On Value-at-Risk and Expected Shortfall of Financial Asset with Stochastic Pricing

We study the problem of measuring market risk of financial asset with stochastic pricing. Market risk metrics under study are Value-at- Risk and Expected Shortfall. The price of the financial asset is assumed to satisfy a given stochastic differential equation with diffusion coefficient being a function of asset price and time. We investigate various models of asset price dynamics, including well-known lognormal Black-Scholes model, shifted lognormal model, Bachelier and Cox-Ross normal models and a new stochastic model with hyperbolic sine function. For stochastic models under study we derive explicit analytic expressions for loss distribution function, Value-at- Risk and Expected Shortfall. Dependence of derived Value-at- Risk and Expected Shortfall functions on confidence level is shown on the plots. The possibility of using derived formulae for market risk estimation for equity traded on financial markets is demonstrated. We show that the highest estimate of market risk is given by hyperbolic sine and Cox-Ross models, the lowest estimate is given by Black-Scholes model and shifted lognormal model with negative model parameter.