Bruno Dupire

Applications of the Root Solution of the

Skorohod Embedding Problem in Finance

Bruno Dupire

Bloomberg LP

CRFMS, UCSB

Santa Barbara, April 26, 2007

Bruno Dupire

• Variance Swaps capture volatility independently of S

• Payoff:

Variance Swaps

/S

0

2

1ln VSS

S

i

i

t

t

TT

T SdS

dSSS

0

2

00 )ln(2

1lnln

• Vanilla options are complex bets on

• Replicable from Vanilla option (if no jump):

Realized Variance

Bruno Dupire

Options on Realized Variance

L

S

S

i

i

t

t

2

1ln

2

1lni

i

t

t

S

SL

• Over the past couple of years, massive growth of

- Calls on Realized Variance:

- Puts on Realized Variance:

• Cannot be replicated by Vanilla options

Bruno Dupire

Classical Models

Classical approach:

– To price an option on X:• Model the dynamics of X, in particular its volatility • Perform dynamic hedging

– For options on realized variance:• Hypothesis on the volatility of VS • Dynamic hedge with VS

But Skew contains important information and we will examine

how to exploit it to obtain bounds for the option prices.

Bruno Dupire

• Option prices of maturity T Risk Neutral density of :

Link with Skorokhod Problem

TS

2

2 ])[()(

K

KSK T

E

,)(,0)( 2 vdxxxdxxx

}:inf{ tSut uSW

tSt WS

• Skorokhod problem: For a given probability density function such that

find a stopping time of finite expectation such that the density of a Brownian motion W stopped at is

is a BM, and

• A continuous martingale S is a time changed Brownian Motion:

Bruno Dupire

• If , then is a solution of Skorokhod

Then satisfies

• solution of Skorokhod:

Solution of Skorokhod Calibrated Martingale

~W

)/( tTtt WS ~WST

~TS TS

~TS SWWT

as

Bruno Dupire

• Possibly simplest solution : hitting time of a barrier

ROOT Solution

Bruno Dupire

BKtKKK

BTKK

TKT

C

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

on),(2

1),(

0

2

2

Barrier Density

Barrier}T)(K,|T{K,B

WDensity of

PDE:

BUT: How about Density Barrier?

Bruno Dupire

• Then for ,

• If , satisfies

• Given , define

PDE construction of ROOT (1)

dxxKxKf )(||)(

sss dWsXdX ),( ||),( KXtKf t E

02

),(2

22

K

ftK

t

f

1t)(K, )(),( KftKf K

Ktt (K))t)(K,( 0t)(K, ff

)(),(,2

1 ¯2

2

KftKfK

f

t

f

• Apply the previous equation with

until

• Variational inequality:

with initial condition: ||)0,( 0 KXKf

Bruno Dupire

• Thus , and B is the ROOT barrier

• Define as the hitting time of

PDE computation of ROOT (2)

tt WX ,0}t)(K, :t){(K,B

|][||][|),( KWKXtKf tt EE

|][|),(lim),( KWtKfKft

E

W

• Then

Bruno Dupire

PDE computation of ROOT (3)

Interpretation within Potential Theory

Bruno Dupire

ROOT Examples

Bruno Dupire

][)( 2LT WLS

EmaxEmin

TSRV

)( L

LdWWWWL ttL

22

])[(][ 2 LWv L EE

• Realized Variance

• Call on RV:

Ito:

taking expectation,

• Minimize one expectation amounts to maximize the other one

Bruno Dupire

For our purpose, identified by

the same prices as X: for all (K,T)

where generates

• For , one has and

• satisfies

• Suppose , then define

Link / LVM τ

ttt dWdY |][|),( KYTKf TY E

Yf 2

22 ]|[

2

1

K

fKY

T

f Y

TT

Y

E

Let be a stopping time.

tt WX tt 1 ]|[]|[ 2 KXTtKX TTT PE

ttt dWtYdY ),( ]|[),( yXtty t P

),(),( TKfTKf XY

]|[t)(x, xXt t P

Bruno Dupire

and satisfies:

Optimality of ROOT

dKKWKWW LL |||][||| 02

EE

|| KW L E 2LW E

])[( LE

),(|][| LKfKW L E fxWtttxa t ],|[),( P

As

to maximize

to minimize

to maximize

0for0

0for),(2

1

2

112

2

af

aftxaf

is maximum for ROOT time, where in CB and in f 1a 0a B

Bruno Dupire

Application to Monte-Carlo simulation

• Simple case: BM simulation• Classical discretization:

with N(0,1)

– Time increment is fixed.– BM increment is gaussian.

nnntt gttWWnn

11 ng

tnt 1nt 2nt

Bruno Dupire

BM increment unbounded

Hard to control the error in Euler discretization of SDE No control of overshoot for barrier options :

and

No control for time changed methods

LWnt

tnt 1nt

L

LWLW t

tttt

nnn

1! ;

min

Bruno Dupire

• Clear benefits to confine the (time, BM) increment to a bounded region :

1. Choose a centered law that is simple to simulate

2. Compute the associated ROOT barrier :

3. and, for , draw

The scheme generates a discrete BM with the additional information that in continuous time, it has not exited the bounded region.

ROOT Monte-Carlo

)(Wf

00

tW nX

ntt

nnn

XWW

Xftt

nn

1

)(1

0n

Bruno Dupire

•

•

• : associated Root barrier

•

Uniform case

00

tW

]1,1[UU n

ntt

nnn

UWW

Uftt

nn

1

)(1

)( nUfnU

ntW

1ntW

nt 1nt t

f

1

-1

Bruno Dupire

Uniform case

ntt

nnn

UWW

Uftt

nn

1

)(21

5.05.1

1

• Scaling by :

Bruno Dupire

Example

0t 1t 2t 3t

0tW

1tW

2tW

3tW

t4t

1. Homogeneous scheme:

Bruno Dupire

Example

t

Case 1

0t 1t 2t 3t

L

4t 0t 1t 2t 3t

L

t4t 5t 6t

Case 2

2. Adaptive scheme:

2a. With a barrier:

Bruno Dupire

Example

0t 1t 2t 3t t4t T

2. Adaptive scheme:

2b. Close to maturity:

Bruno Dupire

Example

tnt T1nt

L

t

1%

99%

nt1nt

Close to barrier Close to maturity

2. Adaptive scheme:

Very close to barrier/maturity : conclude with binomial

50%

50%

Bruno Dupire

Approximation of f

35.02139.0 xxg

)(xf

xfxg

x

• can be very well approximated by a simple functionf

Bruno Dupire

Properties

• Increments are controlled better convergence

• No overshoot

• Easy to scale

• Very easy to implement (uniform sample)

• Low discrepancy sequence apply

Bruno Dupire

CONCLUSION

• Skorokhod problem is the right framework to analyze range of exotic prices constrained by Vanilla prices

• Barrier solutions provide canonical mapping of densities into barriers

• They give the range of prices for option on realized variance

• The Root solution diffuses as much as possible until it is constrained

• The Rost solution stops as soon as possible

• We provide explicit construction of these barriers and generalize to

the multi-period case.