Data Mining in Finance, 1999

March 8, 1999

Extracting Risk-Neutral Densities from Option Pricesusing Mixture Binomial Trees

Christian Pirkner

Andreas S. Weigend

Heinz Zimmermann

Version 1.0

DMF 99

Outline

Introduction

Model

Application

Motivation Butterfly-Spread Implied Binomial Tree

Mixture Binomial Tree Optimization Graph

Density Extraction: 1 Day Density Extraction over Time Conclusion

Part 1

Part 2

Part 3

Introduction

Application

Model

DMF 99

1. Introduction

- Motivation -

An European equity call option (C) is the right to …– buy– an underlying security, S– for a specified strike price, X– at time to expiration, T

payoff function: max [ST - X, 0]

Goal:– What can we learn from market prices of traded options?

Extract expectations of market participants– Use this information for decision making!

Exotic option pricing, risk measurement and trading

Introduction

Application

Model

DMF 99

1. Introduction

- … a butterfly-spread -

Introduction

Model

X

7

8

9

10

11

12

13

C

3.354

2.459

1.670

1.045

0.604

0.325

0.164

+1.670

-2.095

+0.604

0 0 0 1 2 3 4

0 0 0 0 -2 -4 -6

0 0 0 0 0 1 2

Payoff if ST = ...

7 8 9 10 11 12 13

0 0 0 1 0 0 00.184

C

-0.895

-0.789

-0.625

-0.441

-0.279

-0.161

(C)

0.106

0.164

0.184

0.162

0.118

Costbsp

Buy 1 C(X=9)Sell 2 C(X=10)Buy 1 C(X=11)

S=10

Application

vj

DMF 99

1. Introduction

- … risk-neutral probabilities -

Introduction

Model

S=10

Application

X

7

8

9

10

11

12

13

C

3.354

2.459

1.670

1.045

0.604

0.325

0.164

(C)

0.106

0.164

0.184

0.162

0.118

vj

Valuing an option with payoffs j using vj:

j

jj vC

Buying all vj’s: riskless investment

j

jrT ve 00.1

Alternative way to value derivative:

jjj

rT PeC

jrT

j Pev Defining Pj’s: “risk-neutral probabilities”:

TSTrT dSPXSeC

T 0,max

0

DMF 99

1. Introduction

- Density extraction techniques -

Parametric

NonParametric

I.2nd Derivative ofcall price function

II.Estimating

density directly

•Linear•Logit•Polynomial

•Several tanh

•Kernel regression

•Gauss•Gamma•Edgeworth expansion

•Smoothness•Mixture models

•Kernel density

III. Recovering parameters of assumed stochastic process of the underlying security.

Introduction

Model

Application

DMF 99

1. Introduction

- Standard & implied trees -

Introduction

Model

Instead of building a ...

standard binomial tree– starting at time t=0– resting on the assumption of

normally distributed returns

and constant volatility

We build an …

implied binomial tree:– starting at time T– and flexible modeling of end-

nodal probabilities

0 10 20-1

-0.5

0

0.5

1Stock return tree

steps

(log

)-re

turn

s

0 1 2-1

-0.5

0

0.5

1End-nodal probabilities

probability(l

og)-

retu

rns

0 10 20-1

-0.5

0

0.5

1Stock return tree

steps

(log

)-re

turn

s

0 1 2-1

-0.5

0

0.5

1End-nodal probabilities

probability

(log

)-re

turn

s

0 10 20-1

-0.5

0

0.5

1Stock return tree

steps

(log

)-re

turn

s

0 1 2-1

-0.5

0

0.5

1End-nodal probabilities

probability

(log

)-re

turn

s0 10 20

-1

-0.5

0

0.5

1Stock return tree

steps

(log

)-re

turn

s

0 1 2-1

-0.5

0

0.5

1End-nodal probabilities

probability(l

og)-

retu

rns

Application

DMF 99

2. Model

- Mixture binomial tree -

… where we optimize for the lowest

absolute mean squared error in

option prices

m

0k market

elmod

,,g C

C1

m

1min

L

1l

2lll

T ,gF

FlnP

Subject to constraint:

1gx

The weights of all mixture components

are positive and add up to one0gx

Introduction

Model

We propose to model end-nodal

probabilities with a mixture of

Gaussians ...

Application

DMF 99

2. Model

- Mixture binomial tree -

Introduction

Model

0 10 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Stock return tree

steps

(log

)-re

turn

s

0 1 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Mixture probabilities

probability

(log

)-re

turn

s

Application

DMF 99

02/06 03/06 04/10 05/08 06/12 07/10 08/07 09/11 10/09 11/06 12/11 01/08

950

1000

1050

1100

1150

1200

1250

1300

S&P 500 Future [9903]

Date

Futu

re p

rice

3. Application

- Data: S&P 500 futures options -

Introduction

Model

Application

DMF 99

3. Evaluation & Analysis

- February 6, 1 Gauss & Error -

Introduction

Model

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0

1

2

3

4

5Implied Density & Pricing Error: 1-Gaussian (02/06)

(log)-returns

impl

ied

prob

abili

ty

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-40

-20

0

20

40

moneyness

%-e

rror

: tru

e vs

mod

el

Application

DMF 99

3. Evaluation & Analysis

- February 6, 3 Gauss & Error -

Introduction

Model

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0

1

2

3

Implied Density & Pricing Error: 3-Gaussians (02/06)

(log)-returns

impl

ied

prob

abili

ty

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-40

-20

0

20

40

moneyness

%-e

rror

: tru

e vs

mod

el

Application

DMF 99

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

1

2

3

4

5

6

7February

(log)-returns

impl

ied

prob

abil

ity

3. Evaluation & Analysis

- February -

Introduction

Model

02/06 03/06 04/10 05/08 06/12 07/10 08/07 09/11 10/09 11/06 12/11 01/08

950

1000

1050

1100

1150

1200

1250

1300

S&P 500 Future [9903]

Date

Futu

re p

rice

Application

DMF 99

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

1

2

3

4

5

6

7May [vs previous]

(log)-returns

impl

ied

prob

abil

ity

3. Evaluation & Analysis

- May -

Introduction

Model

02/06 03/06 04/10 05/08 06/12 07/10 08/07 09/11 10/09 11/06 12/11 01/08

950

1000

1050

1100

1150

1200

1250

1300

S&P 500 Future [9903]

Date

Futu

re p

rice

Application

DMF 99

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

1

2

3

4

5

6

7July [vs previous]

(log)-returns

impl

ied

prob

abil

ity

3. Evaluation & Analysis

- July -

Introduction

Model

02/06 03/06 04/10 05/08 06/12 07/10 08/07 09/11 10/09 11/06 12/11 01/08

950

1000

1050

1100

1150

1200

1250

1300

S&P 500 Future [9903]

Date

Futu

re p

rice

Application

DMF 99

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

1

2

3

4

5

6

7August [vs previous]

(log)-returns

impl

ied

prob

abil

ity

3. Evaluation & Analysis

- August -

Introduction

Model

02/06 03/06 04/10 05/08 06/12 07/10 08/07 09/11 10/09 11/06 12/11 01/08

950

1000

1050

1100

1150

1200

1250

1300

S&P 500 Future [9903]

Date

Futu

re p

rice

Application

DMF 99

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

1

2

3

4

5

6

7October [vs previous]

(log)-returns

impl

ied

prob

abil

ity

3. Evaluation & Analysis

- October -

Introduction

Model

02/06 03/06 04/10 05/08 06/12 07/10 08/07 09/11 10/09 11/06 12/11 01/08

950

1000

1050

1100

1150

1200

1250

1300

S&P 500 Future [9903]

Date

Futu

re p

rice

Application

DMF 99

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

1

2

3

4

5

6

7January [vs previous]

(log)-returns

impl

ied

prob

abil

ity

3. Evaluation & Analysis

- January -

Introduction

Model

02/06 03/06 04/10 05/08 06/12 07/10 08/07 09/11 10/09 11/06 12/11 01/08

950

1000

1050

1100

1150

1200

1250

1300

S&P 500 Future [9903]

Date

Futu

re p

rice

Application

DMF 99

Conclusion Introduction

Model

Learning from option prices Extracting market expectations

Use information for decision making Exotic option pricing

Use extracted kernel to price non-standard derivatives:

consistent with liquid options Risk measurement

Calculate “Economic Value at Risk” Trading

Take positions if extracted density differs from own view

Application