Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility

We perform a classification of the Lie point symmetries for the Black--Scholes--Merton Model for European options with stochastic volatility, $\sigma$, in which the last is defined by a stochastic differential equation with an Orstein--Uhlenbeck term. In this model, the value of the option is given by a linear (1 + 2) evolution partial differential equation in which the price of the option depends upon two independent variables, the value of the underlying asset, $S$, and a new variable, $y$. We find that for arbitrary functional form of the volatility, $\sigma(y)$, the (1 + 2) evolution equation always admits two Lie point symmetries in addition to the automatic linear symmetry and the infinite number of solution symmetries. However, when $\sigma(y)=\sigma_{0}$ and as the price of the option depends upon the second Brownian motion in which the volatility is defined, the (1 + 2) evolution is not reduced to the Black--Scholes--Merton Equation, the model admits five Lie point symmetries in addition to the linear symmetry and the infinite number of solution symmetries. We apply the zeroth-order invariants of the Lie symmetries and we reduce the (1 + 2) evolution equation to a linear second-order ordinary differential equation. Finally, we study two models of special interest, the Heston model and the Stein--Stein model.Comment: Published version, 14pages, 4 figure

Lie symmetries of (1+2) nonautonomous evolution equations in Financial Mathematics

We analyse two classes of $(1+2)$ evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study these equations for the case in which they are autonomous and for the case in which the parameters of the equations are unspecified functions of time. For the autonomous Black-Scholes Equation we find that the symmetry is maximal and so the equation is reducible to the $(1+2)$ Classical Heat Equation. This is not the case for the nonautonomous equation for which the number of symmetries is submaximal. In the case of the two-factor equation the number of symmetries is submaximal in both autonomous and nonautonomous cases. When the solution symmetries are used to reduce each equation to a $(1+1)$ equation, the resulting equation is of maximal symmetry and so equivalent to the $(1+1)$ Classical Heat Equation.Comment: 15 pages, 1 figure, to be published in Mathematics in the Special issue "Mathematical Finance

Some extensions of the Black-Scholes and Cox-Ingersoll-Ross models

In this thesis we will study some financial problems concerning the option pricing in complete and incomplete markets and the bond pricing in the short-term interest rates framework. We start from well known models in pricing options or zero-coupon bonds, as the Black-Scholes model and the Cox-Ingersoll-Ross model and study some their generalizations. In particular, in the first part of the thesis, we study a generalized Black-Scholes equation to derive explicit or approximate solutions of an option pricing problem in incomplete market where the incompleteness is generated by the presence of a non-traded asset. Our aim is to give a closed form representation of the indifference price by using the analytic tool of (C0) semigroup theory. The second part of the thesis deals with the problem of forecasting future interest rates from observed financial market data. We propose a new numerical methodology for the CIR framework, which we call the CIR# model, that well fits the term structure of short interest rates as observed in a real market

Essays in financial asset pricing

Three essays in financial asset pricing are given; one concerning the partial differential equation (PDE) pricing and hedging of a class of continuous/generalized power mean Asian options, via their (optimal) Lie point symmetry groups, leading to practical pricing formulas. The second presents high-frequency predictions of S&P 500 returns via several machine learning models, statistically significantly demonstrating short-horizon market predictability and economically significantly profitable (beyond transaction costs) trading strategies. The third compares profitability between these [(mean) ensemble] strategies and Asian option Δ-hedging, using results of the first. Interpreting bounds on arithmetic Asian option prices as ask and bid values, hedging profitability depends largely on securing prices closer to the bid, and settling midway between the bid and ask, significant profits are consistently accumulated during the years 2004-2016. Ensemble predictive trading the S&P 500 yields comparatively very small returns, despite trading much more frequently. The pricing and hedging of (arithmetic) Asian options are difficult and have spurred several solution approaches, differing in theoretical insight and practicality. Multiple families of exact solutions to relaxed power mean Asian option pricing boundary-value problems are explicitly established, which approximately satisfy the full pricing problem, and in one case, converge to exact solutions under certain parametric restrictions. Corresponding hedging parameters/ Greeks are derived. This family consists of (optimal) invariant solutions, constructed for the corresponding pricing PDEs. Numerical experiments explore this family behaviorally, achieving reliably accurate pricing. The second chapter studies intraday market return predictability. Regularized linear and nonlinear tree-based models enjoy significant predictability. Ensemble models perform best across time and their return predictability realizes economically significant profits with Sharpe ratios after transaction costs of 0.98. These results strongly evidence that intraday market returns are predictable during short time horizons, beyond that explainable by transaction costs. The lagged constituent returns are shown to hold significant predictive information not contained in lagged market returns or price trend and liquidity characteristics. Consistent with the hypothesis that predictability is driven by slow-moving trader capital, predictability decreased post-decimalization, and market returns are more predictable midday, on days with high volatility or illiquidity, and during financial crises