1. Introduction

2. Methodology: Solving Option Pricing PDEs Using PINNs:

3. Benchmarking

3.3. Bench Marking with One-Dimensional Path-Dependent Option

Pricing Asian average rate call Options we introduce a new variable $A\left(t\right)$

$$A\left(t\right)={\displaystyle \frac{1}{t}}{\int }_0^{t}x\left(s\right)ds$$

and solve the system of PDEs

$${\displaystyle \frac{\partial \mathcal{V}}{\partial t}}+{\displaystyle \frac{1}{2}}{\sigma }^2{x}^2{\displaystyle \frac{{\partial }^2\mathcal{V}}{\partial x^2}}+rx{\displaystyle \frac{\partial \mathcal{V}}{\partial x}}-r\mathcal{V}=0$$

(9)

and

$${\displaystyle \frac{dA}{dt}}-{\displaystyle \frac{x\left(t\right)-A\left(t\right)}{t}}=0$$

(10)

subject to the terminal condition of option price:

$$\mathcal{O}(t=T)=\max (A-x(T),0)$$

and initial condition of average

$$A\left(t\right)=x\left(t\right)$$

boundary conditions for this average rate call options are:

$$\mathcal{V}(x,A,t)=(S-A)atx=0$$

$$\mathcal{V}(x,A,t)=0atx=L$$

$$\mathcal{V}(x,A,t)=g(x,A)=\max (S-A,0)att=T$$

where L is some suitably large value by truncating the original $(0,\infty )$ domain to $(0,L]$. We take $L=5S$.

A singularity exists in the equation for A at $t=0$ (i.e., today). However, at $t=0$ the term ${\displaystyle \frac{U-A}{t}}*{\displaystyle \frac{\partial \mathcal{O}}{\partial A}}$ also goes to zero leaving us with a BSPDE for the European option which has an exact solution. so we leave out t=0 and take our time domain as $(0,T]$ or $[0.0001,T]$ ([33])

We solve this model problem with parameters:

$$S=1,r=0.04,\sigma =0.03,T=1$$

Note that for an Asian option, we use the actual time $\left(t\right)$ instead of introducing $\tau$ because t appears explicitly in this pde. We have two inputs (scaled stock price and time to expiry) and two outputs (average price, and option price). We use a fully connected neural network with depth 4 (ie 3 hidden layers) and width 20. Thus there are 942 weights. We utilize 2000*10 training residual points sampled within the spatio-temporal domain, 200*10 training points sampled on the boundary, and 100*10 residual points for the initial conditions. The activation function used is the hyperbolic tangent (tanh). The training begins with the Adam optimizer for 2000 iterations at a learning rate of 0.001, after which we switch to the L-BFGS optimizer. L-BFGS does not require a learning rate, and the neural network is trained until it converges.

The "Adam" optimizer had a train loss of 2.94e-01, and training took 13.90 seconds; The "L-BFGS" optimizer had a train loss of 2.76e-01 and training took 4.64 seconds. So both the optimizers together took training time of 18.54 seconds

For error comparison we randomly generated 1000 data points from the input domain of our NN i.e. $(0,L]x(0,T]$. The first coordinate of the out point represents the initial value of stock$\left(U_i\right)$ and the other represents a time to expiry(t). Then we calculated the option value $\left(V_i^{mc}\right)$ for those points using the Monte Carlo method with 100000 paths and 256 time steps. We also calculate the values our model predicts $\left(V_i^{nn}\right)$ for each point. The relative error is calculated using the formula:

Figure 4. Plots related to 0ne-underlying Asian option.

Figure 4. Plots related to 0ne-underlying Asian option.

$${\mathrm{error}}_i={\displaystyle \frac{\left(V_i^{mc}\right)-\left(V_i^{nn}\right)}{\left(V_i^{mc}\right)}}$$

We plot an error histogram, to evaluate our results [22]. The root-mean-squared error (RMSE) is 0.0041, which may be compared to the fact that the strike prices are all normalized to $1. Hence the average error is less than ±0.4% of the strike. The average percentage pricing error (error divided by option price) is 0.0220 i.e., almost 2%. The histogram of pricing errors is shown Figure 5a. We see that the errors are sufficiently small. Further, we estimated a regression of the model values and obtained R2 = 0.9981, i.e. almost 99.8% which is quite acceptable.

4. Pricing of Rainbow Options

4.1. Pricing of European Call Options with Four Underlying Stocks

The scaled PDE for the European call option that will act as the loss function of our PINNs will be:

$${\displaystyle \frac{\partial \mathcal{V}}{\partial \tau }}-{\displaystyle \frac{1}{2}}\sum _{i=1}^d\sum _{j=1}^4{\sigma }_i{\sigma }_j{\rho }_{ij}{x}_i{x}_j{\displaystyle \frac{{\partial }^2\mathcal{V}}{\partial x_i\partial x_j}}-r\sum _{i=1}^4{\displaystyle \frac{\partial \mathcal{V}}{\partial x_i}}{x}_i+r\mathcal{V}=0$$

(11)

which is obtained by substituting $d=4$ in equation(3). The initial condition for the max call option is:

$$g(x_1,x_2,x_3,x_4,\tau =0)=\max (\max (x_1,x_2,x_3,x_4)-1,0)$$

Unlike for the one-dimensional or two-dimensional cases, we don’t have clear values at the stock boundary but for the multi-dimension call option we can impose this Boundary condition at boundary $x_i=L$

$$\mathcal{V}(x,\tau )=max(x_i-1,0)*e^{r*\tau }as{x}_i=L$$

for any $i\in \{1,2,3,4\}$

We took all our underlying stocks, $U_i$ to be in the domain $(0,L]$; where L is sufficiently large.( $L=5*S$). Since our input, $x_i$ is scaled as $x_i=U_i/S$ our $x_i$ is in the domain $(0,5]$ for each stock. We take these parameters for our option

• Strike Price $\left(S\right):\$$100 ,
• Maturity $\left(T\right):$ 1 year,
• Risk-free rate $\left(r\right):$ 0.01,
• volatility of stocks $\left({\sigma }_i\right):$ 0.08 ,
• correlation between stocks $\left({\rho }_{ij}\right):$ 0.5

To train the network 2000*10 training residual points are sampled within the spatio-temporal domain, 200*10 training points are sampled on the boundary, and 100*10 residual points for the initial conditions. We have five inputs (price of stock one, initial price of stock two, initial price of stock three, price of stock four and time to expiry) and one output (multi-asset max call option price). We choose a fully connected neural network of depth 3 (i.e. 2 hidden layers) and width 20. So there are 561 weights. We start with "tanh" as the activation switch. After defining the neural network we build the model; choosing "adam" as the optimizer and "${10}^{-}3$" learning rate for "5000" iterations where an iteration is the number of times a certain batch is passed via an algorithm. After training with "adam" we train with "L-BFGS" until convergence.

The "Adam" optimizer had a train loss of 3.89e-01, and training took 60.77 seconds; The "L-BFGS" optimizer had a train loss of 1.89e-03 and training took 248.28 seconds. So both the optimizers together took the training time of 309.05 seconds. Figure 5b shows the test train loss history

Figure 5. 4-asset European option.

Figure 5. 4-asset European option.

For error comparison we generate option values for 1000 data points using another numerical solution, the Monte Carlo method with 100000 paths and 256 time steps. To evaluate our solution we plot the error histogram. The root-mean-squared error (RMSE) is 0.00965, which is very reasonable considering that the strike prices are all normalized to $1. Hence the average error is less than ±1% of the strike. The average percentage pricing error (error divided by option price) is 0.0312, i.e., almost 3.1%. The histogram of pricing errors is shown Figure 5a. We see that the errors are very small. Further, we estimated a regression of the model values and obtained an R2 = 0.9856, which again is quite good.

4.1.1. American Option PINNs for Four Underlying

The scaled PDE for the American call option that will act as the loss function of our PINNs will be:

$${\displaystyle \frac{\partial \mathcal{V}}{\partial \tau }}-{\displaystyle \frac{1}{2}}\sum _{i=1}^d\sum _{j=1}^4{\sigma }_i{\sigma }_j{\rho }_{ij}{x}_i{x}_j{\displaystyle \frac{{\partial }^2\mathcal{V}}{\partial x_i\partial x_j}}-r\sum _{i=1}^4{\displaystyle \frac{\partial \mathcal{V}}{\partial x_i}}{x}_i+r\mathcal{V}.(\mathcal{V}(x,\tau )-g(x,\tau )=0$$

(12)

where g is the initial condition

$$g(x_1,x_2,x_3,x_4,\tau =0)=max(max(x_1,x_2,x_3,x_4)-1,0)$$

The same problem parameters as in the European option’s case are also used for the American option. We also take the same model parameters and training points as for the European 4D call option.

The "Adam" optimizer had a train loss of 4.62e-01, and training took 60.80 seconds; The "L-BFGS" optimizer had a train loss of 6.12e-04 and training took 159.80 seconds. So both the optimizers together took the training time of 220.60 seconds. Figure 6b shows the test train loss history

Just like we did in the case of the European multi-asset option, for error comparison we generate option values for 1000 data points using Monte Carlo with 100000 paths and 256 time steps. To evaluate our result we plot error histogram. The root mean-squared error (RMSE) is 0.0014, which may be compared to the fact that the strike prices are all normalized to $1. Hence the average error is less than ±0.15% of the strike. The average percentage pricing error (error divided by option price) is 0.0138, i.e. 1.4% of option price. The histogram of pricing errors is shown in Figure 6a. which shows that the errors are very small. Further, we estimated a regression of the model values and obtained an R2 = 0.9977 , which is very high. Thus we can conclude that the method gives very good results.

For time comparison we saw that determining the European option’s (with four underlying stocks) values using the Monte Carlo method with 100000 paths and 256 time steps required 661.8950 seconds. Conversely, training and generating the value of an option using PINNs took 312.8564 seconds, meaning a 52.73% improvement. Determining the American option’s (with four underlying stocks) values using the Monte Carlo method with 100000 paths and 256 time steps required 1128.2052 seconds. Conversely, training and generating the value of an option using PINNs took 223.7130 seconds which means an 80.17 improvement. Also employing a trained neural network to calculate the option value always took less than 0.2 seconds which is very fast and can be of great use in case one is racing against time. This substantial enhancement in time efficiency underscores the efficacy of neural networks, Also a very interesting observation is that NN converges faster and better for the American Option than it does for the European option, which points towards the phenomenon that the complexity of the PDE does not affect the speed and performance of PINNs negatively.

5. Conclusion

References

1. Black, F.; Scholes, M. The pricing of options and corporate liabilities. Journal of Political Economy 1973, 81, 637–654. [Google Scholar] [CrossRef]
2. Boyle, P.P. Options: A Monte Carlo approach. Journal of Financial Economics 1977, 4, 323–338. [Google Scholar] [CrossRef]
3. Broadie, M.; Glasserman, P. Estimating security price derivatives using simulation. Management Science 1996, 42, 269–285. [Google Scholar] [CrossRef]
4. Longstaff, F.A.; Schwartz, E.S. Valuing American Options by Simulation: A Simple Least-Squares Approach. Review of Financial Studies 2001, 14, 113–147. [Google Scholar] [CrossRef]
5. Cox, J.C.; Ross, S.A.; Rubinstein, M. Option pricing: A simplified approach. Journal of Financial Economics 1979, 7, 229–263. [Google Scholar] [CrossRef]
6. Boyle, P.P.; Evnine, J.; Gibbs, S. Numerical evaluation of multivariate contingent claims. Review of Financial Studies 1989, 2, 241–250. [Google Scholar] [CrossRef]
7. Moon, K.S.; Kim, W.J.; Kim, H. Adaptive lattice methods for multi-asset models. Computers & Mathematics with Applications 2008, 56, 352–366. [Google Scholar]
8. Courtadon, G. A more accurate finite difference approximation for the valuation of options. Journal of Financial and Quantitative Analysis 1983, 17, 697–703. [Google Scholar] [CrossRef]
9. Liao, W.; Khaliq, A.Q.M. High-order compact scheme for solving nonlinear Black–Scholes equation with transaction cost. International Journal of Computer Mathematics 2009, 86, 1009–1023. [Google Scholar] [CrossRef]
10. Andalaft-Chacur, A.; Ali, M.M.; Salazar, J.G. Real options pricing by the finite element method. Computers and Mathematics with Applications 2011, 16, 2863–2873. [Google Scholar] [CrossRef]
11. Kaya, D. Pricing a Multi-Asset American Option in a Parallel Environment by a Finite Element Method Approach. PhD thesis, Uppsala University, Department of Mathematics, 2011.
12. Seydel, R.U. Tools for Computational Finance, 5th ed.; Springer, 2012.
13. Wang, S.; Zhang, S.; Fang, Z. A superconvergent fitted finite volume method for Black-Scholes equations governing European and American option valuation: SUPERCONVERGENT FITTED FINITE VOLUME METHOD. Numerical Methods for Partial Differential Equations 2014, 31, 1190–1208. [Google Scholar] [CrossRef]
14. Dilloo, M.J.; Tangman, D.Y. A high-order finite difference method for option valuation. Computers & Mathematics with Applications 2017, 74, 652–670. [Google Scholar] [CrossRef]
15. Koffi, R.S.; Tambue, A. A Fitted L-Multi-Point Flux Approximation Method for Pricing Options. Computational Economics 2022, 60, 633–663. [Google Scholar] [CrossRef]
16. Mollapourasl, R.; Fereshtian, A.; Vanmaele, M. Radial Basis Functions with Partition of Unity Method for American Options with Stochastic Volatility. Computational Economics 2019, 53, 259–287. [Google Scholar] [CrossRef]
17. Soleymani, F.; Zhu, S. RBF-FD solution for a financial partial-integro differential equation utilizing the generalized multiquadric function. Computers & Mathematics with Applications 2021, 82, 161–178. [Google Scholar]
18. McCulloch, W.S.; Pitts, W. A Logical Calculus of the Ideas Immanent in Nervous Activity. Bulletin of Mathematical Biophysics 1943, 5, 115–113. [Google Scholar] [CrossRef]
19. Rumelhart, D.; Hinton, G.; Williams, R. Learning representations by back-propagating errors. Nature 1986, 323, 533–536. [Google Scholar] [CrossRef]
20. Hutchinson, J.M.; Lo, A.W.; Poggio, T. A nonparametric approach to pricing and hedging derivative securities via learning networks. The journal of Finance 1994, 49, 851–889. [Google Scholar] [CrossRef]
21. Garcia, R.; Gençay, R. Pricing and hedging derivative securities with neural networks and a homogeneity hint. Journal of Econometrics 2000, 94, 93–115. [Google Scholar] [CrossRef]
22. Culkin, R. Machine Learning in Finance: The Case of Deep Learning for Option Pricing. computer science 2017. [Google Scholar]
23. Han, J.; Jentzen, A.; Ee, W. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 2018, 115, 8505–8510. [Google Scholar] [CrossRef] [PubMed]
24. Eskiizmirliler, S.; Günel, K.; Polat, R. On the solution of the black-scholes equation using feed-forward neural networks. Computational Economics 2021, 58, 915–941. [Google Scholar] [CrossRef]
25. Ruf, J.; Wang, W. Neural networks for option pricing and hedging: a literature review. Journal of Computational Finance 2020, 24, 1–45. [Google Scholar] [CrossRef]
26. Gatta, F.; Di Cola, V.S.; Giampaolo, F.; Piccialli, F.; Cuomo, S. Meshless methods for American option pricing through Physics-Informed Neural Networks. Engineering Analysis with Boundary Elements 2023, 151, 68–82. [Google Scholar] [CrossRef]
27. Wang, X.; Li, J.; Li, J. A Deep Learning Based Numerical PDE Method for Option Pricing. Computational Economics 2023, 62, 149–164. [Google Scholar] [CrossRef]
28. Wilmott, P. Paul Wilmott on Quantitative Finance, second ed.; John Wiley and Sons, 2006.
29. Wilmott, P.; Dewynne, J.; Howison, S. Option Pricing: Mathematical Models and Computation; Oxford Financial Press, 1993.
30. Sola, J.; Sevilla, J. Importance of input data normalization for the application of neural networks to complex industrial problems. IEEE Transactions on Nuclear Science 1997, 44, 1464–1468. [Google Scholar] [CrossRef]
31. Lu, L.; Meng, X.; Mao, Z.; Karniadakis, G.E. DeepXDE: A deep learning library for solving differential equations. SIAM Journal on Scientific Computing 2019, 41, A463–A483. [Google Scholar] [CrossRef]
32. Toit, J.F.D.; Laubscher, R. Evaluation of Physics-Informed Neural Network Solution Accuracy and Efficiency for Modeling Aortic Transvalvular Blood Flow. Mathematical and Computational Applications 2023, 28, 62. [Google Scholar] [CrossRef]
33. Hozman, J.; Tichý, T. DG method for numerical pricing of multi-asset Asian options—The case of options with floating strike. Applied Mathematics 2017, 62, 171–195. [Google Scholar] [CrossRef]