Wo-Chiang Lee

Wo-Chiang LeeDepartment of Finance and Banking ,Aletheia University

AI-ECON Research Group

August 20 ,2004

Computational Intelligence in the Development of Derivative’s Pricing ,Arbitrage and Hedging

Wo-Chiang Lee

Outline

The Basic concept of Option Pricing ModelThe Traditional Option Pricing ModelNNs in the Derivative PricingGP in the Derivative PricingFuzzy in the Derivative PricingConcluding Remark

Wo-Chiang Lee

Computational Intelligence for Financial Engineering and Financial Applications

Wo-Chiang Lee

Wo-Chiang Lee

The Application field of Financial DerivativesThe Application field of Financial Derivatives

Design of Financial Derivatives-Financial Engineering

Pricing

List

Arbitrage

1.Prediction

2.Arbitrage trading

3.Timing

Hedging(Risk management)

1.sd,cv,beta

2.VaR,CaR,EaR

3.Others(delta,…)

Trading

Wo-Chiang Lee

The factors of Option PricingThe factors of Option Pricing

SS :Stock price:Stock price

E E :exercise price:exercise priceP:P:

Option Option pricepriceT : time to T : time to

maturitymaturity

σσ:: volatilityvolatility

rrff: : risk free raterisk free rate

DD:dividend:dividend

Others

Wo-Chiang Lee

Model-Driven Approaches in Option Pricing

Wo-Chiang Lee

Q:Where is the true option pricing model?

Ans:God Knows.

Q:Where is the true option pricing model?

Ans:God Knows.

real data real data

ModelModel--drivendrivenapproachesapproaches

DataData--drivendrivenapproachesapproaches

fit datafit data

Wo-Chiang Lee

Model-Driven Approach Data-Driven Approach

With a certainty model. Adaptive learning

Based on some importantassumptions.

Based on natural selection ,Needn’t rely on some important assumptions.

Serve market price as true price.

By way of evolution, widely search space ,can find optimal solution.

Wo-Chiang Lee

Black-Scholes Option Pricing Formula

WhereS：underlying asset price P : option priceE：exercise priceT：time to maturity σ：volatility r:risk-free rate

TσddTσ

T2)/σ(rln(S/E)d

)N(dEe)SN(dP

12

2

1

2rT

1

−=

++=

−= − The Black-Scholesmodel is the standardapproach used for pricing financial options.

Wo-Chiang Lee

Fischer BlackBorn: 1938 Died: 1995 1959 -- earned bachelor's degree in physics

1964 -- earned PhD. from Harvard in applied math1971 -- joined faculty of University of Chicago Graduate School of Business1973 -- Published "The Pricing of Options and Corporate Liabilities"19XX -- Left the University of Chicago to teach at MIT1984 -- left MIT to work for Goldman Sachs & Co.

Wo-Chiang Lee

Myron ScholesBorn: 1941

1973 -- Published "The Pricing of Options and Corporate Liabilities"Currently works in the derivatives trading group at Salomon Brothers.

Wo-Chiang Lee

The Problems of Black-Scholes Option Pricing Model

Black-Scholes model was derived under strict assumptions that do not hold in the real world and model prices exhibit systematic biases from observed option prices

Assumes normal distribution of prices.Assumes constant volatility.Assumes constant risk-free rate.Although being theoretically strong, option prices valued by the model often differ from the prices observed in the financial markets.

How to solve the problems? Computational Intelligence.

Wo-Chiang Lee

The Volatility Models for Option Pricing

Wo-Chiang Lee

Model 1:Estimating Volatility from Historical Data

1. Take observations S0, S1, . . . , Sn at intervals of τ years

2. Define the continuously compounded return as:

3. Calculate the standard deviation, s , of the ui ’s

4.The historical volatility estimate is:

uS

Sii

i=

−

l n1

τ=σ

sˆ

Wo-Chiang Lee

Model 2:Implied Volatility

The implied volatility of an option is the volatility for which the Black-Scholes price equals the market priceThe is a one-to-one correspondence between prices and implied volatilitiesTraders and brokers often quote implied volatilities rather than dollar prices

τγτσ 2ln ×+

=

XS

I

( ) ( ) ( )( )( )σ

σσσ

fPif

ii′

−−=+1

1.Calculate the initial value of volatility

2.Recurative calculate the volatility.

Wo-Chiang Lee

Model 3:GARCH (Generalized ARCH)

•A high order GARCH(p,q) model

2211

2211

1

2

1

22

........ ptptqtqt

p

jjtj

q

iititt h

−−−−

=−

=−

++++++=

++== ∑∑σβσβεαεαϖ

σβεαϖσ

Ex: A AR(3)-GARCH(1,1) model

21

21

2

321

1219.0847.0000099.0

0085.0013.0034.0021.0

−−

−−−

++=

++−−=

ttt

ttttt rrrr

εσσ

ε

Wo-Chiang Lee

Model 4: GJR-GARCH

∑ ∑∑= =

−−−=

− +++=q

j

r

kktktkjtj

p

iiti D

1 1

22

1

22 εφεγσβασ

<≥

=−

−− newsbadif

newsgoodifD

kt

ktkt 0,1

0,0εε

1.For a leverage effect, we would see φk > 0.If φk ,the news impact is asymmetric.

2.Good news has a impact of γbad news has a impact of γ + φk

211

21

21

2−−−− +++= tttt D εφεγβσασ

EX:GJR-GARCH(1,1)

Wo-Chiang Lee

Donaldson,R.G. and M. Kamstra(1997) provide the artificial neutral networks to describe the nonlinear relationship between variables-ANN-GARCH model.

( )∑∑∑∑==

−−=

−=

− Ψ++++=s

hhth

r

kktktk

q

jjtj

p

iitit zDr

11

2

1

2

1

22 λξεφεσβασ

<≥

=−

−− 01

00

kt

ktkt if

ifD

εε

( )1

1 1,,0,0,exp1,

−

= =−

++=Ψ ∑ ∑

v

d

m

w

wdtwdhhtt Zz λλλ

( )( )( )2ε

εεE

EZ dtdt

−= −

−

[ ]1,1~21

,, +−uniformwdhλ

ANNs modelSigmoid function

Model 5: ANN-GARCH Model

Wo-Chiang Lee

Model 6:Neural network to predict volatility

Wo-Chiang Lee

Model 7:GP-Volatility Model

Chen,S-H. and C.-H.Yeh.(1997).” Using Genetic Programming to Model Volatility in Financial Time Series: The Case of Nikkei 225 and S&P 500,", in Proceedings of the 4th JAFEE International Conference on Investments and Derivatives (JIC'97), Aoyoma Gakuin University, Tokyo, Japan, July 29-31, 1997. pp. 288-306.

Wo-Chiang Lee

Artificial Neural Networks in Derivative’s Pricing ,Arbitrage and

Hedging

Wo-Chiang Lee

ANNs for Financial Applications

Wo-Chiang Lee

Neural Network Topology

Layers of neurons interconnectedNon-linear activation functionsWeights define strength of information flow

Wo-Chiang Lee

Hutchinson. J. A., Lo and T. Poggio (1994), “A Nonparametric Approach to Pricing and Hedge Derivative Structure via Learning Networks,”Journal of Finance, vol. 49, 851-889.

In this paper,they first propose a nonparametric method(data-driven) for estimating the pricing formula of a derivative asset using learning networks .

Wo-Chiang Lee

Advantages:

They don’t rely on restrictive parametric assumption ,such

as lognormality or sample –path contiunity.

They are robust to the specification errors that plague

parametric models.

They are adaptive and respond to structural changes in the

data-generating process in ways that parametric models

can not.

They are flexible enough to encompass a wide range of

derivative securities and fundamental asset price.

dynamics, yet they are relatively simple to implement.

Wo-Chiang Lee

Drawback:

The approach would be inappropriate for thinly traded

derivatives.

It also be inappropriate for newly created derivatives that

Have no similar counterparts among existing securities.

Wo-Chiang Lee

Based on desired output p and the net’s actual output pann-rbf, adjust weights to all layers wto minimize J(w) = ½|| p - pann-rbf ||2

Backpropagation: Generalization of iterative LMS approach—gradient descent on J(w)

NotesStart with random weights“Normalize” inputs to same scaleToo many hidden units can cause overfitting

Learning Networks:Radial Basis Functions(Backpropagation):

),1,/( tTESfP RBFANN −=−

Wo-Chiang Lee

Performance Measure –Tracking Error

|])([|

))()()(()()(

))0()0(()0(

),0()0(

)()(),()()(

)0()0(),0()0()0(

)()()()(

TVEevaluepresenterrorTracking

tVtVtSTVetV

VVV

bsopmFVS

tFtttStV

SFSV

tVtVtVtV

rT

RBFRBFBr

B

CBB

BSC

RBFRBFRBFS

RBFRBFRBF

CBS

−≡

−−∆−∆−−=

+−=

−=∂

∂=∆∆=

∂∂

=∆∆=

++=

ξ

τττ

Let V(t) is the dollar value of replicating portfolio at time t ,we sell one call option and undertake the usual dynamic trading strategy in stock s and bonds to hedge this call during its life.then

Wo-Chiang Lee

ANNs-Call Warrant Pricing Model

1、Traditional ANNs Model

),,,( σTESfPANN =

WhereS：underlying asset pricePANN : ANNsoption priceE：exercise priceT：time to maturity σ：volatility(implied and history )

BS model input variables

Wo-Chiang Lee

0.0417

98.4640

1.2843

0.0191

92.9434

0.5043

σ2

0.0280

40.5971

0.6929

0.0202

100.3249

0.5298

σ1

ANN(H=7)

0.0322

55.4180

0.7963

0.0197

95.8428

0.5189

σ1

ANN(H=5)

92.94343694.8808092.515SSE(Training)

0.50433.96955.5756MAE(Training)

98.4640178.2792253.5605SSE(Test)

0.04171.88832.2519RMSE(Test)

1.28431.58421.8146MAE(Test)

0.01914.14556.1351RMSE(Training)

σ2σ2σ1

MODEL Black-Scholes Option Pricing Model

Lee,Wo-Chiang(2001)，”Use Nonparametric Network to Pricing Reset Warrant”，Proceeding on 2001 Taiwan Finance Association Annual Meeting. pp.1-14.

Comparison of the BS Model and ANNs Model

INDEX

In sample

Out of sample

Wo-Chiang Lee

2.Genetic Adaptive Neural Networks(GANN) Pricing Model

),/)(,,

,,1,(1

LEESST

EvolfPgann

−

= σ

where

vol .is trading volume.

. σ1 is historical volatility .

E: is exercise price.

T is the time to maturity.

S is underlying asset price.

(S-E)/E is moneyness.

L is liquidity.

),,,2,/)(,

,,,1,(2

RPLEESS

TEvolfPgann

σ

σ

−

=

Where

vol .is trading volume

σ2 is implied volatility

E: is exercise price.

T is the time to maturity.

S is underlying asset price.

(S-E)/E is moneyness.

L is liquidity.

P is premium ratio.

R is leverage ratio.ref:Lee ,Wo-Chiang(2002),” Applied Genetic Adaptive Neural Network Approach in the Evaluation of Upper-and-Out Call Warrant", International Conference of Artificial and Computational Intelligence(ACI 2002) , September 25-27,TOKYO,JAPAN.

Wo-Chiang Lee

Genetic Adaptive Neural Networks Algorithms

Topology Optimizationnumber of hidden layers, number of hidden nodes, interconnection pattern

Weights Optimization

Control Parameter Optimization

learning rate, momentum rate, tolerance level,

Input Factors Optimization

Wo-Chiang Lee

ANNs RMSE

Combination of chromesome

Final variables

Convergence?No

Yes

Reproduction ,crossover,mutation

RMSE fitness

1:choice 0:no choice

chromesome population

Flowchart of GANN System(Input factors optimization)

Wo-Chiang Lee

ALGORITHMS ANN(BS variables)

GANN1 GANN2

Final input variables

X2,X3,X4,X5 X2,X3,X4,X6 X3,X5,X6,X7,X9

MEAN(in-sample) 0.662896 0.649376 0.345299

VARIANCE 0.003046 0.003509 1.6E-06

MAX 0.7881 0.7954 0.3521

MIN 0.5662 0.5393 0.3426

MEAN(out-sample) 0.692642 0.691678 0.497016

VARIANCE 0.004269 0.002773 0.007155

MAX 0.908 0.9116 0.7198

MIN 0.5736 0.6045 0.2983

Data drive form Lee(2002)

X1=vol

X2=σ1

X3=E

X4=T

X5=S

X6=(S-E)/E

X7=σ2

X8=L

X9=P

X10=R

Comparison of ANNs,GANN1 and GANN2

Wo-Chiang Lee

Genetic Programming in Derivative’s Pricing ,Arbitrage and

Hedging

Wo-Chiang Lee

An Adaptive Evolutionary Approach to Option Pricing via Genetic ProgrammingN. K. Chidambaran, Chi-Wen Jevons Lee, and Joaquin R. Trigueros1998 Conference on Computational Intelligence for Financial Engineering

Propose a new methodology of Genetic Programming for better approximating the elusive relationship between the option price and its contract terms and properties of the underlying stock price. I.e. to develop an adaptive evolutionary model of option pricing, is also data driven and non-parametric

This method requires minimal assumptions and can easily adapt to changing and uncertain economic environments

strongly encouraging and suggest that the Genetic Programming approach works well in practice.

Wo-Chiang Lee

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Limit the complexity of the problem by setting a maximum depth size of 17 for the trees used to represent formulas. The search space is, however, still very large,

A 17 deep tree is a popular number used to limit the size of tree sizes ractically, we chose the maximum depth size possible without running into excessive computer run times. Note that the Black-Scholes formula is represented by a tree of depth size 12. A depth size of 17, therefore, is large enough toaccommodate complicated option pricing formulas and works in practice.

Wo-Chiang Lee

Wo-Chiang Lee

Wo-Chiang Lee

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it is a non-parametric data driven approach and requires minimal assumptions. We thus avoid the problems associated with making specific assumptions regarding the stock price process.The Genetic Programming method uses options price data and extracts the implied pricing equation directly.

Second, the Genetic Programming method requires less data than other numerical techniques such as Neural Networks (Hutchinson,Lo, and Poggio (1994)).

The flexibility in adding terms to the parameter set used to develop the functional approximation can also be used to examine whetherfactors beyond those used in this study, for example, trading volume, skewness and kurtosis of returns, and inflation, are relevant to option pricing. The self-learning and self-improving feature also makes the method robust to changes in the economicenvironment.

Wo-Chiang Lee

Chen,Shu-Heng,Lee,Wo-Chiang and Yeh-Chia-Shen(1999),” Hedging Derivative Securities with Genetic Programming”, International Journal of Intelligent Systems in Accounting Finance & Management, Vol.4,No.8, pp.237-251

Wo-Chiang Lee

--**

**

r+0.5 r+0.5 σ2 τ

**σ

τ

**

**τ

++

CDFCDF

//

LOGLOG **τ

**sqrtsqrt

τ

S/ES/E CDFCDF

//

++

LOGLOG

S/ES/E

expexp

sqrtsqrt

--rr

S/ES/E

σ

rr--0.5 0.5 σ2

Genetic Programming in Option PricingGenetic Programming in Option Pricing

)()( 21 dNEedSNC fr τ−−=

Wo-Chiang Lee

Wo-Chiang Lee

Wo-Chiang Lee

Data drive from Chen ,Lee and Yeh(1999)

RMSE( In sample and out sample) model In sample Out sample

BS(*) 1.98 6.82

BS(3) 1.79 5.10 BS(12) 0.47 0.85

Linear-1 47.51 17.18

Linear-2 24.70 2.57

ANNBPN(1) 2.01 2.79

ANNBPN(3) 0.19 2.68

ANNBPN(6) 0.98 5.93

ANNBPN(9) 0.33 3.87

GP(7U3) 0.32 1.29

Comparison of BS ,Linear,ANNs and GP Model

*.True historical volatility

3-month historical volatility

12-month historical volatility

( .):Number of hidden unit

Program 7 and 3

In-the money

Out-the-money

Wo-Chiang Lee

Fuzzy Logic in Derivative’s Pricing ,Arbitrage and Hedging

Wo-Chiang Lee

Pricing European options based on the fuzzy patternOf Black-Scholes formula-Hsien-Chung WuComputers&OperationsResearch31(2004) 1069–1081

In the real world, some parameters in the Black–Scholesformula cannot always be expected in a precise sense. For instance, the risk-free interest rate r may occur imprecisely. Therefore, the fuzzy sets theory proposed by Zadeh(1965) may be a useful tool for modeling this kind of imprecise problem.

Wo-Chiang Lee

Fuzzy Black-Scholes random variables

risk-free interest rate : constant or stochastic interest rate. fuzzy interest ratestock price S: is a stochastic process: fuzzy stock price.

Volatility: an imprecise data: fuzzy volatility

under the considerations of fuzzy interest rate, fuzzy volatility and fuzzy stock price, the option price will turn into a fuzzy number.

Wo-Chiang Lee

Fuzzy pattern of Black–Scholes formula

We can solve the put call price Pt by put-call parity

Wo-Chiang Lee

Wo-Chiang Lee

Fuzzy pattern of Black–Scholes formula

Since the strike price K and time T are real numbers, they are displayed as the crisp numbers

Wo-Chiang Lee

Fuzzy pattern of Black–Scholes formula

Wo-Chiang Lee

Wo-Chiang Lee

Wo-Chiang Lee

Thank you for your attention!.

THE END