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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解與a解之間的密切聯系

PDE獲取期權價格或解決其相關BSDE。最近

BSDE方法。在這種方法中，深度學習用于學習一些知識

1 Introduction

the underlying asset has breached a predetermined barrier price. For a simple

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

option for a lower premium. There are different methods to solve option prices,

simulations. Recently, a different approach using machine learning has been

proposed.

method was proposed for solving parabolic partial differential equations with

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

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

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

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

mathematical foundation of this approach is based on a Kolmogorov-Arnold

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

this deep-learning method include  and . In , a different way of simulating

is proposed. Rather than using a neural network to approximate the derivative

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

two new types of structures are proposed in  . These problems are in the

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

by an equivalent BSDE.

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

term is added to the loss function to take into account the free boundary condition in order to solve the RBSDE. Again, machine learning can be used in

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

options differently. Rather than using RBSDE with a penalty function or exercise options at a boundary, we incorporated the boundary conditions as terminal

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

In this paper, we organize as follows. In section 2, we present the standard

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

to a Cauchy-Dirichlet problem. In section 4, we present numerical considerations

2

2 Basic method to solve BSDE by machine learning

We briefly introduced the deep-learning-based numerical BSDE algorithm proposed in . We start from an FBSDE, which is first proposed in .

Xt = X0 +

? t

0

bs(Xs)ds +

? t

0

σs(Xs)dWs

t

? T

t

ZsdWs

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

)(ti+1 ? ti) + σti

(Xti

)(Wti+1 ? Wti

) (1)

Yti+1 ≈ Yti ? fti

(Xti

, Yti

, Zti

(Wti+1 ? Wti

) (2)

Note that we have made the backward process to be forward; this is a commonly

used technique in treating FBSDEs. This set of equations has the following

interpretation on a given path: Xti

is the underlying price; Yti

is the option

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

——富國銀行公司模型風險部

1.

S形函數。

中提出了兩種新型結構。這些問題都存在

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

（Xti）

)（ti+1）? ti）+σti

（Xti）

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

1導言

?您將了解攻擊者在程序不存在漏洞時利用安全漏洞的不同方式

?通過本課程，您將更好地了解如何編寫更安全的程序，如

?您將更深入地了解x86-64指令的編碼方式。

?您將獲得更多使用GDB和OBJDUMP等調試工具的經驗。

2物流

2.1獲取文件

ctarget：易受代碼注入攻擊的可執行程序

hex2raw：用于生成攻擊字符串的實用程序。

2.

2.2要點

?您必須在與生成目標的機器類似的機器上執行任務。

?您的解決方案可能不會使用攻擊繞過程序中的驗證代碼。明確地

–功能touch1、touch2或touch3的地址。

–注入代碼的地址

–小工具場中一個小工具的地址。

?您只能從文件rtarget構建小工具，其地址介于函數start_farm和end_farm的地址之間。

3個目標項目

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1個未簽名的getbuf（）

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