1. Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April 26, 2007
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

2. Variance Swaps Vanilla options are complex bets on
Variance Swaps capture volatility independently of S
Payoff:
Realized Variance
Replicable from Vanilla option (if no jump):
Bruno Dupire

3. Options on Realized Variance
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

4. 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

5. Link with Skorokhod Problem
Option prices of maturity T Risk Neutral density of :
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
A continuous martingale S is a time changed Brownian Motion:
is a BM, and
Bruno Dupire

6. Solution of Skorokhod Calibrated Martingale
Then satisfies
If , then is a solution of Skorokhod as
Bruno Dupire

7. ROOT Solution Possibly simplest solution : hitting time of a barrier
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8. Barrier Density Density of PDE: BUT: How about Density Barrier?
Bruno Dupire

9. PDE construction of ROOT (1)
Given , define
If , satisfies
with initial condition:
Apply the previous equation with
until
Then for ,
Variational inequality:
Bruno Dupire

10. PDE computation of ROOT (2)
Define as the hitting time of
Then
Thus , and B is the ROOT barrier
Bruno Dupire

11. PDE computation of ROOT (3)
Interpretation within Potential Theory
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12. ROOT Examples
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13. Minimize one expectation amounts to maximize the other one
Realized Variance
Call on RV:
Ito:
taking expectation,
Minimize one expectation amounts to maximize the other one
Bruno Dupire

14. Link / LVM Suppose , then define satisfies Let be a stopping time.
For , one has and
where generates
the same prices as X: for all (K,T)
For our purpose, identified by
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15. Optimality of ROOT As to maximize to maximize to minimize
and satisfies:
is maximum for ROOT time, where in
and in
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16. Application to Monte-Carlo simulation
Simple case: BM simulation
Classical discretization:
with  N(0,1)
Time increment is fixed.
BM increment is gaussian.
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17. 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 L
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18. ROOT Monte-Carlo
Clear benefits to confine the (time, BM) increment to a bounded region :
Choose a centered law that is simple to simulate
Compute the associated ROOT barrier :
and, for , draw 
 The scheme generates a discrete BM with the additional information that in continuous time, it has not exited the bounded region.
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19. Uniform case 1 
: associated Root barrier -1
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20. Uniform case
Scaling by :
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21. Example
1. Homogeneous scheme:
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22. Example Adaptive scheme: 2a. With a barrier: L L Case 1 Case 2
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23. Example
2. Adaptive scheme:
2b. Close to maturity:
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24. Example 2. Adaptive scheme:
Very close to barrier/maturity : conclude with binomial 1%
50%
50%
99% L
Close to barrier
Close to maturity
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25. Approximation of can be very well approximated by a simple function
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26. Properties Increments are controlled  better convergence No overshoot
Easy to scale
Very easy to implement (uniform sample)
Low discrepancy sequence apply
Bruno Dupire

27. 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.
Bruno Dupire