Volatility Derivatives Modeling Bruno Dupire Bloomberg NY NYU, January, 2005.
Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno...
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![Page 1: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/1.jpg)
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
![Page 2: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/2.jpg)
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
![Page 3: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/3.jpg)
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
![Page 4: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/4.jpg)
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.
![Page 5: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/5.jpg)
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:
![Page 6: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/6.jpg)
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
![Page 7: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/7.jpg)
Bruno Dupire
• Possibly simplest solution : hitting time of a barrier
ROOT Solution
![Page 8: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/8.jpg)
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?
![Page 9: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/9.jpg)
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
![Page 10: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/10.jpg)
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
![Page 11: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/11.jpg)
Bruno Dupire
PDE computation of ROOT (3)
Interpretation within Potential Theory
![Page 12: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/12.jpg)
Bruno Dupire
ROOT Examples
![Page 13: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/13.jpg)
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
![Page 14: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/14.jpg)
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
![Page 15: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/15.jpg)
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
![Page 16: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/16.jpg)
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
![Page 17: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/17.jpg)
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
![Page 18: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/18.jpg)
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
![Page 19: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/19.jpg)
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
![Page 20: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/20.jpg)
Bruno Dupire
Uniform case
ntt
nnn
UWW
Uftt
nn
1
)(21
5.05.1
1
• Scaling by :
![Page 21: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/21.jpg)
Bruno Dupire
Example
0t 1t 2t 3t
0tW
1tW
2tW
3tW
t4t
1. Homogeneous scheme:
![Page 22: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/22.jpg)
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:
![Page 23: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/23.jpg)
Bruno Dupire
Example
0t 1t 2t 3t t4t T
2. Adaptive scheme:
2b. Close to maturity:
![Page 24: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/24.jpg)
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%
![Page 25: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/25.jpg)
Bruno Dupire
Approximation of f
35.02139.0 xxg
)(xf
xfxg
x
• can be very well approximated by a simple functionf
![Page 26: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/26.jpg)
Bruno Dupire
Properties
• Increments are controlled better convergence
• No overshoot
• Easy to scale
• Very easy to implement (uniform sample)
• Low discrepancy sequence apply
![Page 27: Bruno Dupire Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April.](https://reader036.fdocuments.in/reader036/viewer/2022062300/56649d3a5503460f94a14f48/html5/thumbnails/27.jpg)
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.