Existence and Uniqueness Theory for the
Black-Scholes Equation in Stochastic
Volatility Models
Erik Ekström
Joint work with Johan Tysk
Department of Mathematics, Uppsala University
May 8, 2008, Pitesti
A Motivating Example
Assume that
dX (t) = X 2 (t) dW ,
and that
u(x, t) = Ex,t X (T ).
Then X is a strict local martingale so that u(x, t) < x. The
corresponding BS-equation is
ut + 21 x 4 uxx = 0
u(x, T ) = x.
However, v ≡ x is also a solution!
A Motivating Example
Assume that
dX (t) = X 2 (t) dW ,
and that
u(x, t) = Ex,t X (T ).
Then X is a strict local martingale so that u(x, t) < x. The
corresponding BS-equation is
ut + 21 x 4 uxx = 0
u(x, T ) = x.
However, v ≡ x is also a solution!
Uniqueness of solutions does not hold!
A Classical PDE Result
The PDE
ut (x, t) + a(x, t)uxx (x, t) + b(x, t)ux (x, t) + c(x, t)u(x, t) = 0
u(x, T ) = g(x)
has a unique solution of at most polynomial growth if the
coefficient a is of at most quadratic growth:
0 < a(x, t) ≤ C(1 + x 2 )
(b of linear growth and c bounded).
The Classical Set-Up
Stock price process:
dX (t) = α(X (t), t) dW ,
absorbed at 0. Assume that
|α(x, t)| ≤ C(1 + x).
Given a pay-off function g, define the option price by
u(x, t) = Ex,t g(X (T )).
The corresponding Black-Scholes equation is

 ut + 12 α 2 (x, t)uxx = 0
u(x, T ) = g(x)

u(0, t) = g(0)
Existence of Solutions
Theorem
If |α(x, t)| ≤ C(1 + x), then all moments of X (T ) are finite.
Existence of Solutions
Theorem
If |α(x, t)| ≤ C(1 + x), then all moments of X (T ) are finite.
Corollary
Assume that |α(x, t)| ≤ C(1 + x) and |g(x)| ≤ C(1 + x N ). Then
the function
u(x, t) = Ex,t g(X (T ))
is a classical solution to the corresponding Black-Scholes
equation

 ut + 12 α 2 (x, t)uxx = 0
u(x, T ) = g(x)

u(0, t) = g(0).
Uniqueness of Solutions via the Maximum Principle
Theorem
Assume that |α(x, t)| ≤ C(1 + x). Then u = 0 is the unique
classical solution of at most polynomial growth to the equation
ut + 12 α 2 (x, t)uxx = 0
u(x, T ) = u(0, t) = 0.
Uniqueness of Solutions via the Maximum Principle
Theorem
Assume that |α(x, t)| ≤ C(1 + x). Then u = 0 is the unique
classical solution of at most polynomial growth to the equation
ut + 12 α 2 (x, t)uxx = 0
u(x, T ) = u(0, t) = 0.
Proof.
For the proof one looks for a supersolution h(x, t) which grows
faster than the candidate solution u as x → ∞. Since u is of at
most polynomial growth, say |u(x, t)| ≤ C(1 + x N ), the function
h(x, t) = eMt (1 + x N+1 )
will do. h is indeed a supersolution if M is large enough:
ht = MeMt (1 + x N+1 )
and
1 2
α hxx ∼ α 2 x N−1 ∼ x N+1 .
2
General Local Volatility Models
As before,
dX (t) = α(X (t), t) dW ,
with absorption at 0. WE NO LONGER ASSUME THE LINEAR
BOUND ON α!! Clearly, X is a non-negative local martingale,
hence a supermartingale. Therefore
Ex,t X (T ) ≤ x.
Models in which the stock price is a strict local martingale has
been proposed to model bubbles, see Cox-Hobson (2005) and
Heston-Loewenstein-Willard (2007).
Existence
Theorem
If g is of at most linear growth, then u(x, t) = Ex,t g(X (T )) is a
classical solution to the BS-equation

 ut + 12 α 2 (x, t)uxx = 0
u(x, T ) = g(x)

u(0, t) = g(0).
The Stochastic Solution is the Smallest one
Theorem
(Heston-Loewenstein-Willard) Let g be lower bounded. Then
the stochastic solution u(x, t) = Ex,t g(X (T )) is the smallest
solution to the Black-Scholes PDE which is bounded from
below.
The Stochastic Solution is the Smallest one
Theorem
(Heston-Loewenstein-Willard) Let g be lower bounded. Then
the stochastic solution u(x, t) = Ex,t g(X (T )) is the smallest
solution to the Black-Scholes PDE which is bounded from
below.
Proof.
Let v be a solution to the Black-Scholes PDE which is bounded
from below. By Ito’s Lemma, v (X (s), s) is a local martingale.
Since it is lower bounded, it is a supermartingale. Hence
v (x, t) ≥ Ex,t v (X (T ), T ) = Ex,t g(X (T )) = u(x, t).
The Stochastic Solution is the Smallest one
Theorem
(Heston-Loewenstein-Willard) Let g be lower bounded. Then
the stochastic solution u(x, t) = Ex,t g(X (T )) is the smallest
solution to the Black-Scholes PDE which is bounded from
below.
Proof.
Let v be a solution to the Black-Scholes PDE which is bounded
from below. By Ito’s Lemma, v (X (s), s) is a local martingale.
Since it is lower bounded, it is a supermartingale. Hence
v (x, t) ≥ Ex,t v (X (T ), T ) = Ex,t g(X (T )) = u(x, t).
Corollary
If g is bounded, then the Black-Scholes equation has a unique
bounded solution given by Ex,t g(X (T )).
A General Uniqueness Result
We also have the following stronger uniqueness result.
Theorem
The Black-Scholes equation has a unique solution in the class
of functions of strictly sublinear growth.
A General Uniqueness Result
We also have the following stronger uniqueness result.
Theorem
The Black-Scholes equation has a unique solution in the class
of functions of strictly sublinear growth.
Example
Any pay-off function of the form g(x) = x 1−ε gives a unique
solution (unique in the class of strictly sublinear functions).
A General Uniqueness Result
We also have the following stronger uniqueness result.
Theorem
The Black-Scholes equation has a unique solution in the class
of functions of strictly sublinear growth.
Example
Any pay-off function of the form g(x) = x 1−ε gives a unique
solution (unique in the class of strictly sublinear functions).
Proof.
Note that h(x, t) = eMt (1 + x) is a supersolution. Indeed,
1
ht = MeMt (1 + x) > 0 = α 2 hxx .
2
Thus we have uniqueness in the class of functions which grow
slower than x.
A Condition for X(t) to be a Strict Local Martingale
Uniqueness is lost in the class of linear functions if
x − Ex,t X (T ) > 0.
Theorem
Assume that |α(x, t)| ≥ x 1+δ for large x. If δ > 0, then
Ex,t X (T ) = o(x ε ) as x → ∞
for any ε > 0.
If δ > 1/2, then Ex,t X (T ) is bounded in x.
Proof.
It can be checked that the function
h(x, t) = eMt
x
1 + t nx β
is a supersolution if M, n and β are chosen appropriately. The
result follows since the option price is the smallest solution to
the BS-equation.
Stochastic Volatility Models
Stock price:
dX (t) =
p
Y (t)α(X (t)) dW ,
where the variance process Y follows
dY (t) = β (Y (t)) dt + σ (Y (t)) dV ,
dW dV = ρdt.
Stochastic Volatility Models
Stock price:
dX (t) =
p
Y (t)α(X (t)) dW ,
where the variance process Y follows
dY (t) = β (Y (t)) dt + σ (Y (t)) dV ,
dW dV = ρdt.
We assume that X is absorbed at 0, and that Y stays
nonnegative automatically. X is a nonnegative local martingale,
hence a supermartingale so Ex,y,t X (T ) ≤ x. Linear growth on
the coefficients:
|β (y )| ≤ C(1 + y)
|σ (y )| ≤ C(1 + y)
and
|α(x)| ≤ C(1 + x).
The BS-equation for Stochastic Volatility Models
The option price is defined as
u(x, y, t) = Ex,y,t g(X (T )).
The corresponding Black-Scholes equation is

√
1 2
1
2

 ut + 2 yα (x)uxx + ρ yσ (y)α(x)uxy + 2 σ (y)uyy + β (y)uy = 0

u(x, y, T ) = g(x)
u(0, y , t) = g(0)



ut (x, 0, t) + β (0)uy (x, 0, t) = 0
The BS-equation for Stochastic Volatility Models
The option price is defined as
u(x, y, t) = Ex,y,t g(X (T )).
The corresponding Black-Scholes equation is

√
1 2
1
2

 ut + 2 yα (x)uxx + ρ yσ (y)α(x)uxy + 2 σ (y)uyy + β (y)uy = 0

u(x, y, T ) = g(x)
u(0, y , t) = g(0)



ut (x, 0, t) + β (0)uy (x, 0, t) = 0
The diffusion coefficient grows superquadratically, so neither
existence or uniqueness of solutions to the equation is covered
by the standard PDE-theory!
Existence
We expect to prove:
Theorem
If g is of at most linear growth, then the function
u(x, y , t) = Ex,y,t g(X (T )) is a classical solution of the
BS-equation

√
ut + 21 yα 2 (x)uxx + ρ yσ (y)α(x)uxy + 12 σ 2 (y)uyy + β (y)uy = 0



u(x, y, T ) = g(x)
u(0, y , t) = g(0)



ut (x, 0, t) + β (0)uy (x, 0, t) = 0
Uniqueness
If X is a strict local martingale, then uniqueness of solutions is
lost for linear contracts.
Uniqueness
If X is a strict local martingale, then uniqueness of solutions is
lost for linear contracts.
Theorem
Uniqueness always holds in the class of functions of strictly
sublinear growth in x and polynomial growth in y .
Uniqueness
If X is a strict local martingale, then uniqueness of solutions is
lost for linear contracts.
Theorem
Uniqueness always holds in the class of functions of strictly
sublinear growth in x and polynomial growth in y .
Proof.
The function h(x, t) = eMt (1 + x + y m ) is a supersolution (if M is
large enough).
The Heston Model
In the Heston model,
p
dX (t) = Y (t)X (t) dW p
dY (t) = (b − aY (t)) dt + Y (t) dV .
The corresponding Black-Scholes equation is

2
ut + 12 yx 2 uxx + ρyxuxy + σ2 yuyy + (b − ay )uy = 0



u(x, y, T ) = g(x)

u(0, y , t) = g(0)


ut (x, 0, t) + buy (x, 0, t) = 0.
Results for the Heston Model
Theorem
Uniqueness holds in the class of functions that are linear in x
and polynomial in y.
Results for the Heston Model
Theorem
Uniqueness holds in the class of functions that are linear in x
and polynomial in y.
Proof.
The function
h(x, y , t) = eMt (1 + x ln x + y m + xy)
is a supersolution if M is large enough.
Results for the Heston Model
Theorem
Uniqueness holds in the class of functions that are linear in x
and polynomial in y.
Proof.
The function
h(x, y , t) = eMt (1 + x ln x + y m + xy)
is a supersolution if M is large enough.
Corollary
In the Heston model, X (t) is a true martingale.
The SABR Model
In the SABR model,
dX (t) =
p
Y (t)X γ (t) dW
dY (t) = σ Y (t) dV .
The corresponding BS-equation is

2
u + 1 yx 2γ (x, t)uxx + ρσ y 3/2 x γ uxy + σ2 y 2 uyy = 0


 t 2
u(x, y, T ) = g(x)

u(0,
y , t) = g(0)


ut (x, 0, t) = 0
Results for the SABR model (γ < 1)
Theorem
If γ < 1 in the SABR model, then there is uniqueness for the
BS-equation in the class of functions of at most polynomial
growth.
Results for the SABR model (γ < 1)
Theorem
If γ < 1 in the SABR model, then there is uniqueness for the
BS-equation in the class of functions of at most polynomial
growth.
Proof.
The function
h(x, y, t) = eMt (1 + x n + y m )
is a supersolution provided m and M are chosen large
enough.
Corollary
(Andersen-Piterbarg) If γ < 1, then X (t) is a true martingale.
Results for the SABR Model (γ = 1)
Theorem
Let γ = 1. If ρ ≤ 0, then uniqueness holds in the class of
functions which are linear in x and polynomial in y .
Note: If ρ > 0, then Ex,t X (T ) < x as is shown by Sin (1998).
Thus there is no uniqueness in the linear class in this case.
Results for the SABR Model (γ = 1)
Theorem
Let γ = 1. If ρ ≤ 0, then uniqueness holds in the class of
functions which are linear in x and polynomial in y .
Note: If ρ > 0, then Ex,t X (T ) < x as is shown by Sin (1998).
Thus there is no uniqueness in the linear class in this case.
Proof.
If ρ ≤ 0, then
h(x, y , t) = eMt (1 + x ln x + y m + xy)
is a supersolution.
Corollary
If α = 1 and ρ ≤ 0, then X (t) is a true martingale.
References
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Andersen and Piterbarg: Moment explosions in stochastic
volatility models. Finance Stoch. (2007).
Cox and Hobson: Local martingales, bubbles and option
prices. Finance Stoch. (2005).
E. and Tysk. Bubbles, convexity and the Black-Scholes
equation. Manuscript (2008).
E. and Tysk. Existence and uniqueness theory for the term
structure equation. Manuscript (2008).
E. and Tysk. Existence and uniqueness theory for the
pricing equation in stochastic volatility models. In progress.
Heston, Loewenstein, Willard. Options and bubbles. Rev.
Financial Studies (2007).
Sin. Complications with stochastic volatility models. Adv.
Appl. Probab. (1998).
References
I
I
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I
I
I
I
Andersen and Piterbarg: Moment explosions in stochastic
volatility models. Finance Stoch. (2007).
Cox and Hobson: Local martingales, bubbles and option
prices. Finance Stoch. (2005).
E. and Tysk. Bubbles, convexity and the Black-Scholes
equation. Manuscript (2008).
E. and Tysk. Existence and uniqueness theory for the term
structure equation. Manuscript (2008).
E. and Tysk. Existence and uniqueness theory for the
pricing equation in stochastic volatility models. In progress.
Heston, Loewenstein, Willard. Options and bubbles. Rev.
Financial Studies (2007).
Sin. Complications with stochastic volatility models. Adv.
Appl. Probab. (1998).
Thank you for your attention!