Deep neural network framework based on backward stochastic differential equations for pricing and hedging American options in high dimensions

Abstract

We propose a deep neural network framework for computing prices and deltas of American options in high dimensions. The architecture of the framework is a sequence of neural networks, where each network learns the difference of the price functions between adjacent timesteps. We introduce the least squares residual of the associated backward stochastic differential equation as the loss function. Our proposed framework yields prices and deltas for the entire spacetime, not only at a given point (e.g. t = 0). The computational cost of the proposed approach is quadratic in dimension, which addresses the curse of dimensionality issue that state-of-the-art approaches suffer. Our numerical simulations demonstrate these contributions, and show that the proposed neural network framework outperforms state-of-the-art approaches in high dimensions.

Keywords:

• American options
• Delta hedging
• Neural network
• Stochastic differential equations

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Although one may consider using the Longstaff-Schwartz regressed values as an estimate of the spacetime prices, figure 1 in Bouchard and Warin (2012) shows that using such regressed values as the spacetime solution is inaccurate. Alternatively, one may consider applying the Longstaff-Schwartz method repeatedly on all the spacetime points, where every point requires $M\to \mathrm{\infty }$ samples. However, this is expensive.

2 Here the proposed ‘recursively-defined’ feedforward network is not the same as the Recurrent Neural Network (RNN) in the literature, which will be explained in Section 4.1.

3 To assess $\mathbb{E}\left[{\mathrm{e}}^{-r\mathrm{\Delta }t}{v}^{n+1}\right({\overrightarrow{S}}_m^{n+1}\left)\right]$, we start with a fixed set of $\left\{{\overrightarrow{S}}_m^n\right\}$. For each point of ${\overrightarrow{S}}_m^n$, we generate multiple ${\overrightarrow{S}}_m^{n+1}$'s by (7(7) $\begin{array}{rl}(S_i{)}_m^{n+1}& =(1+(r-{\delta }_i\left)\mathrm{\Delta }t\right)(S_i{)}_m^n+{\sigma }_i(S_i{)}_m^n(\mathrm{\Delta }{W}_i{)}_m^n,\\ & n=0,\dots ,N-1,i=1,\dots ,d.\end{array}$(7) ), denoted as $\{{\overrightarrow{S}}_{m;m^{\prime }}^{n+1}|m^{\prime }=1,\dots ,M^{\prime }\}$; compute $\left\{v\right({\overrightarrow{S}}_{m;m^{\prime }}^{n+1}\left)\right\}$; and then compute the imperial average: $\mathbb{E}\left[{\mathrm{e}}^{-r\mathrm{\Delta }t}{v}^{n+1}\right({\overrightarrow{S}}_m^{n+1}\left)\right]\approx {\mathrm{e}}^{-r\mathrm{\Delta }t}\frac{1}{M^{\prime }}\sum _{m^{\prime }}v\left({\overrightarrow{S}}_{m;m^{\prime }}^{n+1}\right).$

4 Cedar is a Compute Canada cluster. For more details, see https://docs.computecanada.ca/wiki/Cedar and https://docs.computecanada.ca/wiki/Using_GPUs_with_Slurm.

5 We note that solving the equivalent one-dimensional option is not sufficient for computing the d-dimensional delta except at the symmetric points $s_1=\cdots =s_d$. Interested readers can verify this by straightforward algebra.

6 We note that even though finite difference methods yield nearly exact spacetime prices and deltas, due to the finite number of hedging intervals, the resulting relative P&Ls are not a Dirac delta distribution.

Additional information

Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada.