Double barrier American put option pricing under uncertain volatility model

This paper explores the application of Machine Learning techniques for pricing high-dimensional options within the framework of the Uncertain Volatility Model (UVM). The UVM is a robust framework that accounts for the inherent unpredictability of market volatility by setting upper and lower bounds on volatility and the correlation among underlying assets. By leveraging historical data and extreme values of estimated volatilities and correlations, the model establishes a confidence interval for future volatility and correlations, thus providing a more realistic approach to option pricing. By integrating advanced Machine Learning algorithms, we aim to enhance the accuracy and efficiency of option pricing under the UVM, especially when the option price depends on a large number of variables, such as in basket or path-dependent options. In this paper, we consider two approaches based on Machine Learning. The first one, termed GTU, evolves backward in time, dynamically selecting at each time step the most expensive volatility and correlation for each market state. Specifically, it identifies the particular values of volatility and correlation that maximize the expected option value at the next time step, and therefore, an optimization problem must be solved. This is achieved through the use of Gaussian Process regression, the computation of expectations via a single step of a multidimensional tree and the Sequential Quadratic Programming optimization algorithm. The second approach, referred to as NNU, leverages neural networks and frames pricing in the UVM as a control problem. Specifically, we train a neural network to determine the most adverse volatility and correlation for each simulated market state, generated via random simulations. The option price is then obtained through Monte Carlo simulations, which are performed using the values for the uncertain parameters provided by the neural network. The numerical results demonstrate that the proposed approaches can significantly improve the precision of option pricing and risk management strategies compared with methods already in the literature, particularly in high-dimensional contexts.

Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach

The option pricing model with constant volatility in illiquid markets has been expanded by introducing two uncertain volatility models in this paper volatility. To conquer some insufficient existed in some literature, for example the time step should be small enough to satisfy the stability, the implicit difference equation has been established and numerical solution of the modified model with uncertain volatility has been discussed. Numerical results show that the method is nice and the accurate results can be gained with less computation.

We examine, from both analytical and numerical viewpoints, the uncertain volatility model by Hobson-Rogers in the framework of degenerate parabolic PDEs of Kolmogorov type.

The uncertain volatility model has long ago attracted the attention of practitioners as it provides worst-case pricing scenario for the sell-side. The valuation of a financial derivative based on this model requires solving a fully non-linear PDE. One can rely on finite difference schemes only when the number of variables (that is, underlyings and path-dependent variables) is small - in practice no more than three. In all other cases, numerical valuation seems out of reach. In this paper, we outline two accurate, easy-to-implement Monte-Carlo-like methods which hardly depend on dimensionality. The first method requires a parameterization of the optimal covariance matrix and consists in a series of backward low-dimensional optimizations. The second method relies heavily on a recently established connection between second-order backward stochastic differential equations and non-linear second-order parabolic PDEs. Both methods are illustrated by numerical experiments.

The aim of this work is to evaluate the cheapest superreplication price of a general (possibly path-dependent) European contingent claim in a context where the model is uncertain. This setting is a generalization of the uncertain volatility model (UVM) introduced in by Avellaneda, Levy and Paras. The uncertainty is specified by a family of martingale probability measures which may not be dominated. We obtain a partial characterization result and a full characterization which extends Avellaneda, Levy and Paras results in the UVM case.

We study the Option pricing with linear investment strategy based on discrete time trading of the underlying security, which unlike the existing continuous trading models provides a feasible real market implementation. Closed form formulas for Call and Put Option price are established for fixed interest rates and their extensions to stochastic Vasicek and Hull-White interest rates.

The SABR stochastic volatility model with β-volatility β ? (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are expressed using only elementary functions. The option pricing formulae are used to study synthetic and real data. In particular we study a time series (of real data) of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website: http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations, an interactive application and an app that helps the understanding of the paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.

Calibration of the double Heston model and an analytical formula in pricing American put option

The double barrier option is characterized by pay-off with strike K, maturity T, upper Su and lower Sd barrier levels and the corresponding rebates φu and φd which can be time dependent. We divide last four quantities by strike K and introduce new variables x = ln(S/K), xu = ln(Su/K), xd = ln(Sd/K). The value of European double barrier call option U(t, x) satisfies the extended Black-Scholes equation

We first recall the well-known expression of the price of barrier options, and compute double barrier options by the mean of the iterated mirror principle. The formula for double barriers provides an intraday volatility estimator from the information of high-low-close prices. Then we give explicit formulas for the probability distribution function and the expectation of the exit time of single and double barrier options. These formulas allow to price time independent and time dependent rebates. They are also helpful to hedge barrier and double barrier options, when taking into account variations of the term structure of interest rates and of volatility. We also compute the price of rebates of double knock-out options that depend on which barrier is hit first, and of the BOOST, an option which pays the time spent in a corridor. All these formulas are either in closed form or double infinite series which converge like e-α n2.

Determining bid-ask prices for options with stochastic illiquidity and applications to index options

We consider closed-form approximations for European put option prices within the Heston and GARCH diffusion stochastic volatility models with time-dependent parameters. Our methodology involves writing the put option price as an expectation of a Black-Scholes formula and performing a second-order Taylor expansion around the mean of its argument. The difficulties then faced are simplifying a number of expectations induced by the Taylor expansion. Under the assumption of piecewise-constant parameters, we derive closed-form pricing formulas and devise a fast calibration scheme. Furthermore, we perform a numerical error and sensitivity analysis to investigate the quality of our approximation and show that the errors are well within the acceptable range for application purposes. Lastly, we derive bounds on the remainder term generated by the Taylor expansion.

I address the dichotomy between American put option pricing theory and the numerical algorithms designed to estimate American put option prices. The literature has focused only on pricing error. However, early exercise is the essence of American option pricing (exercising) and with it comes the possibility of early exercise error and its opportunity costs. I introduce an economically viable metric that identifies all the errors of American put option pricing algorithms. The accuracy of such algorithms can thereby be fully assessed. A rational option pricing result generalises the usual integral equation and motivates pure pricing error. This provides new intuition for the optimality condition for early exercise. Early exercise error is motivated by comparing discounted expected profits generated by the estimated model vs. the optimal early exercise model. The error measure applies to any put pricing algorithm and any benchmark. We illustrate our total error measure with a new algorithm.

We introduce a consistent method for pricing panels of options through time with a single simulation. The approach is particularly useful for pricing options with early exercise features and allows calibrating flexible option pricing models to large panels of such options. We showcase this by calibrating option pricing models with time-varying volatility and asymmetric features to a sample of more than 25,000 American-style options. Our method leverages the homogeneity of option prices, relies only on polynomial approximations using simulated paths, and could be applied to other problems that require estimating optimal stopping times using dynamic programming.