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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.