Exercise 2.1.2 A Nonhyperbolic Cauchy Problem for the Wave Equation)
9.3 The inverse problem of option pricing
As an important example of inverse problem with single measurements for parabolic equations we will consider determination of so-called volatility coeffi- cientσof a parabolic equation for option prices discovered by Black and Scholes in 1973. This discovery revolutionized financial markets in 1990-s. In 1997 Merton and Scholes were awarded the Nobel prize in economics.
For any stock price, 0<s<∞, and time t,0<t <T, the price u for an option expiring at timeT satisfies the following Black-Scholes partial differential equation
(9.3.1) ∂u/∂t+1/2s2σ2(s,t)∂2u/∂s2+sà∂u/∂s−r u=0.
Here,σ(s,t) is the volatility coefficient that satisfies 0<m< σ(s,t)<M <∞ and is assumed to belong to the H¨older spaceCλ( ¯ω),0< λ <1,on some interval ωand outside this interval, andàandrare, respectively, the risk-neutral drift and the risk-free interest rate assumed to be constants. The backward in time parabolic equation (9.3.1) is augmented by the final condition specified by the payoff of the call option with the strike priceK
(9.3.2) u(s,T)=(s−K)+=max(0,s−K), 0<s.
By using the logarithmic substitutiony=logsand Theorem 9.1 one can show that there is a unique solutionuto (9.3.1), (9.3.2) which belongs toC1((0,∞)×(0,T]) and toC((0,∞)×[0,T]) and satisfies the bound|u(s,t)|<C(s+1).
All coefficients of the equation (9.3.1) except ofσare known. Volatility coeffi- cient is a fundamental characteristic of options market, so it is highly desirable to know it. The inverse problem of option pricing seeks forσ given
(9.3.3) u(s∗,t∗;K,T)=u∗(K), K ∈ω∗.
Heres∗is market price of the stock at timet∗, andu∗(K) denote market price of options with different strikesK for a given expiryT. The additional data (9.3.3) are available from current trading. One can find them on Internet. We will attempt to recover volatility in the same intervalω∗containings∗.
To obtain our results we will use that the option premiumu(., .;K,T) satisfies the equation dual to the Black-Scholes equation (9.3.1):
(9.3.4) ∂u/∂T −1/2K2σ2(K,T)∂2u/(∂K2)+àK∂u/(∂K)+(r−à)u =0.
The equation (9.3.4) was found by Dupire in 1994 and rigorously justified, for example, in [BI].
The logarithmic substitution
y=ln(K/s∗), τ =T −t, U(y, τ)=u(;K,T), a(y, τ)=σ(s∗ey,T −τ), (9.3.5)
9.3. The inverse problem of option pricing 271 transforms the dual equation (9.3.4) and the additional (market) data into the following inverse parabolic problem
∂U/∂τ =1/2a2(y, τ)∂2U/∂y2−(1/2a2(y, τ)+à)∂U/∂y +(à−r)U, y∈R, 0< τ <T,
(9.3.7) U(y,0)=s∗(1−ey)+, y∈R with the final observation
(9.3.8) U(y, τ∗)=U∗(y), y∈ω.
whereωis the intervalω∗iny-variables.
We list available theoretical results.σ(s,t) can be found from the dataU∗(K,T) due to the Dupire’s equation (9.3.4). In many important cases temporal data are not available or sparse. In any case these data are not available for future, when knowledge of volatility is most important for useful predictions. This is why we think it is reasonable to look forσ =σ(s) and that what we will do in the remaining part of section 9.3.
Theorem 9.3.1. Let U1and U2be two solutions to the initial value problem with a =a1and a =a2 and let U1∗,U2∗ be the corresponding final data. Letω0be a non-void open subinterval ofω.
If U1∗(y)=U2∗(y)for y∈ωand a1(y)=a2(y)for y∈ω0, then a1(y)=a2(y) when y∈ω.
If, in addition, a1(y)=a2(y)when y∈ω∪(R\ω)and ifωis bounded, then there is a constant C depending only on|a1|1(ω),|a2|2(ω), ω, ω0, τ∗, λsuch that
|a2−a1|λ(ω)≤C|U2∗−U1∗|2+λ(ω)
PROOF. To show uniqueness we subtract two equations forU2andU1to get
∂U/∂τ=1/2a22(y)∂2U/∂y2−(1/2a22(y)+à)∂U/∂y+(à−r)U+α1(y, τ)f(y) where
U=U2−U1, α1=∂2U1/∂y2−∂U1/∂y, f(y)=1/2(a22(y)−a21(y)).
Besides,U(y,0)=0. By Theorem 9.5 solutions of an initial value parabolic prob- lem with time independent coefficients are time analytic. SinceU(, τ∗)=0, f =0 onω0we conclude from the differential equation that∂U/(∂τ)(, τ∗)=0 onω0. Re- peating this argument we conclude that allτ-derivatives ofUare zero onω0× {τ∗}.
By analyticityU =0 onω0×(0, τ∗). By using (9.3.7) the functionα1 satisfies (in distributional/generalized sense) the initial value problem
∂τα1=1/2(∂2y−∂y)(a12(y)α1)−1/2à∂yα1+(à−r)α1, y∈R, 0< τ <T α1(y,0)=δ(y)
whereδis the Dirac delta-function. Soα1is the Green’s function for this Cauchy problem and henceα(y, τ)>0. By Theorem 9.2.6 f =0 onω.
Stability estimate follows from uniqueness and Theorems 9.1, 9.1.1 by
compactness-uniqueness arguments.
The assumption thata(y) is known on a subinterval ofωis probably not neses- sary. Moreover it prevents from existence results since it severely overdetermines the inverse problem. Masahiro Yamamoto observed that fora∈C∞(R) this as- sumption can be removed.
A feature of the inverse options pricing problem is localization around the underlying prices∗resulting from singularity of the final data. Thus local results make sense.
Theorem 9.3.2. Let|a|λ(ω)<M and a2(y)=σ02(given constant) onR\ω.
Then there isε >0(depending only on s∗, τ∗, σ0) and M such that under the condition ω⊂(−ε, ε) a solution a(y) to the inverse problem (9.3.1)-(9.3.3) is unique.
A proof follows from standard contraction arguments augmented by careful study of singularities of solution and it is based on the study of the linearized inverse problem given below. For details of proofs of Theorems 9.3.1, 9.3.2 we refer to the review paper [BI].
In the remaining part of section 9.3 we will assume that 1/2σ2(s)=1/2σ02+ f∗(s) where f∗is a smallC( ¯ω)-perturbation of constantσ02and f∗=0 outsideω∗. As in section 4.5 one can show thatU =V0+V +v. HereV0solves (9.3.10) witha =σ0 andvis quadratically small with respect to f∗, while the principal linear termV satisfies the equations
∂V/(∂τ)−1/2σ02∂2V/(∂y2)+(σ02/2+à)∂V/(∂y)+(r−à)V =α0f, V(y,0)=0, y∈R, V(y, τ∗)=V∗(y), y∈ω,
where
α0(y, τ)=s∗1/σ0(4πaτ)−1/2e−y2/(2τσ02)+cy+dτ, c=1/2+à/σ02, d = −1/(2σ02)(σ02/2+à)2+à−r andV∗is the principal linear part ofU∗.
The new substitutionV =ecy+dτW simplifies these equations to
∂W/(∂τ)−1/2σ02∂2W/(∂y2)=αf, 0< τ < τ∗, y∈R, (9.3.9) W(y,0)=0, y∈R, W(y, τ∗)=W∗(y), y∈ω, with
α(τ,y)=s∗/(√
2πτσ0)e−y2/(2τσ02),W∗(y)=e−cy−dτ∗V∗(y).
9.3. The inverse problem of option pricing 273 In the remaining part of this section we will assume that f =0 outsideω. From numerical experiments ( in particular, in [BIV]) we can see that values of f outside ωare not essential, due to a very fast decay of the Gaussian kernelαins.
Let us denote byA f the solution to (9.3.9) onω:A f(y)=W(y, τ∗), y∈ω. Lemma 9.3.3. We have
A f(x)=
ωB(x,y;τ∗)f(y)d y, x∈ω, where
B(x,y;τ∗)=s∗/(σ02√ π)
(|x−y|+|y|)/(σ0
√2τ∗)
e−τ2dτ.
Moreover, the linearized inverse problem implies the following Fredholm inte- gral equation
f(x)−1/(2τ∗σ02)
ωe−((|x−y|+|y|)2−|x|2)/(2τ∗σ02)(|x−y| + |y|)f(y)d y
(9.3.10) = −√
πτ∗/√
2s∗σ03e(|x|2)/(2τ∗σ02)∂2/(∂x2)W(x, τ∗), x∈ω PROOF. The well-known representation of the solution to the Cauchy problem (9.3.9) for the heat equation yields
W(x, τ)=
RB(x,y;τ)f(y)d y, B(x,y;τ)=
τ
0
1/(
2π(τ −θ)σ0)e−|x−y|2/(2σ02(τ−θ))s∗/(√
2πθσ0)e−|y|2/(2σ02θ)dθ.
We will simplify B(x,y;τ) by using the Laplace transform(p)=L(φ)(p) ofφ(τ) with respect toτ. Since the Laplace transform of the convolution is the product of Laplace transforms of convoluted functions, we have
LB(x,y; )(p)=s∗/(2πσ02)L(τ−1/2e−|x−y|2/(2σ02τ))L(τ−1/2e−|y|2/(2σ02τ))
=s∗/(2πσ02) π/pe−
√2|x−y|/σ0√ p
π/pe−
√2|y|/σ0√p
=s∗/(2σ02)1/pe−
√2(|x−y|+|y|)/σ0√p,
where we used the formula for the Laplace transform ofτ−1/2e−β/τ. Applying the formula for the inverse Laplace transform of the function 1/pe−α√pwe arrive at the conclusion of Lemma 9.3.3.
A proof of the second statement can be obtained by differentiating the equation
A f =W(, τ∗) onω.
Theorem 9.3.4. Letω=(−b,b). Letθ0be the root of the equation2θ−e−4θ=3.
If
(9.3.11) b2/(τ∗σ02)< θ0,
then a solution f ∈ L∞(ω) to the integral equation (9.3.11) and hence to the linearized inverse option pricing problem (9.3.9) is unique.
PROOF. Due to Lemma 9.3.3 to prove Theorem 9.3.4 it suffices to show uniqueness of solution f of (9.3.10). To do it we observe that
ωe−(|x−y|+|y|)2/(2τ∗σ02)(|x−y| + |y|)d y
=e−x2/(2τ∗σ02)(τ∗σ02+x2)−(τ∗σ02)/2(e−(2b−x)2/(2τ∗σ02)+e−(2b+x)2/(2τ∗σ02)).
(9.3.12)
Returning to uniqueness of f we assume that f is not zero. We can assume that f∞(ω)= f(x0)>0 at somex0∈[−b,b]. From (9.3.10) atx=x0(with zero right side) we have
f∞≤1/(2τ∗σ02)
ωe−(|x0−y|+|y|)2−x02)/(2τ∗σ02)(|x0−y| + |y|)f∞d y
= f∞((τ∗σ02+x02)/(2τ∗σ02)−1/4(e−((2b−x0)2−x20)/(2τ∗σ02) +e−((2b+x0)2−x02)/(2τ∗σ02)))
if we use (9.3.12).
One can show that (9.3.11) implies
g(x)=x2/(τ∗σ02)−1/2(e(2b(x−b))/(τ∗σ02)+e−(2b(x+b))/(τ∗σ02))<1, −b≤x≤b.
Then the previous inequality yields
f∞<f∞(1/2+1/2)= f∞,
and hencef∞(ω)=0.
By Lemma 9.3.3 the linearized inverse option pricing problem implies the inte- gral equation
(9.3.13) A f(x)=W∗(x), x∈ω=(−b,b),
whereW∗is the function defined after (9.3.9). Lemma 9.3.3 and Theorem 9.3.4 guarantee uniqueness a solution f ∈C( ¯ω) to this equation under the condition (9.3.11). It is not known whether this condition is necessary for uniqueness in the linearized inverse problem.
One can show [BIV] that the range ofAhas the codimension 2 inC2+λ( ¯ω). At present we do not know an exact description of the range ofA.
The integral equation (9.3.13) with the dataW∗(x) equal to the difference of the final statese−cy−dτ∗(U∗(y)−U0∗(y)) whereU solves the parabolic equation (9.3.7) witha2(y)=σ02+2f(y) andU0 solves the unperturbed equation (9.3.7) was solved numerically in [BIV] with good results indicating that the condition (9.3.11) of Theorem 9.3.4 can be essential. Moreover, numerics worked very well when the (uniform) deviation ofσ fromσ0=1 did not exceed 0.3.