Applications of the Root Solution of the Skorohod Embedding Problem in Finance

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Applications of the Root Solution of the
Skorohod Embedding Problem in Finance

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

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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)

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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

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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.

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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
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Solution of Skorokhod Calibrated
Martingale

• solution of Skorokhod

Then satisfies

• If , then
  is a solution of Skorokhod

as
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ROOT Solution

• Possibly simplest solution hitting time
  of a barrier

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Barrier Density
Density of
PDE
BUT How about Density Barrier?
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PDE construction of ROOT (1)

• Given , define

• If ,
  satisfies

with initial condition

• Apply the previous equation with

until

• Then for ,

• Variational inequality

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PDE computation of ROOT (2)

• Define as the hitting time of

• Then

• Thus , and B is the ROOT
  barrier

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PDE computation of ROOT (3)
Interpretation within Potential Theory
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ROOT Examples

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• Realized Variance

• Call on RV

Ito
taking expectation,

• Minimize one expectation amounts to maximize
  the other one

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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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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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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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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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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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Uniform case
1

• ?
• associated Root barrier

-1
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Uniform case

• Scaling by

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Example
1. Homogeneous scheme
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Example

• Adaptive scheme
• 2a. With a barrier

L
L
Case 1
Case 2
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Example
2. Adaptive scheme 2b. Close to maturity
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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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Approximation of

• can be very well approximated by a simple function

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Properties

• Increments are controlled ? better convergence
• No overshoot
• Easy to scale
• Very easy to implement (uniform sample)
• Low discrepancy sequence apply

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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.