider the UOU family, which is based on a regular Ornstein-Uhlenbeck process (X t ) t≥0 ∈ I ≡ R. The regular Ornstein-Uhlenbeck process is defined by the in
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