some applications of classical pde in ﬁnance · using hadamard’s method and lie’s symmetry...

Some Applications of Classical PDE in finance

Peter Michael [email protected]

Dipartimento di Matematica and Facolta di Statistica

Universita di Roma I

StockAug2005 – p.1/40

Main aims of this contribution

Joint work with Peter Carr (Bloomberg and Courant) andTai-Ho Wang, National Chung Cheng U., Chia Yi, Taiwan)using Hadamard’s method and Lie’s symmetry groupsin PDE in mathematical finance. To briefly discuss threerecent contributions

Work joint with Peter Carr on multi-asset stochastic localvariance

Work joint with Wang on special solvable quadratic andinverse quadratic interest rate models

Joint work with both Carr and Wang on generating andcharacterizing solvable one-dimensional diffusions


Topic 1 Multi-asset variance contracts, Multi-asset

Generalization of multi-variance contracts to many assets.

What is a variance contract?

Such a contract specifies a fixed strike K > 0 and a fixed maturity date T . Just after T thecontract pays

δ(ST − K) d < ST >T ,

where

d < ST >T = local quadratic variation

and where δ(ST − K) is Dirac’s Delta function. For example if dSt = atdWt ,< dST >T = a2

T dt.

Question What is the natural generalization of this contract in the context of a contract on abasket of assets?

Answer: The answer will be related to finding a new contract whose payoff is a cut-off of afundamental solution for an appropriate elliptic equation in n variables.


Breeden and Litzenberger

C(St, t, K, T ) = e−r(T−t)E[(ST − K)+]

where : E[(ST − K)+] =

∫

<+

(ST − K)+p(St, t, s, T )ds,

and p(st, t, sT , T ) is the transition probability density for passage from St at time t to ST attime T .

So

CKK(K, T ) = e−r(T−t)E[(ST − K)+)KK ] = e−r(T−t)E[δK(S)]

= e−r(T−t)p(St, t, K, T )

ie. Breeden Litzenberger allows us to recover the transition probability from knowledge of

the quoted market prices of call options.StockAug2005 – p.4/40

Breeden-Litzenberger inn-D

Is there a generalization of Breeden and Litzenberger to n

assets?

• Answer: A positive answer (kind of) appeared in the caseof two assets in the book by Lipton (World Scientific, 2002).The answer involves the Radon transform.• The generalization to the n-asset case is straightforwardand also can be formulated in terms of the Radon transform.• However the answer may be of limited practical interest asit assumes liquid market for index options with a continuumof weights.• We will return to it later, time permitting.


Review of Dupire’s construction of a variance contract

For simplicity assume zero rates and no dividends.Under the risk neutral measure we have

dSt = atdWt

This is a stochastic volatility model in that at is allowed todepend on sources of uncertainty above and beyond theuncertainty in the spot level St.

For the quadratic variation we have

d < S >t= a2

t dt


Review, p2.

Dupire showed that the local variance

E0

[a2

T |ST ∈ dK]

can be determined from the initial market value of straddlesstruck around K and maturing at T .Let V0(K,T ) be the values of such straddles at time 0. Then

E0

[a2

T |ST ∈ dK]

= 2∂

∂TV0(K,T )

∂2

∂K2 V0(K,T )

Ratio of a calendar spread of straddles to a butterfly spreadof straddles. StockAug2005 – p.7/40

Review p. 3

• To establish his formula, Dupire uses the Tanaka-Meyer theorem to get

|ST − K| = |S0 − K| +∫ T

0sgn(St − K)dSt +

∫ T

0δK(St)a

2t dt

• Next, take risk-neutral expectations

V0(K, T ) = |S0 − K| +∫ T

0E0

[a2

T δ(ST − K)]dt

E0

[a2

T |ST = K]

=E0

[a2

T δ(ST − K)]

E0 [δ(ST − K)]

=︸︷︷︸

using ∂|S0−K|∂K

=2δ(S0−K)

12E0

[a2

T δ(ST − K)]

∂2

∂K2 V (K, T )


Review p. 4

(In summary)

∂V0(K, T )

∂T=

1

2E0

[a2

T |ST ∈ dK] ∂2

∂K2V (K, T )

So

E0

[a2

T |ST ∈ dK]dt

= 2

∂V0(K,T )∂T

∂2

∂K2 V (K, T )

Therefore the initial expectation of the terminal local variance given that ST = K is twice the ratioof a calendar spread to a butterfly spread.

QED


Synopsis

It is seen that the key element in Dupire’s proof is that

One dimensional Laplacian F = 2δK(S)

where

F = |ST − K|

Therefore the key was that function |ST − K| is afundamental solution of the Laplacian in 1 − D.


Generalization to two assets: Normal Dynamics

First case is where the dynamics is normal as opposed to lognormal

dS1t = atσ1dW1t

dS2t = a2tσ2dW2t t ∈ [0, T ]




dS1t = atσ1dW1t

dS2t = a2tσ2dW2t t ∈ [0, T ]

Here the volatility of the stock Si has an idiosyncratic component σi and a stochasticcomponent at common to both stocks. W1t and W2t are two correlated Brownian motions.The instantaneous variance of dS1t + dS2t at time t.

a2t (σ2

1 + 2ρσ1σ2 + σ22)




dS1t = atσ1dW1t

dS2t = a2tσ2dW2t t ∈ [0, T ]

Here the volatility of the stock Si has an idiosyncratic component σi and a stochasticcomponent at common to both stocks. W1t and W2t are two correlated Brownian motions.The instantaneous variance of dS1t + dS2t at time t.

a2t (σ2

1 + 2ρσ1σ2 + σ22)

The idiosyncratic component of this instantaneous variance is

VI = σ21 + 2ρσ1σ2 + σ2

2


Normal case ct’d

Our objective is to synthesize the bivariate local variance a2t VIδ(S1t − K1, S2t − K2)dt.

Note that this payoff is zero unless (S1T , S2T ∈ (dK1, dK2) and in this event a2T VIdt is the

increment in the quadratic variation of S1 + S2.


Normal case ct’d




Natural candidate to take the place of |S − K| in the 2D case is (up to a constant and anorthogonal transformation) the fundamental solution of the Laplacian in 2 − D.


Normal case ct’d





Ie. g(S1, S2) satisfies

σ21

2

∂2

∂S21

g(S1, S2) + 2ρσ1σ2∂2

∂S1dS2g(S1, S2) +

σ22

2

∂2

∂S22

g(S1, S2) = −VIδ(S1t − K1, S2t − K2)


Normal case ct’d






σ21

2

∂2

∂S21

g(S1, S2) + 2ρσ1σ2∂2

∂S1dS2g(S1, S2) +

σ22

2

∂2

∂S22

g(S1, S2) = −VIδ(S1t − K1, S2t − K2)

Solution known to be:


Normal case ct’d






σ21

2

∂2

∂S21

g(S1, S2) + 2ρσ1σ2∂2

∂S1dS2g(S1, S2) +

σ22

2

∂2

∂S22

g(S1, S2) = −VIδ(S1t − K1, S2t − K2)

Solution known to be:

g(S1, S2) =1

π√

(1 − ρ2)σ1σ2

ln r(S1, S2)

r(S1, S2) ≡

√√√√

1

1 − ρ2

[(S1 − K1

σ1

)2

− 2ρS1 − K1

σ1

S2 − K2

σ2+

(S2 − K2

σ2

)2]


Normal ct’d

We wish to apply Ito’s formula to g just as we did in the 1 − D case for |S − K|. However this isnot possible due to the singularity of g. For now, we illustrate the main idea by proceeding at aformal level. Apply Itô’s formula to the process Gt defined by Gt = g(S1t, S2t) implies:

g(S1T , S2T ) = g(S10, S20) +

∫ T

0

∂

∂S1g(S1t, S2t)

︸︷︷︸

∼S1,t

r2t

dS1t +

∫ T

0

∂

∂S2g(S1t, S2t)

︸︷︷︸

∼S2,t

r2t

dS2t

+

∫ T

0a2

t

σ21

2

∂2

∂S21

g(S1t, S2t) + ρσ1σ2∂2

∂S1∂S2g(S1t, S2t) +

σ22

2

∂2

∂S22

g(S1t, S2t)

︸︷︷︸

−VIδ(S1t−K1,S2t−K2)

dt.(6)

Ito’s formula can be used on a truncated version of g we call gε.


Normal ct’d


g(S1T , S2T ) = g(S10, S20) +

∫ T

0

∂

∂S1g(S1t, S2t)

︸︷︷︸

∼S1,t

r2t

dS1t +

∫ T

0

∂

∂S2g(S1t, S2t)

︸︷︷︸

∼S2,t

r2t

dS2t

+

∫ T

0a2

t

σ21

2

∂2

∂S21

g(S1t, S2t) + ρσ1σ2∂2

∂S1∂S2g(S1t, S2t) +

σ22

2

∂2

∂S22

g(S1t, S2t)

︸︷︷︸


dt.(7)


Idea: Let A = (ΣtΣ)−1 be the inverse of coefficient matrix. Surround singularity (K1, K2)

by an ellipsoid of radius ε, EK(ε) = {X = (x1, x2) : XtAX = ε}


Normal ct’d


g(S1T , S2T ) = g(S10, S20) +

∫ T

0

∂

∂S1g(S1t, S2t)

︸︷︷︸

∼S1,t

r2t

dS1t +

∫ T

0

∂

∂S2g(S1t, S2t)

︸︷︷︸

∼S2,t

r2t

dS2t

+

∫ T

0a2

t

σ21

2

∂2

∂S21

g(S1t, S2t) + ρσ1σ2∂2

∂S1∂S2g(S1t, S2t) +

σ22

2

∂2

∂S22

g(S1t, S2t)

︸︷︷︸


dt.(8)



by an ellipsoid of radius ε, EK(ε) = {X = (x1, x2) : XtAX = ε}Glue the solution g to the solution outside the ball to the solution inside the ball using aquadratic function, while matching both first and second derivatives. Glued together functionis called gε


Normal ct’d


g(S1T , S2T ) = g(S10, S20) +

∫ T

0

∂

∂S1g(S1t, S2t)

︸︷︷︸

∼S1,t

r2t

dS1t +

∫ T

0

∂

∂S2g(S1t, S2t)

︸︷︷︸

∼S2,t

r2t

dS2t

+

∫ T

0a2

t

σ21

2

∂2

∂S21

g(S1t, S2t) + ρσ1σ2∂2

∂S1∂S2g(S1t, S2t) +

σ22

2

∂2

∂S22

g(S1t, S2t)

︸︷︷︸


dt.(9)



by an ellipsoid of radius ε, EK(ε) = {X = (x1, x2) : XtAX = ε}Glue the solution g to the solution outside the ball to the solution inside the ball using aquadratic function, while matching both first and second derivatives. Glued together functionis called gε


Normal Ct’d

• Therefore we obtain

VI

4π√4ε2

∫ T

0a2

t1(S1t,S2t)∈EKε

dt

︸︷︷︸

from approximation to delta function

= gε(S10, S20) − gε(S1T , S2T )

+

∫ T

0

∂

∂S1gε(S1t, S2t)dS1t +

∫ T

0

∂

∂S2gε(S1t, S2t)dS2t.

where VI = σ21 + 2ρσ1σ2 + σ2

2 ,

• Differentiate this result with respect to T

VI

4π√4ε2

a2T 1(S1T ,S2T )∈EK

ε

= − ∂

∂Tgε(S1T , S2T ) +

∂

∂S1gε(S1T , S2T )dS1T +

∂

∂S2gε(S2T , S2T )dS2T

• The left hand side is a payoff that captures an average stochastic volatility in an ε neighborhood

of the strike K = (K1, K2).StockAug2005 – p.14/40

Normal Dynamics

• (from last slide)

VI

4π√4ε2

a2T 1(S1t,S2t)∈EK

ε

= − ∂

∂Tgε(S1T , S2T ) + gε(S1T , S2T )dS1T +

∂

∂S2gε(S2t, S2t)dS2t

• This payoff can be replicated by shorting a calendar spread of contingent ε contingent claims

with maturities T and T + ∆T and by going long ∂gε(S1T ,S2T )∂S1

shares of asset 1 and∂gε(S12,S2T )

∂S2shares of asset 2 during the time interval [T, T + ∆T ]/

• Note the crucial role of the ε approximating payoff. The fundamental solution being unbounded

for n ≥ 2, implies that the untruncated claim would not be a marketable claim.


Normal Dynamics: Conclusion

• Once again, as in 1-asset case, take risk-neutral expectations after having differentiated theresult with respect to T and obtain

VI

4π√4ε2

EQ

[

a2T 1(S1t,S2t)∈EK

ε

]

= − ∂

∂TE0 [gε

T ]

• So that, with the same reasoning as before we obtain

E[

a2T | (S1t, S2t) ∈ EK

ε

]

= Q(

(S1t, S2t) ∈ EKε

) (

− ∂

∂TE [−gε

T ]

)

• So if we know the risk-neutral proability that (S1t, S2t) ∈ EKε we have again the result that

the averaged local variance can be captured by a calendar spread involving our ε-payoff function

gε(S, t).


Modifications needed for Lognormal Dynamics

{(S1t, S2t) : t ∈ [0, T ]}:

dS1t = S1tσ1atdW1t

dS2t = S2tσ2atdW2t, t ∈ [0, T ],

and initial spot prices S10 > 0 and S20 > 0. Here σ1 and σ2 are constants (this assumptionwill be relaxed later) and {W1t : t ∈ [0, T ]} and {W2t : t ∈ [0, T ]} are correlatedone-dimensional standard Brownian motions, with constant correlation coefficient ρ i.e.:

< W1t, W2t >t= ρt.

Letting xit = ln(Sit), i = 1, 2, and applying Itô’s formula, we may write the risk-neutraldynamics in the x variables as:

dxit = −σ2i

2a2

t dt + σiatdWit, t ∈ [0, T ], i = 1, 2.


Solution: Log-normal dynamics

Consider a function U(x1, x2) which solves the following canonical version of the two dimensionalconstant coefficient elliptic equation:

σ21

2

∂2

∂x21

U(x1, x2) + ρσ1σ2∂2

∂x1∂x2U(x1, x2) +

σ22

2

∂2

∂x22

U(x1, x2) + ν1∂

∂x1U(x1, x2) + ν2

∂

∂x2U(x

= −V atmI δ(x1 − k1, x2 − k2),

where ki ≡ ln Ki, i = 1, 2. A fundamental solution can be found in explicit form:

g(x, y, k) ≡ VI

π√4

exp

[

−1

2νtA(x − k)

]

K0

[

Q√

(x − k)tA(x − k)]

,

where 4 again denotes the determinant of the covariance matrix and ν is the vector of drifts:

ν = [2ν1, 2ν2] =[−σ2

1 ,−σ22

]

the matrix A ≡ a−1 is the inverse of a, K0 is the modified Bessel function of the second kind and

degree zero, and Q is the constant νtAν.StockAug2005 – p.18/40

Higher dimension: Lognormal dynamics

Constant coefficients: Everything generalizes naturally.


Variable coefficients

Let us consider again the stochastic differential equations:

dSit = atσijdWijt, i = 1, . . . , n,

where now σij = σij(S1, . . . , Sn) can depend on the stock prices S, but not on time. Withaij = Trace(σσt), and x = ln(S).

We again are lead to consider the following variable coefficient elliptic equation:

Lu = aij(x1, . . . , xn)uxixj + νi(x1, . . . , xn)uxi = −δ(x − k), (10)

in the log variables xi = ln Si, ki = ln Ki, i = 1, . . . , n. Here, as before, νi = −n∑

i=1

σ2ij

2.

In order to explain Hadamard’s approach, we need to recall the idea of geodesic distance.


Riemannian

0.1 Riemannian Metrics and Geodesic Distance

Consider the inverse A = a−1 of the coefficient matrix a. A Riemannian metric is defined by:

ds2 = Aijdxidxj . (11)

The geodesics are curves x(τ) = (x1(τ), . . . , xn(τ)) that minimize the integral:

I =

∫ τ1

τ0

√√√√

n∑

i,j=1

Aij xixjdτ

between fixed limits:

x(τ0) = x, x(τ1) = x.

Consider the square Γ = I2 of the geodesic distance as a function of its upper limit x. It can beshown that this function solves the following first order partial differential equation:

aijΓxiΓxj = 4Γ. (12)StockAug2005 – p.21/40

Riemannian ct’d

Setting pi = Γxi , i, . . . , n, the Hamiltonian associated to this PDE is:

H =1

4aijpipj ,

geodesics are determined by integrating the system of so-called ray equations:

dxi

dt=

1

2aijpj ,

dpi

ds= −1

4

∂ajk

∂xi

pipk, i = 1, . . . , n,

subject to the initial conditions:

xi(0) = xi, pi(0) = qi.

The qi parametrize the direction of the tangent line to the geodesic through the base point x

in the tangent plane to the surface defined by the Riemannian surface associated to ourmetric Aijdsidsj .


Hadamard’s Method for Determining the Fundamental

We wish to solve

aij(x, t)uxixj + bi(x, t)uxi = −δ(x − K),

i.e. to determine a fundamental solution of the elliptic equation with variable coefficients.

There are several methods.

Perhaps the most geometrically appealing and most constructive in flavour is Hadamard’smethod.(Brief Introduction to Hadamard’s Method)

Recall the Riemannian metric associated to the principal part of the el liptic differential operator . Ie.

Aijdsidsj

where

a = a−1 a = {aij}StockAug2005 – p.23/40

Hadamard’s Method Ct’d

There are some technical differences in the method according as to whether the underlyingspatial dimension is even or odd. For reasons of brevity, here we limit our description to thecase of an odd dimension.

( Odd n ≥ 3 )

F =U

Γn−2

2

, (13)

where m ≡ n−22

and where U is an infinite series of the following form:

U0 + U1Γ + U2Γ2 + . . . =

+∞∑

l=1

UlΓl.

where Γ is the square of the geodesic distance in the Riemannian metric defined above.


System of ODE’s along geodesics

Plugging this ansatz into the PDE and collecting terms multiplying the same power of Γ

leads to the iterative determination of solution by solving system of ODE’sWe have:

sdU0

ds+

M − 2n

4U0 = 0

4sdUj

ds+ (M + 4j − 2n)Uj − LUj−1

j − m= 0, j ≥ 1.

By means of the substitution y = ln s and using an integrating factor, the second set ofequations can be written as:

d

ds

(sjdUj

U0

)

=sj−1LUj−1

4(j−m)U0, l ≥ 1,

where here: .

M ≡ −L(Γ).


Solution

The solution of these equations can be iteratively determined in the form:

U0(x, x′) =γ(m)

4πn2

exp

(

−∫ s

0

M(x(u), y) − 2n

4udu

)

Uj(x, y) =U0(x, y)

4(j − m)sj

∫ s

0

uj−1(LUj−1)(x(u), y)

U0(x, y)du, j ≥ 1.

Here, x(u) denotes the path on the unique geodesic joining y and x.

Hadamard shows that for small enough neighborhood of center the resulting seriesconverges, provided the coefficients of the solution are assumed to be analytic.

The geodesic neighborhood must also be taken to be sufficiently small so as to ensure thatgeodesics do not cross.


Idea of Construction of gε payoff in the simplest

.We will illustrate the construction in the simplest 2 − D case where the stocks follow anormal dynamics.

Idea of construction is a natural extension of a similar construction used in one of the proofsof the Meyer-Tanaka formula for local time.

(Tanaka ) Generalized Ito’s formula for convex functions with corner. For illustration considersimpler Tanaka’s formula for taking stochastic differential of g(Wt) where

f(x) = |x|

It says (we saw it at the beginning of the talk):

|Wt| = |W0| +∫ t

0sign(Ws)dWs + Lt(ω)

where Lt, the local time at zero defined by

Lt = limε→0

1

2ε|{s ∈ [0, t]; Ws ∈ (−ε, ε)] limit in L2(P ) sense


Approximating function

One possible proof of Tanaka formula approximates |x| by aquadratic function in the region |x| ≤ ε.

gε(x) =

{

|x| if |x| ≥ ε1

2(ε + x

2

2) if |x| < ε


Illustration Tanaka formula

parabola in this region

f(x) = |x|


Approximating Tanaka formula

gε(x) approximating function we obtain the approxating formula

gε(Wt) = gε(W0) +

∫ t

0(gε)′(Ws)dWs +

1

2ε|{s ∈ [0, t]; Ws ∈ (−ε, ε)}|

What is the analogue to this approach in dimension n ≥ 2 ?Answer: Replace the fundamental solution inside the ellipsoid by a multivariate quadraticfunction in such a way that:i) The function and its first derivatives paste smoothly along the boundary of the ellipsoid

ii) The first derivatives are piecewise C1 and the jumps in the second derivatives occur butthese are globally bounded.

For n = 2 such a quadratic function can be shown to be

g(x, ξ) = − 1

4π√

∆ε2A2(x, ξ) +

1

4π√

∆(1 − ln(ε2)) for n = 2

where A2(x, ξ) = −2∑

i,j=1Aij(xi − ξi) (recall A = a−1).


Radon-Transform Approach to multi-D Breeden

Payoff of a basket option:

(∑

wiSi − K)+

Normalize this payoff

|w|(∑

w′iSi − q)+ where ξi =

wi

|w| , q =K

|w|

Consider C(ξ, q) i.e. the prices of all basket options with ξ ∈ Sn−1, q ∈ [0,∞] . Letp(S1, · · · , Sn) denote the risk-neutral density, so that

C(ξ, q) =

∫

<+

(ξ · S − q)+p(S1, · · · , Sn)dS1 · · · dSn


Radon Ct’d

∂2

∂q2C(ξ, q) =

∫

S·ξ−q=0

p(S1, · · · , Sn)dS1 · · · dSn

ie. C(ξ, q) corresponds to the integral of density p on a hyperplane with normal ξ ∈ Sn−1

and a distance q from the origin.

∂2

∂q2C(ξ, q) = R[p](ξ, q)

where R[p](ξ, q) is the Radon transform of the density p.


Radon-Conclusion

Now key observation is that there is an inversion formula for the Radon transform whichreads

v(ξ, q) = R[p](ξ, q)

⇔

p(S1, · · ·Sn) =

∫

|ξ|=1h(S · ξ, ξ)dξ

Here definition of h depends on whether dimension is n is even or odd:

h(q, ξ) ≡ (i)n−1

2(2π)n−1

∂n−1

∂qn−1R[f ](ξ, q) n odd

h(q, ξ) ≡ (i)n

2(2π)n−1H

[∂n−1

∂pn−1R[f ](ξ, q)

]

(q) n even

where H[g(p)](t) denotes the Hilbert transform of the function g(p):

H[g(p)](t) ≡ 1

π

∫ ∞

−∞

g(p)

p − tdp, (14)


Closed form solutions for option pricingand interest models on two assets

– p.1/58

The result is to appear in

International Journal of Theoretical and Applied Finance.

– p.2/58

Black-Scholes equation

∂C

∂τ+σ2

2S2

∂C2

∂S2+ rS

∂C

∂S− rC = 0

t = T − τ

∂C

∂t=σ2

2S2

∂C2

∂S2+ rS

∂C

∂S− rC

ξ = lnS

∂C

∂t=σ2

2

∂C2

∂ξ2+

(

r − σ2

2

)

∂C

∂ξ− rC

– p.3/58


c(ξ, t) = ertC(ξ, t)

∂c

∂t=σ2

2

∂c2

∂ξ2+

(

r − σ2

2

)

∂c

∂ξ

x = ξ +(

r − σ2

2

)

t

∂c

∂t=σ2

2

∂2c

∂x2

– p.4/58


In total, we have done the transformation

t = T − τ

x = lnS +

(

r − σ2

2

)

(T − τ)

c = er(T−τ)C

which transforms Black-Scholes equation to heat equation.

– p.5/58

Merton

Merton’s arguments in his 1973 paper imply moregenerally that the arbitrage-free value C of manyderivatives satisfies

∂C

∂τ+σ2(S, τ)

2S2

∂C2

∂S2+ b(S, τ)

∂C

∂S− r(S, τ)C = 0

with three variable coefficients σ(S, τ), b(S, τ) and r(S, τ).

Carr-Lipton-Madan(2000) characterized the entire set ofvariable coefficients {σ(S, τ), b(S, τ), r(S, τ)} which permitthe above one state variable valuation PDE to betransformed to the heat equation.

– p.6/58

Group classification

Lie’s group classification of linear second order PDE with 2independent variables

Put +Qux +Ruxx + Su = 0

P,R 6= 0 and P,Q,R, S are functions of t and x.

Any equation is admitted by the generators of the trivial

symmetries u∂

∂uand φ(t, x)

∂

∂u. Any equation can be

reduced to the form

vτ = vyy + Z(τ, y)v

by a transformation as y = α(t, x), τ = β(t) and v = γ(t, x)u

with αx 6= 0 and βt 6= 0.– p.7/58


An equation admits symmetry generators generated by

u∂

∂u, φ(t, x)

∂

∂u,

∂

∂τ

can be reduced to the form

vτ = vyy + Z(y)v

by a transformation.

– p.8/58



u∂

∂u, φ(t, x)

∂

∂u,

∂

∂τ, 2τ

∂

∂τ+ y

∂

∂y

τ2∂

∂τ+ τy

∂

∂y−(

1

4y2 +

1

2τ

)

v∂

∂v


vτ = vyy +A

y2v

by a transformation, where A is a constant.

– p.9/58



u∂

∂u, φ(t, x)

∂

∂u,

∂

∂τ2τ

∂

∂τ+ y

∂

∂y,

τ2∂

∂τ+ τy

∂

∂y−(

1

4y2 +

1

2τ

)

v∂

∂v

∂

∂y, 2τ

∂

∂y− yv

∂

∂v


vτ = vyy

by a transformation.

– p.10/58

Convenient Variables

The main difficulty in integrating a given differential equationlies in introducing convenient variables, which there is norule for finding. Therefore we must travel the reverse pathand after finding some notable substitution, look forproblems to which it can be successfully applied.

- Jacobi, Lectures on Dynamics, 1847.

– p.11/58

Lie’s Symmetry Analysis

From geometric point of view, the great Norwegian mathe-

matician Sophus Lie created a systematic way for analyzing

solutions of differential equations. His theory nowadays has

been known as Lie’s symmetry analysis of differential equa-

tion. Up to now, Lie’s theory has been greatly extended and

used to find closed form solutions, to classify equations up to

some unknown functions, to find fundamental solutions and

so on.

– p.12/58

Symmetry group

A symmetry group of a system S of differential equations isa local group of transformations, G, acting on some opensubset M ⊂ X × U in such a way that "G transformssolutions of S to other solutions of S".

Allowing arbitrary nonlinear transformations of both theindependent and dependent variables in the definition ofsymmetry.

– p.13/58

Prolongation of f

f : X → U , u = f(x) u(n) := pr(n)f(x) : X → U (n): the n-th

prolongation of f is defined by

uJ = ∂Jf(x)

For example, u = f(x, y), the second prolongationu(2) = pr

(2)f(x, y) is given by

(u;ux, uy;uxx, uxy, uyy) =

(

f ;∂f

∂x,∂f

∂y;∂2f

∂x2,∂2f

∂x∂y,∂2f

∂y2

)

.

– p.14/58

Solutions of DE’s

An n-th order differential equation in p independentvariable is given as

∆(x, u(n)) = 0

involving x = (x1, · · · , xp), u and the derivatives of u withrespect to x up to order n.

∆ can be viewed as a function from the jet spaceX × U (n) to R, i.e., ∆ : X × U (n) → R.

Hence, an n-th order differential equation can beregarded as a subvariety S∆ := {∆(x, u(n)) = 0} in then-jet space X × U (n).

– p.15/58

Solutions of DE’s

A smooth solution of the given n-th order differentialequation in this sense is a smooth function u = f(x) suchthat

∆(x,pr(n)f(x)) = 0

whenever x lies in the domain of f .

Hence a solution can be regarded as a function u = f(x)

such that the graph of the n-th prolongation pr(n)f is

contained in the subvariety S∆, i.e.,

Γ(n)f := {(x,pr

(n)f(x))} ⊂ S∆

– p.16/58

Prolongation of group actions

G: local group of transformations acting on M ⊂ X × U

The n-th prolongation pr(n)G of G is an induced local

action of G on the n-jet space M (n) which is defined sothat it transforms the derivatives of functions u = f(x)

into the corresponding derivatives of the transformedfunction u = f(x).

This can be done by calculating the transformation ofn-th order Taylor polynomial.

– p.17/58

Invariance of DEs

∆(x, u(n)) = 0: n-th order differential equation definedover M

S∆ ⊂M (n): corresponding subvariety

G: local group of transformations acting on M whoseprolongation leaves S∆ invariant, i.e.,

(x, u(n)) ∈ S∆ ⇒ pr(n)g · (x, u(n)) ∈ S∆ for all g ∈ G.

G is a symmetry group of the differential equation !

– p.18/58

Prolongation of vector fields

v: vector field on M

exp(εv): corresponding one-parameter group

The n-th prolongation pr(n)

v of v is a vector field on the jetspace M (n) and defined to be the infinitesimal generator ofthe corresponding prolonged group pr

(n)[exp(εv)], i.e.,

pr(n)

v|(x,u(n)) =d

dε

∣

∣

∣

∣

ε=0

pr(n)[exp(εv)](x, u(n))

for any (x, u(n)) ∈M (n).

– p.19/58

Infinitesimal invariance

Lie’s Theorem∆(x, u(n)): n-th order differential equation

Let v be a vector field. Then v generates a oneparameter local group of symmetries if and only if

pr(n)

v[∆(x, u(n))] = 0

whenever ∆(x, u(n)) = 0.

The set of all infinitesimal symmetries form a Lie algebraof vector fields on X × U . Moreover, if this Lie algebra isfinite dimensional, the symmetry group of the equation isa local Lie group of transformations acting on X × U .

– p.20/58

Prolongation formula

The second prolongation pr(2)

v of a vector field v with

v =∑

i

ξi(t, x, u)∂

∂xi+ φ(t, x, u)

∂

∂u

is as

pr(2)

v = v +∑

i

φi ∂

∂uxi+∑

i,j

φij ∂

∂uxixj

where the φ’s are given by the prolongation formulas

– p.21/58

Total differentiation

φi = Di(φ)−∑

j

uxjDi(ξj)

φij = Dj(φi)−

∑

k

uxixkDj(ξk)

and the D’s are the so-called total differentiation.

Di =∂

∂xi+ uxi

∂∂u + uxixk

∂∂u

xk+ · · ·

– p.22/58

Determining equation

The determining equation

pr(n)

v[∆(x, u(n))] = 0 whenever ∆(x, u(n)) = 0

is a linear homogeneous partial differential equation of thesecond order for the unknown functions ξ, τ, φ of the"independent variables" t, x, u.

t, x, u, uxi , uxixj , · · · , are regarded as "independent" ones.

The determining equation decomposes into a system ofseveral equations.

The resulting system is an overdetermined system whichcan be solved analytically.

– p.23/58

1 dim heat equation

ut = uxx

The heat equation can be identified with the linearsubvariety in X × U (2) determined by the vanishing of∆(x, t, u(2)) = ut − uxx.

v = ξ(t, x, u) ∂∂x + τ(t, x, u) ∂∂t + φ(t, x, u) ∂

∂u

pr(2)

v = v + φx ∂∂ux+ φt ∂

∂ut+ φxx ∂

∂uxx+ φxt ∂

∂uxt+ φtt ∂

∂utt

– p.24/58

Infinitesimal invariance

pr(2)v[∆(x, t, u(2))] = φt − φxx = 0

φt = φt − ξtux + (φu − τt)ut − ξuuxut − τuu2t

φxx = φxx + (2φxu − ξxx)ux − τxxut + ξuuu3x − τuuu

2xut + (φu −

2ξx)uxx − 2τxuxt − 3ξuuxuxx − τuutuxx − 2τuuxuxtReplacing ut by uxx

– p.25/58

Determining equation

Monomial Coefficient

uxuxt −2τu = 0uxt −2τx = 0u2xx −τu = −τuu2xuxx 0 = −τuuuxuxx −ξu = −2τxu − 3ξuuxx φu − τt = −τxx + φu − 2ξxu3x 0 = −ξuuu2x 0 = φuu − 2ξxuux −ξt = 2φxu − ξxx

1 φt = φxx

– p.26/58

General solution

ξ(x, t) = c1 + c4x+ 2c5t+ 4c6xt

τ(t) = c2 + 2c4t+ 4c6t2

φ(x, t, u) = (c3 − c5x− 2c6t− c6x2)u+ α(x, t)

c1, · · · , c6 are arbitrary constants and α(x, t) an arbitrarysolution of the heat equation.

– p.27/58

Infinitesimal symmetries

Finite (six) dimensional subalgebra

v1 = ∂x

v2 = ∂t

v3 = u∂u

v4 = x∂x + 2t∂t

v5 = 2t∂x − xu∂u

v6 = 4tx∂x + 4t2∂t − (x2 + 2t)u∂u

Infinite dimensional subalgebra

vα = α(x, t)∂u,

where α is an arbitrary solution of the heat equation.– p.28/58

Symmetry group

G1 : (x+ ε, t, u)

G2 : (x, t+ ε, u)

G3 : (x, t, eεu)

G4 : (eεx, e2εt, u)

G5 : (x+ 2εt, t, u · exp(−εx− ε2t))

G6 :

(

x

1− 4εt ,t

1− 4εt , u√1− 4εte −εx

2

1−4εt

)

Gα : (x, t, u+ εα(x, t))

– p.29/58

Symmetry group

G3, Gα: the linearity of the heat equation

G1, G2: time and space invariance of the equation,reflecting that heat equation is of constant coefficients.

G4: well-known scaling symmetry

G5: Galilean boost to a moving coordinate frame

G6: fundamental solution

– p.30/58

Solutions from symmetry

If u = f(x, t) is a solution of the heat equation, so are thefollowing functions

u1 = f(x− ε, t)

u2 = f(x, t− ε)

u3 = eεf(x, t)

u4 = f(e−εx, e−2εt)

u5 = e−εx+ε2tf(x− 2εt, t)

u6 =1√1 + 4εt

e−εx21+4εtf(

x

1 + 4εt,

t

1 + 4εt)

uα = f(x, t) + εα(x, t)

– p.31/58

Finkel

Finkel has classified the 2 dimensional parabolicequations of the form

ut −1

2∆u+M(x, y)u = 0.

Symmetry operators in the following are of the form

X = τ(t)∂t + ξ(x, y, t)∂x + η(x, y, t)∂y + ϕ(x, y, t)u∂u

where

ξ =τt

2+ γy + ξ0(t)

η =τt

2− γx+ η0(t)

ϕ = − τtt4(x2 + y2)− ξ′0x− η′0y + φ(t)

– p.32/58

Classification by Finkel - 1.1a

Case 1.1a, Dimension of group = 4

M =C0

x2+ by + c0, C0 6= 0

τ = δ2t2 + δ1t, γ = ξ0 = 0, η0 =

bδ2

2t3 +

3bδ1

4t2 + β1t+ β0,

φ = − b2δ2

8t4 − b2δ1

4t3 − ( bβ1

2+ c0δ2)t

2 − (δ2 + c0δ1 + bβ0)t

– p.33/58

Classification by Finkel- 1.1b

Case 1.1b, Dimension of group = 4

M =C0

x2+ cr2 + by + c0, C0 6= 0

τ = δ1e2√

2ct + δ2e−2√

2ct, γ = ξ0 = 0,

η0 =bδ1√2ce2√

2ct − bδ2√2ce−2

√2ct + β1e

√2ct + β2e

−√

2ct,

φ = −(√

2c+ c0 +b2

4c

)

δ1e2√

2ct +

(√2c− c0 −

b2

4c

)

δ2e−2√

2ct

− bβ1√2ce√

2ct +bβ2√2ce−√

2ct

– p.34/58

Classification by Finkel- 1.2a


M = C(θ)1

r2+ c0

τ = δ2t2 + δ1t, γ = ξ0 = η0 = 0, φ = −c0δ2t2 − (δ2 + c0δ1)t,

where C(θ) 6= (C0 cos θ + C1 sin θ)−2 and C ′ 6= 0.

– p.35/58



M = C(θ)1

r2+ cr2 + c0

τ = δ1e2√2ct + δ2e

−2√2ct, γ = ξ0 = η0 = 0,

φ = −(√2c+ c0

)

δ1e2√2ct +

(√2c− c0

)

δ12e−2√2ct

with C(θ) 6= (C0 cos θ + C1 sin θ)−2 and C ′ 6= 0.

– p.36/58

Classification by Finkel- 1.3

Case 1.3, Dimension of group = 1

M = C(λ log r + θ)1

r2+ c0, C ′ 6= 0 6= λ

τ =2γ

λt, ξ0 = η0 = 0, φ = −2c0γ

λt

– p.37/58



M = C01

r2+ ax+ by + c0, C0 6= 0

τ = δ2t2 + δ1t, ξ0 = η0 = 0, φ = −c0δ2t2 − (δ2 + c0δ1)t

with the constraints δ1 = δ2 = 0 if a 6= 0 or b 6= 0.

– p.38/58



M = C0r−2 + cr2 + ax+ by + c0, C0 6= 0

τ = δ1e2√2ct + δ2e

−2√2ctb, ξ0 = η0 = 0,

φ = −(√2c+ c0)δ1e

2√2ct + (

√2c− c0)δ2e

−2√2ct

with the constraint δ1 = δ2 = 0 if a 6= 0 or b 6= 0.

– p.39/58



M = ax+ by + c0

τ = δ2t2 + δ1t,

ξ0 =aδ2

2t3 +

1

4(3aδ1 − 2bγ)t2 + α1t+ α0,

η0 =bδ2

2t3 +

1

4(3bδ1 + 2aγ)t

2 + β1t+ β0,

φ = −18(a2 + b2)δ2t

4 − 1

4(a2 + b2)δ1t

3 − ( 12(aα1 + bβ1) + c0δ2)t

2

−(δ2 + c0δ1 + aα0 + bβ0)t

– p.40/58



M = cr2 + ax+ by + c0

τ = δ1e2√

2ct + δ2e−2√

2ct,

ξ0 =aδ1√2ce2√

2ct +aδ2√2ce−2

√2ct + α1e

√2ct + α2e

−√

2ct +bγ

2c,

η0 =bδ1√2ce2√

2ct +bδ2√2ce−2

√2ct + β1e

√2ct + β2e

−√

2ct − aγ

2c,

φ = −(√

2c+ c0 +a2 + b2

4c

)

δ1e2√

2ct +

(√2c− c0 −

a2 + b2

4c

)

δ2e−2√

2ct

−aα1 + bβ1√2c

e√

2ct +aα2 + bβ2√

2ce−√

2ct

– p.41/58

Classification by Finkel- 1.6

Case 1.6, Dimension of group = 1

M = C(r) + dθ

τ = ξ0 = η0 = 0, φ = dγt

with C(r) 6= C0r−2 + C1r

2 + c0 if d = 0.

– p.42/58



M = cx2 + ax+ by + c0

τ = γ = 0, ξ0 = α1e√2ct + α2e

−√2ct, η0 = β1t+ β0,

φ = − aα1√2ce√2ct +

aα2√2ce−√2ct − bβ1

2t2 − bβ0t

– p.43/58



M = c(x2 − y2) + ax+ by + c0

τ = γ = 0, ξ0 = α1e√

2ct + α2e−√

2ct, η0 = β1e√−2ct + β2e

−√−2ct,

φ = − aα1√2ce√

2ct +aα2√2ce−√

2ct − bβ1√−2c

e√−2ct +

bβ2√−2c

e−√−2ct

– p.44/58



M = C(x) + by

τ = γ = ξ0 = 0, η0 = β1t+ β0, φ = −bβ12t2 − bβ0

with C0x2 + ax+ c0 6= C(x) 6= C0

x2 + c0.

– p.45/58



M = C(x) + cy2 + by

τ = γ = ξ0 = 0, η0 = β1e√2ct + β2e

−√2ct,

φ = − bβ1√2ce√2ct +

bβ2√2ce−√2ct

with C0x2 + ax+ c0 6= C(x) 6= C0

x2 + cx2 + c0.

– p.46/58

Finding Fundamental Solutions

The fundamental solutions are determined by

Restricting the symmetry operator of the equationut − 1

2∆u+Mu = 0 to the initial value problem{

ut − 12∆u+Mu = 0

u(x, 0) = δξ(x)

Finding the invariant solutions to the restricted symmetryoperators.

– p.47/58

Convention

The following convention are used in the following slides

Sign convention

± ={

+ when c > 0

− when c < 0

Notation convention

cot± =

{

coth(√2ct) when c > 0

cot(√−2ct) when c < 0

,

and similarly for sin±, sec±, csc± and tan±.

– p.48/58

Fundamental Solution - 1.1a

ut − 12∆u+ (

C0

x2 + by + c0)u = 0

Fundamental Solution (Transition probability)

G(1a) = K( x

t

)κ e−2 ξxt−c0t

tΦ

(

κ, 2κ, 2ξx

t

)

×

exp

{

− (x− ξ)2 + (y − η)2

2t− b

2(y + η)t+

b2

24t3}

,

κ = 1+√1+8C0

2

Φ(κ, 2κ, z) is a confluent hypergeometric function of order (κ, 2κ), i.e., Φ satisfies thedifferential equation zv′′ + (2κ− z)v′ − κv = 0, with the asymptotic behavior

Φ→ Γ(2κ)

Γ(κ)ezz−κ as z → +∞.

– p.49/58

Fundamental Solution - 1.1b

ut − 12∆u+ (

C0

x2 + c(x2 + y2) + by + c0)u = 0


G(1b) = K(x csc±)κeγx csc± −c0t csc± Φ(κ, 2κ,−2γx csc±)×

exp

{

−√

|c|2cot±(x2 + ξ2)

}

×

exp

[

− cot±{

√

|c|2(y − η sec±)2 ± b

√

2|c|(1− sec±)(y − η sec±)

}]

×

exp

[

b2

4|c|

(√

2

|c| (csc±− cot±)± t

)

−(

bη√

2|c|±√

|c|2(η2)

)

tan±]

– p.50/58


ut − 12∆u+ (C(θ)

1r2 + c0)u = 0

Fundamental Solution (Transition probability) at (0, 0)

G(2a) =Ke−c0t

te−

x2+y2

2t l(θ)

where l is a solution to

l′′(θ) = 2C(θ)l(θ) for θ ∈ [0, 2π]

with the boundary condition l(0) = l(2π)

– p.51/58


ut − 12∆u+ (C(θ)

1r2 + c(x2 + y2) + c0)u = 0


G(2b) =Ke−c0t

sin±exp

[

−√

|c|2cot±(x2 + y2)

]

l(θ),

where l is a solution to

l′′(θ) = 2C(θ)l(θ) for θ ∈ [0, 2π]

with the boundary condition l(0) = l(2π)

– p.52/58


ut − 12∆u+ (

C0

r2 + c0)u = 0


G(4a) =Ke−c0t

t

(

√

x2 + y2

t

)λ

Φ

(

1 + λ, 1 + λ,−x2 + y2

2t

)

λ =√2C0

Φ is the incomplete Gamma functionΓ(−λ, z) =

∫∞z

t−λ−1e−tdt.

– p.53/58


ut − 12∆u+ (

C0

r2 + c(x2 + y2) + c0)u = 0


G(4b) =Ke−c0t

sin±

(

√

x2 + y2

sin±

)λ

e−

√

|c|2(x2+y2) cot±

,

where λ =√2C0

– p.54/58


ut − 12∆u+ (ax+ by + c0)u = 0


G(5a) =Ke−c0t

texp

[

− (x− ξ)2 + (y − η)2

2t−(

a

2(x+ ξ) +

b

2(y + η)

)

t+a2 + b2

24t3]

– p.55/58


ut − 12∆u+ (c(x

2 + y2) + ax+ by + c0)u = 0


G(5b) =Ke−c0t

sin±×

exp

[

a2 + b2

4|c|

(√

2


)

−(

aξ + bη√

2|c|±√

|c|2(ξ2 + η2)

)

tan±]

×

exp

[

− cot±{

√

|c|2(x− ξ sec±)2 ± a

√

2|c|(1− sec±)(x− ξ sec±)

}]

×

exp

[

− cot±{

√

|c|2(y − η sec±)2 ± b

√

2|c|(1− sec±)(y − η sec±)

}]

– p.56/58


ut − 12∆u+ (cx

2 + ax+ by + c0)u = 0


G(7a) =Ke−c0t√t√sin±

exp

[

a2

4|c|

(√

2


)

−(

aξ√

2|c|±√

|c|2ξ2

)

tan±

+b2t3

24− b

2(y + η)t

]

×

exp

[

− cot±{

√

|c|2

(

x− ξ sec±)2 ± a

√

2|c|(

1− sec±)

(x− ξ sec±)

}

− (y − η)2

2t

]

.

– p.57/58


ut − 12∆u+ (c(x

2 − y2) + ax+ by + c0)u = 0


G(7b) =Ke−c0t√sin∓

√sin±

×

exp

[

a2

4|c|

(√

2


)

−(

aξ√

2|c|±√

|c|2ξ2

)

tan±]

×

exp

[

b2

4|c|

(√

2

|c| (csc∓− cot∓)∓ t

)

−(

bη√

2|c|∓√

|c|2η2

)

tan∓]

×

exp

[

− cot±{

√

|c|2

(

x− ξ sec±)2 ± a

√

2|c|(

1− sec±)

(x− ξ sec±)

}]

×

exp

[

− cot∓{

√

|c|2

(

y − η sec∓)2 ∓ b

√

2|c|(

1− sec∓)

(y − η sec∓)

}]

– p.58/58

Topic I

aper I: Closed form solution for quadratic and inverse quadratic mod els of the term structure

ut − ∆u + Pu = 0, fundamental PDE or canonical form

where P is a potential.Typical examples of potentials arising in finance and in mathematicalphysics are

P = 0 heat equation

P = x21 + · + x2

n harmonic oscillator

P =λ

|x|2 inverse square potential


Topic I, ct’d

The reason that the PDE ut − ∆u + Pu = 0 is so importantderives from it’s being a prototype of a canonical form for asecond order parabolic differential equation with variable

coefficients . Indeed in the case of one spatial dimension it is the

canonical form. I.e.

Proposition 1 Any parabolic differential equation in one spatialvariable

ut −1

2a(x, t)uxx − b(x, t)ux + c(x, t)u = 0

can, by a change of independent and dependent variables, be re-

duced to canonical form.StockAug2005 – p.35/40

Topic 1, ct’d

This result goes back to Lie and it’s importance was redis-

covered in modern times by Bluman


Finkel’s classification of diffusions on<2 × [0, T ]

A Lie symmetry transformation is a transformation of theindependent variables x, y, t and dependent variable y of theform

(x, y, t, u) 7→ (x, y, t, u)

x = α1(x, y, t) y = α2(x, y, t)τ = β(t) v = γ(x, y, t)β(x, y, t

such that the equation

ut −1

2∆u + Pu = 0

is has the same form in the new variables.StockAug2005 – p.37/40

Use of Lie’s symmetry analysis

Lie’s symmetry analysis can be used to find similarityvariables which allow one, depending on the size of the Liesymmetry group to either

Reduce the PDE to an ODE.

Or

Integrate the PDE in closed form


Application of Finkel’s classification

An interesting application of Lie’s method is in expressing aparticular solution of the PDE in closed form, namely thefundamental solution.The application in mathematical finance is in expressing thefundamental solution of the zero coupon bond P (t, T )

P (t, T ) = E

[

exp

(∫ T

t

−r(X1, X2, t)ds

)]

in a situation where the pricing equation of P (t, T ) under theforward measure is for instance of the form

Pt −1

2∆P + a(t)X2

1 + b(t)X1X2 + c(t)X2

2 = 0,

a special case of the so-called quadratic term structuremodels. Much more general quadratic models have been


Conclusions

It is possible to define a multi-asset local variance contract.

In the context of multi-dimensional local volatility models, the construction of thecorresponding contingent claims involves solving for the fundamental solution of a variablecoefficient elliptic operator.

The construction is only local. Can it be extended ?

How many terms in the Hadamard expansion are needed to accurately approximate thefundamental solution in practice?

It is possible to extend Breeden and Litzenberger’s formula to n assets given the knowledgeof baskets with a continuum of strikes and of weights.

Since there are not sufficiently many weights and strikes, how can to best exploit thegeneralized Breeden and Litzenberger formula.


some applications of classical pde in ﬁnance · using hadamard’s method and lie’s symmetry...

Documents