On Approximation of Transition Densities in Calibration of 1-Dimensional Stochastic Models of Asset Prices

Abstract

For a 1-dimensional stochastic differential equation (SDE) of the type $dS_{t}=\tilde{\mu}(S_{t},t;\mathbf{q})dt+\tilde{\sigma}(S_{t},t;\mathbf{q})dW_{t}$, where $S_{t}$ is e.g. the price of a certain financial asset, $W_{t}$ a Brownian motion and $\mathbf{q}$ is a finite-dimensional vector of unknown model parameters, we provide a survey of methods for estimating the vector $\mathbf{q}$ using the historical time series of $S_{t}$. In quantitative finance terms, the SDEs under consideration are called parametric local volatility models, and the family of parameter estimation methods utilizing the historical time series is called ‘‘real-measure model calibration methods’’, as opposed to ‘‘risk-neutral model calibration methods’’ based on the prices of non-linear derivatives (options) with the underlying price $S_{t}$. We are primarily focused on the classical Maximum Likelihood Estimation (MLE) framework, and provide extensions of the previously-known MLE methods in two major respects. First, we show that the likelihood values can be computed using the probability density functions (PDFs) for 3-variate distributions of Open, High, Low and Close (OHLC) prices for each historical interval, instead of commonly-used 1-variate PDFs of Open and Close prices only. This allows us to take into account additional market information, i.e. ranges of $S_{t}$ values as opposed to just point-wise values. Second, in order to construct the required 3-variate OHLC PDFs for general non-constant coefficients $\tilde{\mu}$ and $\tilde{\sigma}$, we apply results from the Heat Kernel theory. The constructed method may be applicable to a wide class of realistic SDEs in quantitative finance. In particular, this method can be applied to models of crypto-asset prices (e.g., for the purpose of price prediction in algorithmic trading), whereas risk-neutral (derivatives-based) methods would not be applicable due to insufficient liquidity of the crypto-options market. We consider an example of a reversion-based algorithmic trading strategy as an application of this method.