SUMMER RESEARCH PROJECT

DANIEL GUET TAwith PROF. PAUL GLASSERMAN

Detecting Bubbles Using Option Prices

Bubbles

What is a Bubble?

Bubbles are often associated with a large increase in the asset price followed by a collapse when the bubble “bursts”.

In the context of financial markets, bubbles refer to asset prices that exceed the asset's fundamental, intrinsic value possibly because those that own the asset believe that they can sell the asset at a higher price in the future.

What is a Bubble?

“Asset Price Bubbles in Complete Markets”, Jarrow, Protter & Shimbo, 2007

“Asset Price Bubbles in Incomplete Markets”, Jarrow, Protter & Shimbo, 2010

A Very (Very, Very) Short Introduction to Financial Math

Financial Mathematics

tS

t

Google Stock – 1st January 2007 to 1st January 2011

t

d ( , ) d ( , ) dt t t t t t

S S t S t S t S Wm s= +

Financial Mathematics

d ( , ) d ( , ) dt t t t t t

S S t S t S t S Wm s= +

d ( , ) dt t t t

S S t S Ws=

First Fundamental Theorem of Asset Pricing

Price Distributions

Price Distributions

Price Distributions

Price Distributions

Price Distributions

Price Distributions

1T

Price Distributions

1T

2T

The Kolmogorov Forward Equation

22 2

2

1( ) ( , ) ( )

2t tx x t x x

t xf s f

¶ ¶ é ù= ê úë û¶ ¶

d ( , ) dt t t t

S S t S Ws= ( )t t

Sf

Detecting Bubbles

The Bubble Test

Assumption:

d ( , ) dt t t t

S S t S Ws=

“How to Detect an Asset Bubble”, Jarrow, Kchia & Protter, March 2011

The Bubble Test

2 d

( )

xx

xa s

¥< ¥ò

Bubble exists in the asset price St

St is a strict local martingale

Assumption:

d ( ) dt t t t

S S S Ws=

“How to Detect an Asset Bubble”, Jarrow, Kchia & Protter, March 2011

Using Options to Find

What is an Option?

( ),K T

When time T comes along, the call option gives its owner the right, but not the obligation, to buy one unit of the financial asset at price K.

Strike Maturity

Pricing Options

( )( )Payoff at maturity

max[ ,0]

max( ,0) ( )

( , )

dT

T

S K

x

C K T

K x xf

= -

=

=

-ò

E

E

KT

S

Payoff

Magic!

2

2

2

2

( , )

d( , ) step( ) ( ) d

dd

( , )

max( ,0)

( ) ( ) dd

d( , ) ( )

d

( ) d

T

T

T

T

C K T

C K T x K x xK

C K T x K x xK

C K T KK

x K x x

f

f

d f

f

= - -

= -

=

= -

ò

ò

ò

x

The Dupire Equation

22 2

2

1( ) ( , ) ( )

2t tx x t x x

t xf s f

¶ ¶ é ù= ê úë û¶ ¶

2

2

d( , ) ( )

d T

CK T K

Kf=+

2

221

2 2

( , )

CTK T

CK

K

s

¶¶=

¶

¶

=

Kolmogorov Forward Equation

The Dupire Equation

Reality

Optio

n

price

MaturityStrike

1st September 2006, Options on the S&P 500

Local Least Squares

“Arbitrage-free Approximation of Call Price Surfaces and Input Data Risk”, Glaser and

Heider, March 2010

Local Least Squares

Optio

n

price

Maturity

Strike

1st September 2006, calls

Local Least Squares

Optio

n

price

Maturity

Strike

Local Least Squares

Optio

n

price

Maturity

Strike

Local Least Squares

Optio

n

price

Maturity

Strike

Local Least Squares

Optio

n

price

Maturity

Strike

Local Least Squares

Optio

n

price

Maturity

Strike

21 2 3 4

( , )C K T a a K a K aT= + + +

Local Least Squares

Optio

n

price

Maturity

Strike

1st September 2006, calls

Local Least Squares

Optio

n

price

Maturity

Strike

1st September 2006, calls

The Local Volatility

2(K

,T)

K

T

1st March 2004, calls

The Local Volatility

2(K

,T)

KT

2nd July 2007, calls

The Local Volatility

2(K

,T)

KT

2nd July 2007, puts

Results

2 d B b

)l

(u b e

xx

xa s

¥Û< ¥ò

Bubble Indicator

Date

Bubble Indicator

Date

VIX

In

dex

Correlation coefficient: 0.15

Bubble Indicator

Date

S&

P 5

00

2 d B b

)l

(u b e

xx

xa s

¥Û< ¥ò

Correlation coefficient: 0.01

Concluding Remarks

Conclusions

A promising approach to implementing the bubble test.

The non-parametric approach we used might have been slightly too ambitious.

Fitting options prices rather than volatilities might have compounded the problem.

Other Approaches

Use some sort of spline (“Reconstructing the Unknown Volatility Function”, Coleman, Li and Verma, “Computation of Deterministic Volatility Surfaces”, 2001. Jackson, Suli and Howison, 1999. “Improved Implementation of Local Volatility and Its Application to S&P 500 Index Options”, 2010.)

Estimate the local volatility via the implied volatility.

Other Approaches

Assume the volatility is piecewise constant, and solve the Dupire Equation to find the “best” constants. (“Volatility Interpolation”, Andreasen and Huge, 2011).

Assume some sort of parametric pricing model (such as Heston or SABR), fit to option price data and then deduce local volatility.

Questions