A deep learning method for pricing high-dimensional American-style options via state-space partition

Abstract

This paper proposes a deep learning approach for solving optimal stopping problems and high-dimensional American-style options pricing problems. Through state-space partition, the method does not require recalculation of the structure of networks when the price of the asset changes, which makes tracking valuation more efficient. This paper also offers theoretical proof for the existence of a deep learning network that can determine the optimal stopping time via state-space partition. We present convergence proofs for the estimators and also test the method on Bermuda max-call options as examples.

Appendices

Appendix A: Structure of deep neural networks and proof of Lemmas

Since the stopping decision functions $f_i^j, j=1, \ldots , a_i, i=0,1, \ldots , N-1,$ can only take discrete values, based on the notion of piecewise smooth functions proposed by Imaizumi and Fukumizu (2019) and boundary fragment classes developed by Dudley (1974), we introduce a special class of functions that take 1 on some regions of the state space whose boundaries are formed by a series of smooth functions. For that purpose, we introduce the following definition of $\left( p,C\right)$-smoothness and edge function.

Definition 1

Let $p=q+s$ for some $q \in {\mathbb {N}}_0$ and $0<s \le 1$. A function $m: {\mathbb {R}}^d \rightarrow {\mathbb {R}}$ is called $\left( p, C\right)$-smooth, if for every $\alpha =\left( \alpha _1, \ldots , \alpha _d\right) \in {\mathbb {N}}_0^d$ with $\sum _{j=1}^d \alpha _j=q$, the partial derivative $\partial ^q m /\left( \partial x_1^{\alpha _1} \ldots \partial x_d^{\alpha _d}\right)$ exists and satisfies

$$\begin{aligned} \left| \frac{\partial ^q m}{\partial x_1^{\alpha _1} \ldots \partial x_d^{\alpha _d}}\left( {\textbf{x}}\right) -\frac{\partial ^q m}{\partial x_1^{\alpha _1} \ldots \partial x_d^{\alpha _d}}\left( {\textbf{z}}\right) \right| \le C\Vert {\textbf{x}}-{\textbf{z}}\Vert ^s, \end{aligned}$$

for all ${\textbf{x}}, {\textbf{z}} \in {\mathbb {R}}^d$, where $\Vert \cdot \Vert$ denotes the Euclidean norm.

Definition 2

Let $D \in \{1, \ldots , d\}$, $x \in {\mathbb {R}}^d$. A function is called D-edge function, if it is a element of the set

$$\begin{aligned} \begin{aligned} {\mathcal {H}} \left( p, D\right) :=\,&\{I_{\{f(x)>0\}}: f(x)=g\left( x_{i_2}, \ldots x_{i_D}\right) -x_{i_1},\\&g \text{ is } \left. \left( p, C\right) \text {-smooth, and}\left\{ i_1, \ldots , i_D\right\} \subset \{1, \ldots , d\}\right\} . \end{aligned} \end{aligned}$$

For $\lambda , J, R \in {\mathbb {N}}$ and ${\mathcal {P}} \subseteq [1, \infty ) \times {\mathbb {N}}$, we also define

$$\begin{aligned}{} & {} {\mathcal {G}}\left( R, {\mathcal {P}}\right) :=\left\{ f\left( x\right) =\prod _{i=1}^R f_i\left( x\right) : f_i\left( x\right) \in {\mathcal {H}} \left( p_i, D_i\right) \text{ and } \left( p_i, D_i\right) \in {\mathcal {P}}\right\} ,\\{} & {} {\mathcal {K}}\left( \lambda , J, {\mathcal {P}}\right) :=\left\{ f\left( x\right) =\max _{1 \le j \le J} f_j(x): f_j(x) \in {\mathcal {G}}\left( R_j, {\mathcal {P}}_j\right) , \lambda =\sum _{j=1}^J R_j, {\mathcal {P}}_j \subseteq {\mathcal {P}}\right\} . \end{aligned}$$

Obviously, the functions in the ${\mathcal {K}}\left( \lambda , J, {\mathcal {P}}\right)$ take 1 for some blocks and 0 for the rest of the regions and the functions themselves are not differentiable or even discontinuous at the edges of these regions. Thus, $f_{i}^{j}\left( x\right) \in {\mathcal {K}}\left( \lambda _i, J_i, {\mathcal {P}}\right)$, where $\lambda _i, \ J_i \in {\mathbb {N}}$ and ${\mathcal {P}} \subseteq [1, \infty ) \times {\mathbb {N}}$. It is worth noting that each edge function in the definition of $f_i^j(x)$ can have a different smoothness $p_{n, i}=$ $q_{n, i}+s_{n, i}$ and a different input dimension $D_{n, i}$, where $0 \le n \le N, \ 1 \le i \le \lambda _n, \ \left( p_{n, i}, D_{n, i}\right) \in {\mathcal {P}}$.

Lemma 8

Let $\lambda , J \in {\mathbb {N}}$, ${\mathcal {P}} \subseteq [1, \infty ) \times {\mathbb {N}}$. For arbitrary $f\left( x \right) \in {\mathcal {K}}\left( \lambda , J, {\mathcal {P}}\right)$, there exists a set of $\left( p, C\right)$-smooth functions $f_i\left( x \right) , i=1,\ldots ,$ $\sum _{j=1}^J R_j$, such that

$$\begin{aligned}{} & {} \phi _{1, i}=f_i\left( x \right) ,\ \phi _{2, i}={\textbf{1}}_{\left( 0, \infty \right) }\left( \phi _{1, i}\right) , \ \phi _{3, j}=\prod _{i=\sum _{m=1}^{j-1} R_m+1}^{\sum _{m=1}^j R_m} \phi _{2, i},\\{} & {} \phi _4=\sum _{j=1}^J \phi _{3, j}, \ f\left( x \right) ={\textbf{1}}_{(0, \infty )}\left( \phi _4 \right) . \end{aligned}$$

Proof

Using the definition of ${\mathcal {K}}\left( \lambda , J, {\mathcal {P}}\right)$, the proof of Lemma 8 can be obtained directly. $\square$

Lemma 9

For $x \in \left[ 0,1\right] ^d$, there exists a neural network $f_{\text {mult}}$ with the network architecture $\left( \left\lceil \llceil \log _2 d\right\rceil \rrceil , 18 d\right)$ such that

$$\begin{aligned} f_{\text{ mult } }(x) {\left\{ \begin{array}{ll}>0, &{} \text{ if } x_{i}>0, \forall \ 0 \le i \le d, \\ =0, &{} \text{ otherwise. } \end{array}\right. } \end{aligned}$$

Proof

Based on Lemma 4 and Lemma 20 proposed by Kohler and Langer (2021), there exists a network $f_{sq}$ with the network architecture $\left( 1,18\right)$ satisfying $f_{sq}\left( m, n\right) =|m+n|-|m-n|$, for $m,n \in \left[ 0,1\right]$. Let

$$\begin{aligned} w=\left\lceil \llceil \log _2 d\right\rceil \rrceil , \ \left( z_1, \ldots , z_{2^w}\right) =\left( x^{1}, x^{2}, \ldots , x^{d}, 1, \ldots , 1\right) . \end{aligned}$$

In the first layer of $f_{\text {mult}}$, we compute

$$\begin{aligned} f_{sq}\left( z_1, z_2\right) , \ f_{sq}\left( z_3, z_4\right) , \ldots , f_{ q}\left( z_{2 w-1}, \ z_{2 q}\right) , \end{aligned}$$

which can be done by $18 \cdot 2^{w-1} \le 18 d$ neurons. The output of the first layer is a vector of length $2^{w-1}$. This process is repeated for the output vectors until the output is a one-dimensional vector. If $m=0$, then we have $f_{sq}\left( m, n\right) =|n|-|n|=0$. Based on mathematical induction, if a element of the vector x is equal to 0, $f_{\text{ mult } }(x)=0$. $\square$

Lemma 10

For $x \in \left[ 0,1\right]$, there exists a neural network $f_{\text {demo}}$ with the network architecture $\left( 3,2\right)$ such that

$$\begin{aligned} f_{\text {demo}}\left( x\right) ={\textbf{1}}_{\left( \frac{1}{c_2}, \infty \right) }\left( x\right) , \end{aligned}$$

where $c_2 \ge 1$ is a constant.

Proof

Let $f_{\text {demo}}\left( x\right) =-1 \sigma \left( -c_2\sigma \left( x\right) +1\right) +1$, where $\sigma \left( x\right)$ is the ReLU activation function, finishing the proof. $\square$

Lemma 11

Under Assumption 2, let $f\in {\mathcal {K}}(\lambda , J, {\mathcal {P}})$,where $\lambda , J \in {\mathbb {N}}$, ${\mathcal {P}} \subseteq [1, \infty ) \times {\mathbb {N}}$. Then let $Q_i \in {\mathbb {N}}, \ i=1,\ldots ,\lambda$, sufficiently large, and there exist $D_{i }, \ p_{i}, \ q_{i}, \ R_{i}, \ i=1,\ldots ,\lambda ,$ and a neural network $f^\theta \in {\mathcal {F}} \left( L,r\right)$ with the property that

$$\begin{aligned} \Vert f(x)-f^\theta \left( x\right) \Vert _{2,\left[ -c_1,c_1\right] ^d} \le c_{3} c_1^{4\left( \max _{0 \le i \le \lambda }q_{i }+1\right) } \max _{0 \le i \le \lambda } Q_i^{-2 p_i}, \end{aligned}$$

where $c_3$ is a constant,

$$\begin{aligned} \begin{aligned} L =&\max _{1 \le i \le \lambda } \left[ 5 Q_i^{D_i}+\left\lceil \llceil \log _4\left( Q_i^{2 p_i+4{D_i} \left( q_i+1\right) } e^{4\left( q_i+1\right) \left( Q_i^{D_i}-1\right) }\right) \right\rceil \rrceil \right. \\&\left. \left\lceil \llceil \log _2(\max \{q_i, {D_i}\}+1)\right\rceil \rrceil +\left\lceil \llceil \log _4\left( Q_i^{2 p_i}\right) \right\rceil \rrceil \right] +4+\left\lceil \llceil \log _2 d\right\rceil \rrceil , \\ r =&\sum _{i=1}^\lambda 132 \cdot 2^{D_i} \left\lceil \llceil e^{D_i}\right\rceil \rrceil \left( \begin{array}{c}{D_i}+q_i \\ {D_i}\end{array}\right) \max \left\{ q_i+1, {D_i}^2\right\} . \end{aligned} \end{aligned}$$

Proof

According to Lemma 8, there exists a set of functions $f_i\left( x \right)$, $i=1,\ldots ,$ $\sum _{j=1}^J R_j,$ which is $\left( p_i,C \right)$-smooth, respectively, $p_i=q_i+s_i$ for some $q_i \in {\mathbb {N}}_0$ and $0<s_i \le 1$. Based on Theorem 2 introduced by Kohler and Langer (2021), there exists ${\hat{\phi }}_{1, i}$ with the network architecture $\left( L_i,r_i\right)$ such that

$$\begin{aligned} \left\| f_i(x)-{\hat{\phi }}_{1, i}\left( x\right) \right\| _{\infty ,\left[ -c_1,c_1\right] ^d} \le c^{i}_{2} {c_1}^{4\left( q_{i}+1\right) } Q_i^{-2 p_i}, \end{aligned}$$

where $c_2^i$ is a constant,

$$\begin{aligned} \begin{aligned} L_i =\,&5 Q_i^{D_i}+\left\lceil \llceil \log _4\left( Q_i^{2 p_i+4{D_i} \left( q_i+1\right) } e^{4\left( q_i+1\right) \left( Q_i^{D_i}-1\right) }\right) \right\rceil \rrceil \left\lceil \llceil \log _2(\max \{q_i, {D_i}\}+1)\right\rceil \rrceil \\&+\left\lceil \llceil \log _4\left( Q_i^{2 p_i}\right) \right\rceil \rrceil , \\ r_i =\,&132 \cdot 2^{D_i} \left\lceil \llceil e^{D_i}\right\rceil \rrceil \left( \begin{array}{c}{D_i}+q_i \\ {D_i}\end{array}\right) \max \left\{ q_i+1, {D_i}^2\right\} . \end{aligned} \end{aligned}$$

Without loss of generality, for $1 \le u \le D_{i}$, let $x_{i_1}=x_{u}$, $g_i=f_i+x_1$ and $\phi _{1, i}^{\prime }={\hat{\phi }}_{1, i}+x_{1}$,

$$\begin{aligned} \begin{aligned}&\left\| {\textbf{1}}_{\left\{ f_i>0\right\} }-{\textbf{1}}_{\left\{ {\hat{\phi }}_{1, i}>0\right\} }\right\| _{2,\left[ -c_1,c_1\right] ^d}^2 \\&\quad =\int \left( {\textbf{1}}_{\left\{ f_i>0\right\} }-{\textbf{1}}_{\left\{ {\hat{\phi }}_{1, i}>0\right\} }\right) ^2 d x \\&\quad =\int \int _{-c_1}^{c_1} {\textbf{1}}_{\left\{ f_i>0, {\hat{\phi }}_{1, i}\le 0\right\} }+{\textbf{1}}_{\left\{ f_i\le 0, {\hat{\phi }}_{1, i}>0\right\} } d x_{1} d x_{2} \ldots d x_{D_i}. \\ \end{aligned} \end{aligned}$$

For fixed $\left( x_{2}, \ldots , x_{D_i}\right) \in \left[ -c_1, c_1\right] ^{D_i-1}$, ${\textbf{1}}_{\left\{ f_i>0, {\hat{\phi }}_{1, i}<0\right\} }={\textbf{1}}_{\left[ \phi _{1, t}^{\prime }, g_i\right) }$. Thus,

$$\begin{aligned} \int _{-a}^a {\textbf{1}}_{\left\{ f_i>0, \phi _{1, i} \le 0\right\} } d x_{1} \le \left( g_i-\phi _{1, i}^{\prime }\right) \vee 0. \end{aligned}$$

Similarly,

$$\begin{aligned} \int _{-a}^a {\textbf{1}}_{\left\{ f_i \le 0, \phi _{1, i}>0\right\} } d x_{1} \le \left( \phi _{1, i}^{\prime }-g_i\right) \vee 0. \end{aligned}$$

Notice that $\left( b \vee 0\right) +\left( -b \vee 0\right) =|b|$, for any $x \in \left[ -c_1, c_1\right] ^d$, we have

$$\begin{aligned} \begin{aligned}&\int \int _{-c_1}^{c_1} {\textbf{1}}_{\left[ f_i>0, {\hat{\phi }}_{1, i} \le 0\right\} }+{\textbf{1}}_{\left\{ f_i \le 0, {\hat{\phi }}_{1, i}>0\right\} } d x_{1} d x_{2} \ldots d x_{D_i} \\&\quad \le \int \left( \left( g_i-\phi _{1, i}^{\prime }\right) \vee 0\right) +\left( \left( \phi _{1, i}^{\prime }-g_i\right) \vee 0\right) d x_{2} \ldots d x_{D_i} \\&\quad \le c_{4}^i {c_1}^{4\left( q_i+1\right) } Q_i^{-2 p_i}, \end{aligned} \end{aligned}$$

for a constant $c_4^i>0$.

Let

$$\begin{aligned} f^\theta \left( x \right) ={\textbf{1}}_{(0, \infty )} \left( \sum _{j=1}^J \left( \prod _{i=\sum _{i=1}^{j-1} R_i+1}^{\sum _{i=1}^j R_i} \left( {\textbf{1}}_{(0, \infty )} \left( {\hat{\phi }}_{1, i}\left( x\right) \right) \right) \right) \right) . \end{aligned}$$

By Lemma 9 and Lemma 10, given a sufficiently small $c_2$, $\prod \left( {\textbf{1}}_{(0, \infty )}\left( x \right) \right)$ and ${\textbf{1}}_{(0, \infty )}\left( x \right)$ can be implemented through networks $f_{\text{ mult } }(x)$ and $f_{\text {demo}}\left( x\right)$. Therefore, $f^\theta \in {\mathcal {F}} \left( L,r\right)$, where

$$\begin{aligned} \begin{aligned} L =&\max _{1 \le i \le \lambda } \left[ 5 Q_i^{D_i}+\left\lceil \llceil \log _4\left( Q_i^{2 p_i+4{D_i} \left( q_i+1\right) } e^{4\left( q_i+1\right) \left( Q_i^{D_i}-1\right) }\right) \right\rceil \rrceil \right. \\&\left. \left\lceil \llceil \log _2(\max \{q_i, {D_i}\}+1)\right\rceil \rrceil +\left\lceil \llceil \log _4\left( Q_i^{2 p_i}\right) \right\rceil \rrceil \right] +4+\left\lceil \llceil \log _2 d\right\rceil \rrceil , \\ r =&\sum _{i=1}^\lambda 132 \cdot 2^{D_i} \left\lceil \llceil e^{D_i}\right\rceil \rrceil \left( \begin{array}{c}{D_i}+q_i \\ {D_i}\end{array}\right) \max \left\{ q_i+1, {D_i}^2\right\} . \end{aligned} \end{aligned}$$

Moreover,

$$\begin{aligned} \begin{aligned}&\Vert f(x)-f^\theta \left( x\right) \Vert _{2,\left[ -c_1,c_1\right] ^d} \\&\quad \le \sum _{j=1}^J\left\| \prod _{i=\sum _{i=1}^{j-1} R_i+1}^{\sum _{i=1}^j R_i} I_{\left\{ f_i>0\right\} }(x)-\prod _{i=\sum _{i=1}^{j=-1} R_i+1}^{\sum _{i=1}^j R_i} I_{\left\{ {\hat{\phi }}_{1, i}>0\right\} }(x)\right\| _{2,\left[ -c_1,c_1\right] ^d}\\&\quad \le c_{5} \max _{1 \le i \le \lambda }\left\| f_i(x)-{\hat{\phi }}_{1, i}\left( x \right) \right\| _{2,\left[ -c_1,c_1\right] ^d} \\&\quad \le c_{3} a^{4\left( \max _{1 \le i \le \lambda }q_{i}+1\right) } \max _{1 \le i \le \lambda } Q_i^{-2 p_i}, \end{aligned} \end{aligned}$$

where $c_3,c_5$ are constants. $\square$

Remark 4

Lemma 11 illustrates that a sufficiently deep neural network can approximate the decision function very well. Based on Theorem 2 introduced by Kohler and Langer (2021), a sufficiently wide neural network can also yield similar results. More specifically, replacing L and r with

$$\begin{aligned} \begin{aligned} L =\,&9+\left\lceil \llceil \log _4\left( \max _{1 \le i \le \lambda } Q_i^{2 p_i}\right) \right\rceil \rrceil \left( \left\lceil \llceil \log _2\left( \max _{1 \le i \le \lambda }\left\{ D_{i }, q_{i }\right\} +1\right) \right\rceil \rrceil +1\right) \\&+\left\lceil \llceil \log _2 \max _{1 \le j \le J}R_{j }\right\rceil \rrceil , \\ r =\,&\sum _{i=1}^\lambda 2^{D_i} 64\left( \begin{array}{c} D_i+q_i \\ D_i \end{array}\right) D_i^2\left( q_i+1\right) Q_i^{D_i}, \end{aligned} \end{aligned}$$

the Lemma 11 still holds.

Appendix B: Flow chart and pseudocode for DLSSPM

Fig. 2

Main steps in a DLSSPM implementation

In Fig. 2, a flow chart is given to help understand the main steps in the implementation of DLSSPM. Detailed pseudocode for the DLSSPM implementation for multi-asset maximum call options is also provided, where underlying assets follow correlated geometric Brownian motions. This pseudocode is given based on the following assumptions:

• The time to maturity is uniformly divided into N intervals

• Points in a Sobol sequence are available by calls to the function Sobol

• Random standard normal numbers are available by calls to the function RSN

• Inverse to a given standard normal is available by calls to the function InvN

• Cholesky factorization can be implemented by calls to the function Cholesky

• Gradient-based optimal parameters are available by calls to the function GBP

• Discount factors are available by calls to the function Discount

• Finding the position of the closest element in the sequence to a given number can be achieved by calls to the function Near

• Fully connected deep neural networks with ReLU as the activation function, where ${\textbf{1}}_{\left( 0,\infty \right) }$ is applied to the output, are available by calls to the function DNNs

The list of data variables used in the pseudocodes are as follows:

• T is the maturity time of the option

• K is the strike price of the option

• d is the number of underlying assets for the option

• $S_0$ is a d-vector holding initial asset prices

• $\sigma$ is a d-vector holding asset volatilities

• $\delta$ is a d-vector holding dividend yields

• n is the number of simulated price paths for training

• N is the number of uniform time intervals for [0, T]

• $K_L$ is the number of simulated price paths for obtaining an option estimate

• a is a N-vector holding the number of bundles for each time node

• $\rho$ is a $d \times d$ array holding the correlation matrix

• r is the interest rate

• $\alpha$ is the learning rate

• L is a N-vector holding the length of the deep neural networks for each time node

• $\lambda$ is a $N \times \left( \max a\right)$ array for holding the edges of the deep neural networks for each time node and each bundle

• S is a $n \times N \times d$ array containing the Monte Carlo simulated price paths

• ${\widetilde{S}}$ is a $\left( \max a\right) \times N \times d$ array containing the Sobol-based Monte Carlo simulated price paths

• ${\overline{S}}$ is a $K_L \times N \times d$ array containing the Monte Carlo simulated price paths

• h is a $n \times N$ array for holding the dimension-reduced version of the Monte Carlo simulated price paths S

• ${\widetilde{h}}$ is a $\left( \max a\right) \times N$ array for holding the dimension-reduced version of the Sobol-based Monte Carlo simulated price paths ${\widetilde{S}}$

• ${\overline{h}}$ is a $K_L \times N$ array for holding the dimension-reduced version of the Monte Carlo simulated price paths ${\overline{S}}$

• p is a $n \times N$ array for payoffs at each price point of Monte Carlo simulated price paths S

• ${\overline{p}}$ is a $K_L \times N$ array for payoffs at each price point of Monte Carlo simulated price paths ${\overline{S}}$

• ${\mathcal {A}}$ is a $n \times N$ array for holding the path indexes belonging to bundles for the Monte Carlo simulated price paths S

• $\overline{{\mathcal {A}}}$ is a $K_L \times N$ array for holding the path indexes belonging to bundles for the Monte Carlo simulated price paths ${\overline{S}}$

• $\tau$ is a $n \times N$ array holding the estimated optimal stopping time along each Monte Carlo simulated price paths S and each time node

• ${\overline{\tau }}$ is a $K_L$-vector holding the estimated optimal stopping time along each Monte Carlo simulated price paths ${\overline{S}}$ at the initial moment

• $\theta$ is a $N \times \left( \max a\right)$ array holding pointers to the deep neural networks for each bundle and each time node

• V is the option price estimator

Algorithm 1

Main program of DLSSPM

Algorithm 2

Function $MCPaths\left( T, d, S_0, \sigma , \delta , \rho , n, N\right)$

Algorithm 3

Function $SobolPaths\left( T, d, S_0, \sigma , \delta , \rho , a, N\right)$

Algorithm 4

Function $DReduceN\left( S,n,N\right)$

Algorithm 5

Function $DReducea\left( S,a,N\right)$

Algorithm 6

Function $Pay \left( S, K,n,N\right)$

Algorithm 7

Function $Index\left( h, \widetilde{h},n,N \right)$

Algorithm 8

Function $Optimal\left( k,i, \mathcal {A}\left( \cdot ,i:N\right) , S\left( k,\cdot ,\cdot \right) , \theta \left( i:N, \cdot \right) \right)$

Algorithm 9

Function $Price\left( K_L,\overline{\tau },\overline{p},r\right)$