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Deep-learning based numerical BSDE method for barrier options Bing Yu? , Xiaojing Xing? , Agus Sudjianto? April 15, 2019

As is known, an option price is a solution to a certain partial differential equation (PDE) with terminal conditions (payoff functions). There is a close association between the solution of PDE and the solution of a backward stochastic differential equation (BSDE). We can either solve the PDE to obtain option prices or solve its associated BSDE. Recently a deep learning technique has been applied to solve option prices using the BSDE approach. In this approach, deep learning is used to learn some deterministic functions, which are used in solving the BSDE with terminal conditions. In this paper, we extend the deep-learning technique to solve a PDE with both terminal and boundary conditions. In particular, we will employ the technique to solve barrier options using Brownian motion bridges.

PDE獲取期權價格或解決其相關BSDE。最近

1 Introduction

A barrier option is a type of derivative where the payoff depends on whether

barrier case, an analytical pricing formula is available (see [1]). Because barrier

options have additional conditions built in, they tend to have cheaper premiums

barrier is unlikely to be reached, they may prefer to buy a knock-out barrier

ranging from an analytical solution, solving PDE numerically, and Monte Carlo

proposed.

method was proposed for solving parabolic partial differential equations with

terminal conditions, which we will call the standard framework hereafter. In this

?Corporate Model Risk, Wells Fargo

?Corporate Model Risk, Wells Fargo

1

new method, the PDE is formulated as a stochastic control problem through

a Feymann-Kac formula. In this formulation, a connection between PDE for

BSDE rather than solving PDE. The solution to the BSDE is represented by

two deterministic functions. One innovation (shown in [2]) is the use of a neural

network and deep-learning technique to learn these deterministic functions. The

be approximated by a finite composition of continuous functions of a single

realization of the theorem and he provided a concrete implementation using a

sigmoid function.

second-order backward stochastic differential equation. Other works related to

the processes in the forward-backward stochastic differential equation (FBSDE)

of a PDE solution, the network is used to directly approximate the PDE solution

and the derivative is calculated using automatic differentiation. A number of

different choices for building the neural network and learning structure and

framework of a PDE with some terminal conditions. These PDE can be solved

In these works, a BSDE is replaced by a reflected BSDE (RBSDE). A penalty

solving these problems. This approach is used in [7] to solve American options.

Bermudan Swaptions is solved by exercising the option at a boundary in [8]. In

our work, we consider barrier options. We treat boundary conditions of barrier

conditions. To our best knowledge, this approach has not previously been done.

we extend the standard framework to handle barrier options, which corresponds

concluding remarks in section 5.

2

Xt = X0 +

? t

0

bs(Xs)ds +

? t

0

σs(Xs)dWs

Yt = h(XT ) + ? T

t

fs(Xs, Ys, Zs)ds ?

? T

t

ZsdWs

Here, {Ws}0<s<T is a Brownian motion and h(XT ) is the terminal condition.

The pair (Y, Z)0<t<T solves the BSDE. It is known that there exists a deterministic function u = u(t, x) such that Yt = u(t, Xt), Zt = ?u(t, Xt)σt(Xt) and

we can use Euler scheme to approximate:

(Xti

(Xti

)(Wti+1 ? Wti

) (1)

Yti+1 ≈ Yti ? fti

(Xti

, Yti

, Zti

)(ti+1 ? ti) + Zti

(Wti+1 ? Wti

) (2)

interpretation on a given path: Xti

price and Zti

.

In the deep-learning-bas

1導言

?公司模型風險，富國銀行，必應。yu@wellsfargo.com

——富國銀行公司模型風險部

1.

S形函數。

2.

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?t

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fs（Xs，Ys，Zs）ds?

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u（t，x）解一個擬線性偏微分方程。無論是向前還是向后，

（Xti）

)（ti+1）? ti）+σti

（Xti）

)（Wti+1）? Wti

15-213, Fall 20xx The Attack Lab: Understanding Buffer Overflow Bugs Assigned: Tue, Sept. 29 Due: Thu, Oct. 8, 11:59PM EDT Last Possible Time to Turn in: Sun, Oct. 11, 11:59PM EDT

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