Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks

Abstract

Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). The classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied here. 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). In the short note, previously published, a brief introduction to this work was given, where some ideas to price vanilla options by ANNs were presented, and only European options were addressed. In the current work, the methodology is introduced in much more detail.

Efficient pricing and hedging of high-dimensional American options using deep recurrent networks

“…We note that this is feasible only if d ≤ 3. We make comparison results to the unsupervised method of Salvador et al [2019], and also look at the far out-the-money(OTM) and far in-the-money(ITM) performance of our method, the DRL method and the Longstaff-Schwartz method in Experiment 3. In many cases we do not make a comparison to the Longstaff-Schwartz method as it has been done in Chen and Wan [2021].…”

“…In many cases we do not make a comparison to the Longstaff-Schwartz method as it has been done in Chen and Wan [2021]. In addition, we remark that the comparison is not made with the other methods referenced in the introduction, such as Guler and Laignelet [2019], Salvador et al [2019], Sirignano andSpiliopoulos [2018], E et al [2017], Beck et al [2018] and Han and E [2016]. This is because the work of Sirignano and Spiliopoulos [2018] was compared in Chen and Wan [2021].…”

“…We also compare the output of the proposed method to the unsupervised method using the L-BFGS optimizer and the loss function outlined in Salvador et al [2019]. We implemented the unsupervised method as in Salvador et al [2019], where an 101 × 101 × 75 spacetime grid was used to run the unsupervised method.…”

“…We also compare the output of the proposed method to the unsupervised method using the L-BFGS optimizer and the loss function outlined in Salvador et al [2019]. We implemented the unsupervised method as in Salvador et al [2019], where an 101 × 101 × 75 spacetime grid was used to run the unsupervised method. We let the method run until the running time is close to the times given in Table 8 and we found the prices for initial price of s 0 = 90, 100, 110; the resulting value output by the unsupervised method, V unsupervised = 25.72, 25.00, 24.59, respectively.…”

“…The use of a neural network grid has been proposed by Sirignano and Spiliopoulos [2018]. The work of Salvador et al [2019] uses a neural network to learn option prices of the linear complementary problem of option pricing in an unsupervised manner. More recently, authors of Herrera et al [2021] proposed a randomized reinforcement learning using recurrent networks to solve the American option problem.…”