Pricing Model of Financial Engineering Fang-Bo Yeh System Control Group Department of Mathematics...

Pricing Model of Financial Engineering

Fang-Bo Yeh

System Control Group

Department of Mathematics

Tunghai Universitywww.math.thu.tw/~fbyeh/

2

葉芳柏 教授 英國 Glasgow 大學 數學博士 專長 控制工程理論、科學計算模擬、飛彈導引、泛函分析、財務金融工程 現任 東海大學數學系教授 國立交通大學應用數學研究所 , 財務金融研究所兼任教授 亞洲控制工程學刊編輯 .

歷任   1. 英國 Glasgow 大學數學系客座教授 2. 英國 Newcastle 大學數學統計系客座教授   3. 英國 Oxford 大學財務金融中心研究 4. 荷蘭國立 Groningen 大學資訊數學系客座授    5. 日本國立大阪大學電子機械控制工程系客座教授 6. 成功大學航空太空研究所兼任教授 7. 航空發展中心顧問 8. 東海大學數學系主任、所長、理學院院長、教務長    9. 國科會中心學門審議委員、諮議委員 10. 教育部大學評鑑委員 11. 國際數學控制學刊編輯委員 學術獎勵 1. 國際電機電子工程師學會獎 IEEE M. Barry Carlton Award 2. 國際航空電子系統傑出論文獎 3. 國科會傑出研究獎 

Contents

1. Classic and Derivatives Market

2. Derivatives Pricing

3. Methods for Pricing

4. Numerical Solution for Pricing Model

Classic and Derivatives Market

• Underlying Assets

Cash Commodities ( wheat, gold )

Fixed income ( T-bonds )

Stock Equities ( AOL stock )

Equity indexes ( S&P 500 )

Currency Currencies ( GBP, JPY )

• Contracts

Forward & Swap :

FRAs ,

Caps, Floors,

Interest Rate Swaps

Futures & Options :

Options,

Convertibles Bond Option, Swaptions

5

Derivative Securities

• Forward Contract :

is an agreement to buy or sell.• Call Option : gives its owner the right but not the obligation to buy a specified asset on or before a specified date for a specified price. European, American, Lookback, Asian, Capped, Exotics…..

6

Call Option on AOL Stock

on Sep. 8, you buy one Nov.call option contract written on AOL

contract size: 100 shares

strike price: 80

maturity: December 26

option premium: 71/8 per share

on Sep. 8,…• you pay the premium of

$712.50 at maturity on December 26,…

• if you exercise the option, you take delivery of 100

shares of AOL stock and pay the strike price of

$8,000• otherwise, nothing happens

7

Call Option on AOL Stock

denote by ST the price of AOL stock on December 26 date Sep. 8 December 26 scenario (if ST < 80) (if ST 80) exercise option? no yes cash flows (on per-share basis) pay option premium -7.125 receive stock ST

pay strike price -80

8

Call Option on AOL Stock

0AOL stock price

on December 2660 8070 10090

pay-off profit

7.125

pay-off net profit

Fang-bo Yeh

9

Maximal Losses and Gains on Option Positions

Mathematics Finance 2003 Option Markets

Fang-Bo Yeh Tunghai Mathematics

0

long callmaximal gain: unlimitedmaximal loss: premium

short callmaximal gain: premiummaximal loss: unlimited

long putmaximal gain: strike minus premiummaximal loss: premium

0

0 0

short putmaximal gain: premiummaximal loss: strike minus premium

10

Simple Option Strategies: Covered Call

covered call:• the potential loss on a short call

position is unlimited• the worst case occurs when the

stock price at maturity is very high and the option is exercised

• the easiest protection against this case is to buy the stock at the same time as you write the option

this strategy is called “covered call”

• covered call pay-offs:

• Cost of strategy: you receive the option

premium C while paying the stock price S

the total cost is hence S-C

Mathematics Finance 2003 Option Markets

Fang-Bo Yeh Tunghai Mathematics

cash flows at maturity

case: ST < K ST KShort call - K-ST long stock ST ST

total: ST K

11

Simple Option Strategies: Covered Call

Mathematics Finance 2003 Option Markets

Fang-Bo Yeh Tunghai Mathematics

short call

longstock

covered call

K

+

=

Kpay-off

profit

K ST

premium

0

12

Simple Option Strategies: Protective Putprotective put:• suppose you have a long

position in some asset, and you are worried about potential capital losses on your position

• to protect your position, you can purchase an at-the-money put option which allows you to sell the asset at a fixed price should its value decline

this strategy is called “protective put”

• protective put pay-offs:

• cost of strategy: the additional cost of

protection is the price of the option, P

the total cost is hence S+P

Mathematics Finance 2003 Option Markets

Fang-Bo Yeh Tunghai Mathematics

cash flows at maturity

case: ST < K ST Klong stock ST ST long put K-ST -

total: K ST

13

Simple Option Strategies: Protective Put

Mathematics Finance 2003 Option Markets

Fang-Bo Yeh Tunghai Mathematics

longstock

longput

protective put

K

+

=

K

profit

K ST

0

pay-off

premium

14

Financial Engineering

• Bond + Single Option

S&P500 Index Notes

• Bond + Multiple Option

Floored Floating Rate Bonds, Range Notes

• Bond + Forward (Swap) ;Structured Notes

Inverse Floating Rate Note

• Stock + Option

Equity-Linked Securities, ELKS

Main Problem:

What is the fair price for the contract?

Ans:

(1). The expected value of the discounted future stochastic payoff

(2). It is determined by market forces which is impossible have a theoretical price

Main result:

• It is possible

• have a theoretical price which is consistent with the underlying prices given by the market

• But

• is not the same one as in answer (1).

Methods Assume efficient market

• Risk neutral valuation and solving conditional expectation of the random variable

• The elimination of randomness and solving diffusion equation

Problem Formulation

Contract F :

Underlying asset S, return

Future time T, future pay-off f(ST)

Riskless bond B, return

Find contract value

F(t, St)

tt

t dZdtS

dS

dtrB

dB

t

t

Differentiable Not differentiable

Deterministic Stochastic

20

Deterministic Function

21

Stochastic Brownian Motion

tZ

tZ

22

From Calculus to Stochastic Calculus

Calculus Stochastic Calculus

Differentiation Ito Differentiation

Integration Ito Integration

Statistics Stochastic Process

Distribution Measure

Probability Equivalent Probability

Assume

1). The future pay-off is attainable: (controllable)

exists a portfolio

such that

2). Efficient market: (observable)

If then

),( tt

ttttt S B

ttttt ddSd B

),( TT STF ),( tt StF

By assumptions (1)(2)

Ito’s lemma

The Black-Scholes-Merton Equation:

dZ S σdt F]r S r)[(μ

B d S d S)dF(t,

δδ

αδ

ZdS

FS dt

S

FS

FS

t

FS)dF(t, σσμ

2

222

21

S

Fr S

FS

S

FSr

t

F2

222

21 σ

)f(S)SF(T, TT

European Call Option Price:

tTdd

tT

tTrd

dKNedNSStF

KS

tTrttc

t

12

221

1

2)(

1

))((ln

)( )(),(

Martingale Measure

CMG

Drift Brownian Motion Brownian Motion

,

,***

*

tt

trt

t

dZSdS

SeS

t

***

*

ttt

trt

t

dZSd

e

t

ttt dZdtdZdt-r

dZ *

) ,(~* ttNZ rpt

) ,0(~ tNZ pt

) ,0(~ ** tNZ pt

)(

)( )(*

**

*

221

Tpt

tZpp

E

YeEYE t

Where

)]([

)]([),(

)]([

*

*)(

*

Tpr

TptTr

tt

Trt

ptrt

SfEe

SfEeStF

SfeEe

*221 )(

0

*

tZtrt

tt

t

eSS

dZrdtS

dS

dyyxefextF yrr )()(),(21

221 )(

)1 ,0(~ N

Main Result

)]([),( *)(

TptTr

t SfEeStF

The fair price is

the expected value of the

discounted future stochastic payoff under

the new martingale measure.

29

From Real world to Martingale world

Discounted Asset Price & Derivatives Price

Under Real World Measureis not Martingale But Under Risk Neutral Measure is Martingale

30

Numerical Solution

Methods

Finite Difference Monte Carlo Simulation

• Idea: Idea:

Approximate differentials Monte Carlo Integration

by simple differences via Generating and sampling

Taylor series Random variable

31

Introduction to Financial Mathematics (1)

Topics for 2003:

1. Pricing Model for Financial Engineering.

2. Asset Pricing and Stochastic Process.

3. Conditional Expectation and Martingales.

4. Risk Neutral Probability and Arbitrage Free Principal.

5. Black-Scholes Model : PDE and Martingale

and Ito’s Calculus.

6. Numerical method and Simulations.

32

References

• M. Baxter, A. Rennie , Financial Calculus,Cambridge university press, 1998

• R.J. Elliott and P.E. Kopp, Mathematics of Financial Markets, Springer Finance, 2001

• N.H. Bingham and R. Kiesel , Risk Neutral Evaluation, Springer Finance, 2000.

• P. Wilmott, Derivatives, John Wiley and Sons, 1999.• J.C. Hull , Options, Futures and other derivatives, Prentice

Hall. 2002.• R. Jarrow and S. Turnbull, Derivatives Securities, Souther

n College Publishing, 1999.