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
• Vanilla options are complex bets on
S /
• Variance Swaps capture volatility independently of S
 S ti 1
• Payoff:   ln
 St
i

2

  VS 0


Realized Variance
• Replicable from Vanilla option (if no jump):
ln S T  ln S 0  
T
0
Bruno Dupire
dS 1 T
  (d ln S ) 2
S 2 0
Options on Realized Variance
•
Over the past couple of years, massive growth of
- Calls on Realized Variance:
  S
  ln ti 1
   S
ti
 




2

 L 



- Puts on Realized Variance:

 L   ln S ti 1
 S

ti






2





• 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
Link with Skorokhod Problem
• Option prices of maturity T  Risk Neutral density of S T
:
 2 E[( S T  K )  ]
 (K ) 
K 2
• Skorokhod problem: For a given probability density function such that
2
x

(
x
)
dx

0
,
x

  ( x)dx  v  ,
 of finite expectation such that the density of a Brownian
 is
find a stopping time
motion W stopped at
• A continuous martingale S is a time changed Brownian Motion:
Wt  S inf{u: S u t } is a BM, and S t  W S t
Bruno Dupire
Solution of Skorokhod
•
 solution of Skorokhod:
W ~ 
Then S t  W t /(T t )satisfies
• If

 Calibrated Martingale
ST  W ~ 
ST ~ , then   S T is a solution of Skorokhod
as W  W S   S T ~ 
T
Bruno Dupire
ROOT Solution
• Possibly simplest solution  : hitting time of a barrier
Bruno Dupire
Barrier


B  {K, T | (K, T)  Barrier}
  Density of W
PDE:
 
1  2
C
(
K
,
T
)
on
B
 ( K , T ) 
T
2 K 2

 ( K ,0)   0 ( K )
  ( K , t )   ( K )
on B
 Barrier?
BUT: How about Density 
Bruno Dupire
Density
PDE construction of ROOT (1)
• Given  , define f ( K )   | x  K |  ( x)dx
• If dX s   ( X s , s)dWs ,
f ( K , t )  E| X t  K | satisfies
f  2 ( K , t )  2 f

0
t
2
K 2
with initial condition: f ( K ,0) | X 0  K |
• Apply the previous equation with
 (K, t)  1
until f ( K , t K )  f ( K )
• Then for
t  t K,  (K, t)  0 ( f (K, t)  f (K))
• Variational inequality:
Bruno Dupire
f 1  2 f
¯

,
f
(
K
,
t
)

f
(K )
2
t 2 K
PDE computation of ROOT (2)
• Define

as the hitting time of
B  {(K, t) :  (K, t)  0}, X t  W t
• Then f ( K , t )  E[| X t  K |]  E[| W t  K |]
f ( K , )  lim f ( K , t )  E[| W  K |]
t 
• Thus W  , and B is the ROOT barrier
Bruno Dupire
PDE computation of ROOT (3)
Interpretation within Potential Theory
Bruno Dupire
ROOT Examples
Bruno Dupire

min E( S  T  L) 
 max E[W
2
 L
RV  S T  
• Realized Variance
• Call on RV: (  L) 
Ito:
W  W
2
taking expectation,
2
 L

  Wt dWt      L
 L
v  E[W2 L ]  E[(  L)  ]
• Minimize one expectation amounts to maximize the other one
Bruno Dupire
]
Link
• Suppose dYt
Y
• f satisfies
Let

  t dWt , then define
τ / LVM
f Y ( K , T )  E[| YT  K |]
f Y 1
2 f Y
2
 E[ T | YT  K ]
T
2
K 2
be a stopping time.
• For X t  W t , one has  t 
1
t
2
and E[ T | X T  K ]  P[t  T | X T  K ]
 dYt   (Yt , t ) dWt where  ( y, t )  P[  t | X t  y] generates
the same prices as X:
f Y (K ,T )  f X (K ,T )
 For our purpose,  identified by
Bruno Dupire
for all (K,T)
 (x, t)  P[  t | X t  x]
Optimality of ROOT

  E[| W
2
As E | W  L | 
to maximize
 L
 K |] | W0  K |dK
E| W  L  K |



to maximize

to minimize E[(  L)  ]
E W2 L
E[| W  L  K |]  f ( K , L) and a( x, t )  P[t  t | Wt  x], f satisfies:
1

 f 2  a 2 ( x, t ) f11 for a  0

2
 f 2  0
for a  0
f is maximum for ROOT time, where a  1 in B C and a  0 in B
Bruno Dupire
Application to Monte-Carlo
simulation
•
•
Simple case: BM simulation
Classical discretization:
Wtn1  Wtn  tn1  tn g n with
tn
t n 1
t n2
– Time increment is fixed.
– BM increment is gaussian.
Bruno Dupire
g n  N(0,1)
t
BM increment unbounded


Hard to control the error in Euler discretization of SDE
No control of overshoot for barrier options :
Wtn  L and Wtn !  L 
min Wt  L
tt n ;t n1 
L
tn
 No control for time changed methods
Bruno Dupire
t n 1
t
ROOT Monte-Carlo
•
1.
2.
3.
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 : f (W )
Wt0  0 and, for n  0 , draw X n  
tn1  tn  f ( X n )
Wtn1  Wtn  X n
 The scheme generates a discrete BM with the
additional information that in continuous time, it
has not exited the bounded region.
Bruno Dupire
Uniform case
•
Wt0  0
1
Un
•
•
f (U n )
Wt n1
U n  U [1,1]
f
Wt n
: associated Root barrier
• t n 1
 tn  f (U n )
Wtn1  Wtn  U n
-1
tn
Bruno Dupire
t n 1
t
Uniform case
•
Scaling by  : tn 1  tn   2 f (U n )
Wtn1  Wtn  U n
 1
  0.5
Bruno Dupire
  1.5
Example
1.
Homogeneous scheme:
Wt 2
Wt0
Wt3
Wt1
t0
Bruno Dupire
t1
t2
t3
t4
t
Example
2.
Adaptive scheme:
2a. With a barrier:
L
L
t0
t1
t2
Case 1
Bruno Dupire
t3 t4
t
t0
t1
t2
t3t4 t5
Case 2
t6
t
Example
2. Adaptive scheme:
2b. Close to maturity:
t0
Bruno Dupire
t1
t2
t 3 t 4T
t
Example
2. Adaptive scheme:
Very close to barrier/maturity : conclude with binomial
1%
50%
99%
L
t n 1
tn
Close to barrier
Bruno Dupire
50%
t
t n 1
tn T
Close to maturity
t
Approximation of f
•
f
can be very well approximated by a simple function
f (x)

g x   0.39 1  x 2

0.35
g x   f x 
x
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.
Bruno Dupire