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Register a new userOption Valuation through Deep Learning of Transition Probability Density
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Transition probability densities are fundamental to option pricing. Advancing recent work in deep learning, we develop novel transition density function generators through solving backward Kolmogorov equations in parametric space for cumulative probability functions, using neural networks to obtain accurate approximations of transition probability densities, creating ultra-fast transition density function generators offline that can be trained for any underlying. These are single solve , so they do not require recalculation when parameters are changed (e.g. recalibration of volatility) and are portable to other option pricing setups as well as to less powerful computers, where they can be accessed as quickly as closed-form solutions. We demonstrate the range of application for one-dimensional cases, exemplified by the Black-Scholes-Merton model, two-dimensional cases, exemplified by the Heston process, and finally for a modified Heston model with time-dependent parameters that has no closed-form solution.
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Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). In this work, the classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied. Instead of using numerical techniques based on finite element or difference methods, we address the problem using ANNs in the context of unsupervised learning. As a result, the ANN learns the option values for all possible underlying stock values at future time points, based on the minimization of a suitable loss function. For the European option, we solve the linear Black-Scholes equation, whereas for the American option, we solve the linear complementarity problem formulation. Two-asset exotic option values are also computed, since ANNs enable the accurate valuation of high-dimensional options. The resulting errors of the ANN approach are assessed by comparing to the analytic option values or to numerical reference solutions (for American options, computed by finite elements).
When solving the American options with or without dividends, numerical methods often obtain lower convergence rates if further treatment is not implemented even using high-order schemes. In this article, we present a fast and explicit fourth-order compact scheme for solving the free boundary options. In particular, the early exercise features with the asset option and option sensitivity are computed based on a coupled of nonlinear PDEs with fixed boundaries for which a high order analytical approximation is obtained. Furthermore, we implement a new treatment at the left boundary by introducing a third-order Robin boundary condition. Rather than computing the optimal exercise boundary from the analytical approximation, we simply obtain it from the asset option based on the linear relationship at the left boundary. As such, a high order convergence rate can be achieved. We validate by examples that the improvement at the left boundary yields a fourth-order convergence rate without further implementation of mesh refinement, Rannacher time-stepping, and/or smoothing of the initial condition. Furthermore, we extensively compare, the performance of our present method with several 5(4) Runge-Kutta pairs and observe that Dormand and Prince and Bogacki and Shampine 5(4) pairs are faster and provide more accurate numerical solutions. Based on numerical results and comparison with other existing methods, we can validate that the present method is very fast and provides more accurate solutions with very coarse grids.
We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution, which is a challenge for many other methods. Our approach is based on a computable series expansion in terms of a small parameter. As an example, we treat in detail the important case of the SABR PDE for $beta = 1$, namely $partial_{tau}u = sigma^2 big [ frac{1}{2} (partial^2_xu - partial_xu) + u rho partial_xpartial_sigma u + frac{1}{2} u^2 partial^2_sigma u , big ] + kappa (theta - sigma) partial_sigma$, by choosing $ u$ as small parameter. This yields $u = u_0 + u u_1 + u^2 u_2 + ldots$, with $u_j$ independent of $ u$. The terms $u_j$ are explicitly computable, which is also a challenge for many other, related methods. Truncating this expansion leads to computable approximations of $u$ that are in closed form, and hence can be evaluated very quickly. Most of the other related methods use the time $tau$ as a small parameter. The advantage of our method is that it leads to shorter and hence easier to determine and to generalize formulas. We obtain also an explicit expansion for the implied volatility in the SABR model in terms of $ u$, similar to Hagans formula, but including also the {em mean reverting term.} We provide several numerical tests that show the performance of our method. In particular, we compare our formula to the one due to Hagan. Our results also behave well when used for actual market data and show the mean reverting property of the volatility.
We introduce a general decision tree framework to value an option to invest/divest in a project, focusing on the model risk inherent in the assumptions made by standard real option valuation methods. We examine how real option values depend on the dynamics of project value and investment costs, the frequency of exercise opportunities, the size of the project relative to initial wealth, the investors risk tolerance (and how it changes with wealth) and several other choices about model structure. For instance, contrary to stylized facts from previous literature, real option values can actually decrease with the volatility of the underlying project value and increase with investment costs.
We model investor heterogeneity using different required returns on an investment and evaluate the impact on the valuation of an investment. By assuming no disagreement on the cash flows, we emphasize how risk preferences in particular, but also the costs of capital, influence a subjective evaluation of the decision to invest now or retain the option to invest in future. We propose a risk-adjusted valuation model to facilitate investors subjective decision making, in response to the market valuation of an investment opportunity. The investors subjective assessment arises from their perceived misvaluation of the investment by the market, so projected cash flows are discounted using two different rates representing the investors and the markets view. This liberates our model from perfect or imperfect hedging assumptions and instead, we are able to illustrate the hedging effect on the real option value when perceptions of risk premia diverge. During crises periods, delaying an investment becomes more valuable as the idiosyncratic risk of future cash flows increases, but the decision-maker may rush to invest too quickly when the risk level is exceptionally high. Our model verifies features established by classical real-option valuation models and provides many new insights about the importance of modelling divergences in decision-makers risk premia, especially during crisis periods. It also has many practical advantages because it requires no more parameter inputs than basic discounted cash flow approaches, such as the marketed asset disclaimer method, but the outputs are much richer. They allow for complex interactions between cost and revenue uncertainties as well as an easy exploration of the effects of hedgeable and un-hedgeable risks on the real option value. Furthermore, we provide fully-adjustable Python code in which all parameter values can be chosen by the user.
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