pricing model of financial engineering

8/8/2019 Pricing Model of Financial Engineering

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Pricing Model of

Financial Engineering

Fang-Bo Yeh

System Control GroupDepartment of Mathematics

Tunghai University

www.math.thu.tw/~fbyeh/


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2

Glasgow

t

,t

.

1. Glasgow 2. Newcastle

3. Oxfordt 4.Groningen

5.

6.

7.

8.

9.

10.

11.

1. IEEE M. Barry Carlton Award

2.

3.


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Contents

1. Classic and Derivatives Market

2. Derivatives Pricing

3. Methods for Pricing

4. Numerical Solution for Pricing Model


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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


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5

Derivative Securities

Forward Contract :

is an agreement to buy orsell. Call Option :

gives its owner the rightbut not the obligation to

buy a specified asset on or before a specified date

for a specified price.European, American, Lookback, Asian, Capped,

Exotics..


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6

Call Option on AOL Stockon Sep. 8, you buy oneNov.call option contractwritten on AOL

contract size:100 shares

strike price:

80

maturity:

December 26

option premium:

71/8per 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 100shares of AOL stock and

pay the strike price of$8,000

otherwise, nothinghappens


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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 u 80)exercise option? no yes

cash flows (on per-share basis)

pay option premium -7.125 receive stock STpay strike price -80


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8

Call Option on AOL Stock

0AOL stock price

on December2660 8070 10090

pay-off

profit

7.125

pay-off net profit

Fang-bo Yeh


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9

Maximal Losses and Gains on Option Positions

Mathematics Finance 2003 Option Markets

Fang-Bo Yeh Tunghai Mathematics

0

long callmaximal gain:

unlimited

maximal loss:

premium

short callmaximal gain:

premium

maximal loss:

unlimited

long put

maximal gain:

strike minuspremium

maximal loss:

premium

0

0 0

short put

maximal gain:

premium

maximal loss:

strike minus

premium


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10

Simple Option Strategies: Covered Call

covered call:

the potential loss on a shortcall position is unlimited

the worst case occurs when

the stock price at maturity isvery high and the option isexercised

the easiest protection againstthis case is to buy the stock

at the same time as you writethe option

this strategy is called

covered call

covered call pay-offs:

Cost of strategy:

you receive the optionpremium C while payingthestock price S

the total cost is hence S-C



cash flows at maturity

case: ST < K ST u K

Short call - K-STlong stock ST ST

total: ST K


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11

Simple Option Strategies: Covered Call



short

call

long

stock

covered

call

K

+

=

Kpay-off

profit

K ST

premium

0


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12

Simple Option Strategies: Protective Put

protective put:

suppose you have a longposition in some asset, andyou are worried about

potential capital losses onyour position

to protect your position, youcan purchase an at-the-money

put option which allows youto 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 additionalcost ofprotection is the price ofthe option, P

the total cost is hence S+P



cash flows at maturity

case: ST < K ST u K

long stock ST STlong put K-ST -

total: K ST


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Simple Option Strategies: Protective Put



long

stock

long

put

protective

put

K

+

=

K

profit

K ST

0

pay-off

premium


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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


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Main Problem:What is the fair price for the

contract?

Ans:

(1). The expected value ofthe discountedfuture

stochasticpayoff(2). It is determined by marketforces which is

impossible have a theoreticalprice


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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).


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MethodsAssumeefficientmarket

Risk neutral

valuation and solving

conditional

expectation of the

random variable

The elimination of

randomness and

solving diffusion

equation


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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)

t

t

t dZdt

S

dSWQ !

dtrB

dB

t

t !


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Differentiable Not differentiable

Deterministic Stochastic


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Deterministic Function


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21

Stochastic Brownian Motion

t

t


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22

From Calculus to Stochastic Calculus

Calculus Stochastic Calculus

Differentiation Ito Differentiation

Integration Ito Integration

Statistics Stochastic Process

Distribution Measure

Probability Equivalent Probability


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Assume

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

exists a portfolio

such that

2). Efficient market: (observable)

If then

),( tt EH

ttttt S EHT !

ttttt ddSd EHT !

),(TTST!T ),( tt StF!T


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By assumptions (1)(2)

Itos lemma

The Black-Scholes-Merton Equation:

dZSdtF]rSr)[(

BdSdS)dF(t,

!

!

ddtt

)d (t, 2

222

21

xx

-

xx

xx

xx!

S

FrSFS

SFSr

tF 2

2

2221 !xxxxxx

)()(T, TT !


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European Call Option Price:

tTdd

tT

tTrd

dKNedNSStF

K

S

tTr

ttc

t

!

!

!

W

W

W

12

2

21

1

2

)(

1

))((ln

)()(),(


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Martingale Measure

CMG

Drift Brownian Motion Brownian Motion

,

,

***

*

tt

t

rt

t

dZSdS

SeS

tW!

!

***

*

ttt

trt

t

dZSd

e

tWHT

TT

!

!

ttt dZdtdZdt-r

dZ (! PW

Q*

),(~* ttNZr

pt WQ ),0( tZ pt

),0(** tZ pt

)(

)()(

*

*

*

*

2

21

Tpt

tZ

pp

E

YeEYE t

TT

PP

!

(


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Where

)]([

)]([),(

)]([

*

*)(

*

Tp

r

Tpt

Tr

tt

T

rt

pt

rt

SfEe

SfEeStF

SfeEe

X

T

T

!

!!

!

*2

21 )(

0

*

tZtr

t

t

t

t

eSS

dZrdtSdS

WW

W

!

!

dyyxefextFyrr )()(),(

21221 )(

NWXXWX

g

g

!

)1,0(~NN


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Main Result

)]([),( *)(

Tp

tTr

t SfEeStF

!

The fair price is

the expected value of the

discounted future stochastic payoff underthe new martingale measure.


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From Real world to Martingale world

Discounted Asset Price

& Derivatives Price

Under Real World Measure

is not Martingale

But

Under Risk Neutral Measure

is Martingale


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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


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31

Introduction to

Financial Mathematics (1)

Topics for2003:

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 Martingaleand Itos Calculus.

6. Numerical method and Simulations.


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References

M. Baxter, A. Rennie , Financial Calculus,Cambridgeuniversity 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, JohnWiley and Sons, 1999.

J.C. Hull , Options, Futures and other derivatives, PrenticeHall. 2002.

R. Jarrow and S. Turnbull,Derivatives Securities,Southern College Publishing, 1999.

pricing model of financial engineering

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