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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.