Recovery of volatility coefficient by linearization - CiteSeerX

RE S E A R C H PA P E R

Q U A N T I T A T I V E F I N A N C E V O L U M E 2 (2002) 257–263 INSTITUTE O F PHYSICS PUBLISHING

quant.iop.org

Recovery of volatility coefficient by linearization Ilia Bouchouev1 , Victor Isakov2 and Nicolas Valdivia2 1

Koch Industries Inc., PO Box 2256, Wichita, KS 67201, USA Department of Mathematics and Statistics, Wichita State University, Wichita, KS 67260-0033, USA

2

E-mail: [email protected] and [email protected] Received 7 January 2002, in final form 24 May 2002 Published 2 August 2002 Online at stacks.iop.org/Quant/2/257

Abstract We study the problem of reconstruction of the asset price dependent local volatility from market prices of options with different strikes. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black–Scholes formula and of an explicit functional which is linear in perturbation of volatility. We obtain an integral equation for this functional and we show that under some natural conditions it can be inverted for volatility. We demonstrate the stability of the linearized problem, and we propose a numerical algorithm which is accurate for volatility functions with different properties.

1. Introduction. Basic results The Black–Scholes formula [6] provides us with an elegant and simple method to price financial derivatives under the assumption that the stock price is log-normally distributed. However, the actual distribution of most assets is rarely lognormal, and theoretical prices of options with different strikes generated by the Black–Scholes formula differ from observed market prices [9]. One way to reconcile the differences is to replace the log-normal process with constant volatility by a more general diffusion that allows the instantaneous volatility to vary along with the underlying asset. This approach is very popular among the practitioners as it preserves the risk-neutrality without introduction of additional unhedgeable risks. The main challenge of the method lies in its implementation, as it is usually difficult to achieve the unique and stable fitting of the model to actual market prices. While in the log-normal it is sufficient to calculate a single volatility number from each option quote to calibrate the model, in the general diffusion framework the whole volatility function must be restored from collection of simultaneous option quotes with different strikes. Even though many numerical algorithms for this inverse problem have already been published [1, 3, 5, 13], 1469-7688/02/040257+07$30.00

© 2002 IOP Publishing Ltd

their convergence properties have not been satisfactory for practitioners. We refer to [4] for a survey of available theory and of numerical methods. Recently, in the time-dependent case a relation between implied and local volatilities was discovered [2]. Instead of applying ad hoc regularization techniques to a nonlinear inverse problem with unknown solvability properties, we replace it by a linearized version and find conditions which ensure uniqueness of its solution. We illustrate the accuracy and the robustness of our fast numerical algorithm for volatility functions of various shapes. For any stock price, 0 < s < ∞, and time, 0 < t < T , the price u for an option expiring at time T satisfies the following partial differential equation: ∂u 1 2 2 ∂ 2 u ∂u + s σ (s) 2 + sµ − ru = 0. ∂t 2 ∂s ∂s

(1.1)

Here, σ (s) is the volatility coefficient that satisfies 0 < m < σ (s) < M < ∞ and is assumed to belong to the H¨older space C λ (ω), ¯ 0 < λ < 1 on some interval ω and outside this interval, and µ and r are, respectively, the risk-neutral drift and the risk-free interest rate assumed to be constants. The backward-in-time parabolic equation (1.1) is augmented by the final condition specified by the payoff of the call option

PII: S1469-7688(02)32484-9

257

Q UANTITATIVE F I N A N C E

I Bouchouev et al

with the strike price K: u(s, T ) = (s − K)+ = max(0, s − K),

0 < s. (1.2)

It is known (e.g. see [4] for H¨older σ ) that there is a unique solution u to (1.1), (1.2) which belongs to C 1 ((0, ∞) × (0, T ]) and to C((0, ∞) × [0, T ]) and satisfies the bound |u(s, t)| < C(s + 1). The inverse problem of option pricing seeks for σ given u(s ∗ , t ∗ ; K, T ) = u∗ (K), x ∈ ω∗ . (1.3) Here s ∗ is market price of the stock at time t ∗ , and u∗ (K) denote market price of options with different strikes K for a given expiry T . We also assume that the data are given for a certain moneyness x = ln( sK∗ ) ∈ ω∗ , and attempt to recover volatility in the same interval. In this paper we assume that 1 2 σ (s) 2

= 21 σ02 + f∗ (s)

(1.4)

¯ of constant σ02 and where f∗ is a small C(ω)-perturbation f∗ = 0 outside ω∗ . Then the option price can be computed (up to a quadratically small error) as the sum of the Black–Scholes formula with volatility σ0 and of a certain linear operator V (f∗ ). To obtain our results we will use that the option premium u(., .; K, T ) satisfies the equation dual to the Black–Scholes equation (1.1) with respect to the strike price K and expiry time T : ∂u 1 ∂ 2u ∂u − K 2 σ 2 (K) + (r − µ)u = 0. + µK 2 ∂T 2 ∂K ∂K

(1.5)

Equation (1.5) was found by Dupire [7] and rigorously justified, for example, in [4]. Given the volatility function σ (s) the problem (1.5), (1.2) has a unique solution satisfying the bound |u(K, T )| < C(K + 1). In section 2 we use the standard linearization procedure and derive a partial differential equation for V . After a change of variables this equation is reduced to the heat equation with the right-hand side linear with respect to f . We drop ∗ in new variables. We then write the well known integral representation for the solution V , and apply the Laplace transform to explicitly evaluate an integral with respect to time. As a result we arrive at the integral equation for f of the following form:
B(x, y)f (y) dy = F (x), x∈ω (1.6) ω

with the kernel B(x, y) =

s∗ 2√

σ0


π

e−τ dτ 2

|x−y|+|y| √ σ0 2τ∗

given by the error function and the right-hand side F (x) computed from the market data (1.3). In section 3 we prove that the integral equation (1.6) has not more than one solution under a certain constraint on ω which is determined by the range of strikes for which the data (1.3) are given. It follows as a corollary that for a small change in F that can be viewed as a market bid–ask spread or any other noise in the market data (1.3), solution changes

258

remain small as well. In other words, the linearized version of the volatility reconstruction is stable. Section 4 discretizes the integral equation above and provides several numerical examples of recovering volatility functions of various shapes. For a given volatility function we generate data by solving the partial differential equation with the variable coefficient σ (s). Using the generated data and the representation (1.4) we reconstruct σ (s) from the integral equation (1.6). The accuracy and stability of our numerical results are consistent with the stability conditions established in section 3. Our numerical results demonstrate a very accurate reconstruction of the local volatility for a certain range of underlying prices around the at-the-money level. The reconstruction becomes less reliable for the tails of the local volatility function. Deep out-the-money options have a much smaller time value, and therefore their prices contain less information about the underlying stochastic process. In other words, the tails of two volatility functions can significantly differ from each other, but at the same time they both can lead to a very similar set of option prices. For longer maturities the domain where reconstruction is accurate and stable is growing along with the time value of options. Numerical results obtained by other authors and by the first two authors of this paper using different methodologies [1, 3, 13] exhibit reconstruction properties which are similar to the one produced by our algorithm. In [1, 13] they look for the volatility σ = σ (t, s) from the data given for several t ∗ . We believe that values of the volatility for t ∗ < t < T (future) are of most interest. But in the time-dependent case the available data are not sufficient to identify it. That is why we think that the best one can do is to assume that σ does not depend on t at least when t ∗ < t < T and to try to find it from today’s market data (1.3). Our contribution is twofold. First, we explicitly describe (theorem 3.1) the domain where the stable reconstruction is ensured for the linearized inverse problem. The closest result in this direction was obtained in [4] where the linearization was replaced by an approximate linearization. The approximate linearization is not as exact as the linearization, so in simplifying the problem one loses precision. A uniqueness result in [3] is for the exact nonlinear inverse problem, but in a heavily overdetermined formulation. In this paper the answer is obtained in terms of the ratio of the spatial variable to the square root of time to maturity. It is consistent with trader’s intuition, as the ratio is a key component in determining the time value of the option. We show that outside of a certain range the reconstruction is not reliable. It is interesting to note that option traders tend to set bounds on their implied volatility matrices by a somewhat similar ratio of moneyness to the square root of time of maturity. They claim that options with the moneyness that exceeds a certain threshold determined by time to maturity have a small enough time value such that the option premium remains within the market bid–ask spread regardless of how accurate the volatility entry is. Our second contribution is the actual reconstruction methodology. Algorithms proposed in [1, 5, 13] attempt to use standard approaches (regularized best L2 -fit) to a

Q UANTITATIVE F I N A N C E

Recovery of volatility coefficient by linearization

numerical solution of inverse problems. They involve very heavy computations with poor convergence properties, which is typical for inverse parabolic problems. The numerical algorithm in [3] is based on some features of the inverse option pricing problem. In fact, it is a second-order correction of the Black–Scholes formula. This correction lacks a rigorous justification and numerically it is not very efficient. In [5] they used perturbation methods which look reasonable when volatility is close to a constant. A main computational idea in [5] is to solve the inverse problem linearized at a constant, but the corresponding linear integral equation remains complicated as well as its proposed numerical inversion. Besides, it is not tested numerically. In our particular case, the special structure of the initial data and the assumption about time independence of σ allowed us to significantly simplify the inverse problem. In fact, we were able to transform it to a one-dimensional integral equation with an explicit kernel. This led to a very simple and stable numerical algorithm for the reconstruction of the local volatility. The problem was essentially reduced to an inversion of a simple matrix whose properties can be easily analysed for a given range of strike prices and maturities. The results of sections 2 and 3 are obtained by Bouchouev and Isakov, while numerics is due to Valdivia.

the equations 1 ∂ 2V ∂V − σ02 2 + ∂τ 2 ∂y



 σ02 ∂V + (r − µ)V = α0 f, +µ ∂y 2

V (y, 0) = 0, ∗

y ∈ R,

∗

V (y, τ ) = V (y),

y ∈ ω, (2.11)

where α0 (y, τ ) = s ∗

√

1

σ0 4π aτ

e

−

y2 2τ σ02

+cy+dτ

 2 2 σ0 1 µ 1 c = + 2, +µ +µ−r d=− 2 σ0 2σ 2 2 and V ∗ is the principal linear part of U ∗ . One can completely justify this linearization by using standard theory of parabolic boundary value problems [8, 12], as was done in [4] for the inverse option pricing problem or in [11, section 4.5] for some elliptic inverse problems. The new substitution V = ecy+dτ W

1 ∂ 2W ∂W − σ02 2 = αf, ∂τ 2 ∂y 0 < τ < τ ∗,

The substitution 

K y = ln ∗ , s ∗ y

τ = T − t,

α(τ, y) = √ (2.7)

∗ y

a(y) = σ (s e ),

transforms the equation (1.5) and the initial data (1.2) into   ∂U 1 ∂ 2U 1 2 ∂U − a 2 (y) 2 + a (y) + µ ∂τ 2 ∂y 2 ∂y (2.8) y∈R (2.9)

Here ω is the transformed interval ω∗ (ω∗ in y-variables (2.7)). Observe that τ ∗ = T − t ∗ . Equations (2.8) and (2.9) for functions U (τ, y), a(y) form the so-called inverse parabolic problem with the final overdetermination. The known uniqueness conditions for this problem [10, section 6.2], [11, section 9.2], are not satisfied in our particular situation, and we are not aware of any uniqueness result. To derive the linearized inverse problem we observe that due to the assumption (1.4), 1 2 a (y) 2

= 21 σ02 + f (y)

where f is C(ω)-small ¯ and f = 0 outside ω. So U = V0 + V + v.

y ∈ R, s∗ 2π τ σ0

W (y, 0) = 0,

U (y, τ ) = u(s e , τ + t)

+ (r − µ)U = 0U (y, 0) = s ∗ (1 − ey )+ , U (y, τ ∗ ) = U ∗ (y), y ∈ ω.

(2.12)

simplifies (2.11) to

2. Linearization at constant volatility 

,

(2.10)

Here V0 solves (2.8) with a = σ0 and v is quadratically small with respect to f∗ , while the principal linear term V satisfies

∗

e

−

(2.13)

,

y ∈ R,

∗

W (y, τ ) = W (y), with

y2 2τ σ02

y ∈ ω, ∗

W ∗ (y) = e−cy−dτ V ∗ (y)

In the remaining part of this section we assume that f = 0 outside ω. From numerical experiments in section 4 and in [13] we can see that values of f outside ω are not essential, due to a very fast decay of the Gaussian kernel α in s. Let us denote by Af the solution to 2.13 on ω: Af (y) = W (y, τ ∗ ), y ∈ ω. Lemma 2.1. We have
Af (x) = B(x, y; τ ∗ )f (y) dy,

x ∈ ω,

(2.14)

ω

where B(x, y; τ ∗ ) =

s∗ 2√

σ0


π

∞

e−τ dτ. 2

|x−y|+|y| √ σ0 2τ ∗

Proof. The well known representation [8] of the solution to the Cauchy problem 2.13 for the heat equation yields
W (x, τ ) = B(x, y; τ )f (y) dy, R




B(x, y; τ ) =

√ 0

− |x−y| − |y|2 1 s∗ 2 e 2σ0 (τ −θ) √ e 2σ0 θ dθ. 2π(τ − θ)σ0 2π θσ0 2

τ

2

259

Q UANTITATIVE F I N A N C E

I Bouchouev et al

We will simplify B(x, y; τ ) by using the Laplace transform (p) = L(φ)(p) of φ(τ ) with respect to τ . Since the Laplace transform of the convolution is the product of Laplace transforms of convoluted functions, we have  2   2  1 − |x−y| 1 − 2σ|y|2 τ s∗ 2σ02 τ 0 LB(x, y; )(p) = L L e e √ √ τ τ 2πσ02  √ √ ∗  √ √ 2|x−y| s π − σ p π − σ2|y| p 0 0 = e e p 2πσ02 p √ ∗ √ s 1 − 2(|x−y|+|y|) p σ0 = e 2 p 2σ0 where we used the formula for the Laplace transform of β 1 for the τ − 2 e− τ [14, p 526, formula 16]. Applying the formula √ inverse Laplace transform of the function p1 e−α p [14, p 528, formula 42], we arrive at the conclusion of lemma 2.1.  

Corollary 2.2. The linearized inverse problem implies the following Fredholm integral equation:
2 −|x|2 − (|x−y|+|y|) 1 2τ ∗ σ02 e (|x − y| + |y|)f (y) dy f (x) − ∗ 2 2τ σ0 ω √ 2 π τ ∗ 3 2τ|x|∗ σ 2 ∂ 2 σ0 e 0 W (x, τ ∗ ), x∈ω (2.15) = − √ ∂x 2 2s ∗ Proof. Differentiating the equation Af = W (, τ ∗ ) on ω, ∂ |x − y| = sign(x − y) we using (2.14) and the formula ∂x will have
2 − (|x−y|+|y|) 1 2τ ∗ σ02 e sign(x − y)f (y) dy − 3√ σ0 2π τ ∗ ω ∂ = W (x, τ ∗ ), x ∈ ω. ∂x Differentiating once more and multiplying by we get the Fredholm integral equation (2.15). The proof is complete.

One can check numerically that 1.5012 < θ0 < 1.5013. Recall that f ∞ (ω) is the essential supremum of |f | over ω. In particular, when f is continuous on ω is it just max |f (x)| over x ∈ ω. Proof. Due to corollary 2.2 to prove theorem 3.1 it suffices to show the uniqueness of solution f of (2.15), i.e. to assume that the right side is zero and conclude that f = 0. To do this we observe that
2 − (|x−y|+|y|) ∗ 2 e 2τ σ0 (|x − y| + |y|) dy ω  2 2  2 − x∗ 2 − (2b+x) τ ∗ σ02 − (2b−x) ∗ 2 ∗ 2 = e 2τ σ0 (τ ∗ σ02 + x 2 ) − e 2τ σ0 + e 2τ σ0 . (3.17) 2 This can be verified by direct calculations, using that for 0 < x, |x − y| + |y| is 2y − x when x < y, it is x when 0 < y < x, and it is x − 2y when y < 0. For x < 0, |x − y| + |y| is 2y − x when 0 < y, it is −x when x < y < 0 and it is x − 2y when y < x. Returning to uniqueness of f we assume that f is not zero. We can assume that f ∞ (ω) = f (x0 ) > 0 at some x0 ∈ [−b, b]. From (2.15) at x = x0 (with zero right side) we have
|x −y|+|y|)2 −x02 − 0 1 2τ ∗ σ02 e (|x0 − y| + |y|)f ∞ dy f ∞
∗ 2 2τ σ0 ω  ∗ 2  (2b−x )2 −x2 (2b+x0 )2 −x02  − τ σ0 + x02 1 − 2τ 0∗ σ 2 0 2τ ∗ σ02 0 = f ∞ − + e e 4 2τ ∗ σ02 if we use (3.17). We will show that   − 2b(x+b) 1 2b(x−b) x2 ∗ 2 ∗ 2 e τ σ0 + e τ σ0 < 1, g(x) = ∗ 2 − 2 τ σ0 − b
x
b.

(3.18)

Then the previous inequality yields

√ |x|2 σ3 πτ∗ ∗ 2 − 0 √2 e 2τ σ0

f ∞ < f ∞ ( 21 + 21 ) = f ∞ ,

 

and hence f ∞ (ω) = 0. To prove (3.18) we will show that maximum of g on ω∗ is at x = b. Then  2  − 4b 1 b2 ∗ 2 1 + e τ σ0 < 1, g(b) = ∗ 2 − 2 τ σ0

The purpose of this section is to prove the following theorem.

if 2 τ ∗bσ 2 − e τ σ0 < 3. Since the function 2θ − e−4θ is 0 increasing, the last inequality holds when θ < θ0 , where 2θ0 − e−4θ0 = 3. To complete the proof it suffices to prove that supω g = g(b). We will achieve it by a careful analysis of g, g  and g  . Indeed  (x−b)  2b ∗ 2 −2b (x+b) 2x b  τ σ0 τ ∗ σ02 , −e g (x) = ∗ 2 − ∗ 2 e τ σ0 τ σ0   (x−b)  2b ∗ 2 −2b (x+b) 2b2 1 ∗ 2 g  (x) = ∗ 2 2 − ∗ 2 e τ σ0 + e τ σ0 , τ σ0 τ σ0

3. Uniqueness for the linearized inverse problem

2

Theorem 3.1. Let ω = (−b, b). Let θ0 be the root of the equation 2θ − e−4θ = 3. If b2 < θ0 , (3.16) ∗ τ σ02 then a solution f ∈ L∞ (ω) to the integral equation (2.15) and hence to the inverse option pricing problem (2.11) is unique.

260

−

4b2 ∗ 2

Q UANTITATIVE F I N A N C E and finally

Recovery of volatility coefficient by linearization

 (x−b)  2b ∗ 2 −2b (x+b) 4b3 τ σ0 τ ∗ σ02 g (x) = − ∗3 6 e . −e τ σ0 

Obviously, g  (x) < 0 if x > 0 and g  (x) > 0 if x < 0. Hence maxω g  = g  (0) > 0, due to the condition b2 < 1.5013 and to the elementary inequality xe−x
1. τ ∗ σ02 Then g  (x)  0, if −b∗ < x < b∗ and g  (x) < 0; −b < x < −b∗ , −b∗ < x < −b. Here 0 < b∗
b. Returning to g  , we conclude that 0
g  (x) if 0
x
b∗∗ and g  (x) < 0 if b∗∗
x
b, where b∗∗ is a number in (b∗ , b), also g  is an odd function. Since − ∗4b 2  2b b  g  (b) = ∗ 2 − ∗ 2 1 − e τ σ0 > 0 τ σ0 τ σ0 we have b∗∗ = b. So xg  (x) > 0. Hence x = b is a maximum point for g(x). The proof is complete.   By lemma 2.1 the linearized inverse option pricing problem is equivalent to the integral equation ∗

Af (x) = W (x),

x ∈ ω = (−b, b)

(3.19)

∗

where W is the function defined after (2.13). Corollary 2.2 and theorem 3.1 guarantee the uniqueness of a solution f ∈ C(ω) to this equation under the condition (3.16). It is not known whether this condition is necessary for uniqueness in the linearized inverse problem. It is not hard to show that uniqueness for (3.19) holds under the same condition for f ∈ L2 (ω). The integral equation (3.19), theorem 3.1, and simple properties of integral operators imply the following stability estimate: f ∞ (ω)
CW ∗ ∞ (ω). (3.20) This shows that our inverse problem is stable, as observed in [4, 11, section 9.1]. However, at this point we do not have an explicit bound on C in (3.20). It is clear from known properties of solutions to parabolic problems (and it can be seen from equation (2.15)) that f ∈ C λ (ω) ¯ implies that u(, t ∗ ) ∈ C 2+λ (ω). ¯ So the operator A maps C λ (ω) ¯ into the space C 2+λ (ω) ¯ of function with H¨older continuous second-order derivatives. By theorem 3.1 the Fredholm integral equation (2.15) has a unique solution f in C λ (ω) ¯ for any right side in the same space, i.e. when W ∗ ∈ 2+λ C (ω). ¯ From lemma 2.1 and corollary 2.2 it follows that ∂2 ∂2 ∗ ∗ Af = ∂x 2 2 W , and hence Af (x) = W (x) + C1 + C2 x, x ∈ ∂x ω, for some constants C1 , C2 . Observe that we cannot conclude that C1 = 0, C2 = 0, so while equation (2.15) follows from the equation Af = W ∗ , it is not equivalent to this equation. In any event, the range of A has codimension not greater than 2 in C 2+λ (ω). ¯ We will show that it is exactly 2. To do this we observe that equation (1.6) has no solution when W ∗ = W1∗ , W2∗ where W1∗ (x) = 1, W2∗ (x) = x. Let fj be this solution. In both cases Wj∗ (x) = 0, so the solution fj to the equation (2.15) is zero due to uniqueness guaranteed by theorem 3.1. But then Wj∗ = Afj = 0 and we have a contradiction. At present we do not know an exact description of the range of A.

Figure 1. Exact and recovered (from the data at different times) linearly perturbed volatilities.

4. Numerical algorithm and its testing We now propose and test numerically an algorithm for reconstruction of volatility from the data (1.3). We will solve the integral equation (3.19) with the data W ∗ (x) equal to the ∗ difference of the final states e−cy−dτ (U ∗ (y)−U0∗ (y)) where U solves the parabolic equation (2.8) with a 2 (y) = σ02 + 2f (y) and U0 solves the unperturbed equation (2.8). We consider the interval ω∗ = (−1, 1), s ∗ = 20 and we let µ = 0 and r = 0.05. On this interval we use uniformly distributed grid points. Observe that we are solving numerically a linear inverse problem, using the data generated by the original nonlinear problem (2.8). Of course, this generates data errors, due to the linearization. Otherwise, the data are (numerically) exact. The direct problem (2.8) was solved numerically by the finite differences method (the Crank–Nicholson scheme with 80 grid points on the interval (−1.5, 1.5) with artificial zero (Dirichlet) boundary conditions at y = −1.5 and 1.5). The integral operator (1.6) is discretized by using standard tables for the error function errfc as follows:   |xj − yk | + |xj | ∗ Kj,k = s errfc δy τ∗ where yj , j = 1, . . . , N are points of ω used in solving the direct problem. As a result, N = 54. The points xj are the measurements points. Their collection coincides with the points y1 , . . . , y54 . We consider five examples illustrated by figures 1–5 where we let σ0 = 1 and we use different observation times τ ∗ = 0.1, 0.3, 0.5 and 0.7. As perturbations f (y) of constant volatility we take functions f1 (y) = 0.3y, f2 (y) = 0.3y2 , f3 (y) = 0.5y 2 − 0.25y, f4 (y) = 0.3 sin(2πy), and f5 (y) = 0.3 sin(4πy). The functions f4 , f5 are oscillating functions and they are not typical for financial problems. We included them to test how robust is the numerical algorithm. Observe that perturbations can reach 0.3 − 0.7 of the magnitude of the unperturbed constant coefficient.

261

I Bouchouev et al

Figure 2. Exact and recovered quadratically perturbed symmetric volatilities.

Q UANTITATIVE F I N A N C E

Figure 4. Exact and recovered oscillating volatilities.

Figure 5. Exact and recovered rapidly oscillating volatilities. Figure 3. Exact and recovered quadratically perturbed asymmetric volatilities.

From figures 1 and 2 we can see that the reconstruction is near-perfect on the whole ω when τ ∗ is 0.5; 0.7. This is in agreement with condition (3.16), which in these examples simplifies to 0.6667 < τ ∗ . The greater is τ ∗ , the better is the reconstruction on the whole interval (−1, 1), so τ ∗ = 0.5 or 0.7 corresponds to the recovered volatilities closest to the given ones. On the other hand, for smaller time τ ∗ = 0.1 the reconstructed f starts to deteriorate near endpoints (on the intervals (−1, −0.6) and (0.7, 1) in figure 1). For τ ∗ = 0.3 the deterioration is visible, but not as strong. Now we compare different shapes of f . The worst reconstruction is for asymmetric f3 and for oscillating f4 and f5 , but even in these cases the recovered function closely resembles its original. The deterioration of the images for oscillating functions is expected, and in the asymmetric case a

262

possible explanation is linked to a relatively large perturbation f3 . An interesting feature of these examples is the space localization: most likely due to the fast decay of the Gaussian kernel α , the values of the data at distant points seem not to influence reconstruction near y = 0 (s = s ∗ ). Indeed, for smaller τ ∗ recovered volatility practically coincides with the given one on a small interval (−0.2, 0.2) in figures 1, 2, 4 and 5. Observe that this localization occurs in a nonlinear (inverse) problem. A reason could be the well-posedness of the (linearized) inverse problem and the above-mentioned properties of the Gaussian kernel. which Similar local reconstruction (with σ (s) = 15 s resembles the perturbation f1 ) was observed in [13]: it is remarkably better on the interval around s ∗ where the ratio of volatility to its value at s ∗ is less than 0.3. On the other hand, there is a difference between the numerical experiment in [13] and our tests: the data in [13] are exact, while ours are approximate (due to linearization). Their reconstruction looks better on a smaller interval (after rescaling) and for smaller τ ∗ .

Q UANTITATIVE F I N A N C E

Recovery of volatility coefficient by linearization

5. Conclusion

Acknowledgments

The uniqueness result of this paper can be certainly generalized to the many-dimensional case and probably to some parabolic equations with variable coefficients. The inverse option pricing problem is a particular case of the more general inverse diffusion problem which has a probabilistic interpretation. We are not aware of any uniqueness results about recovery of diffusion rate from probability of distribution at a fixed moment of time. The method of this paper can be applied at least to a linearized version of this inverse probabilistic problem. The proposed reconstruction algorithm is expected to perform very well when volatility is not changing fast with respect to stock price s and is changing very slowly with respect to time. Sudden and dramatic changes of market situations most likely cannot be properly described by our model and more generally by the Black–Scholes equation. Probably, a minor modification of the proposed model (replacing R in (2.8) by a finite interval) can eliminate difficulties with the existence theorem and generate even better numerical algorithms. We did not test the algorithm on real market data, but we cannot see any problem with that. Observe that to find continuous f the data F must be at least twice differentiable on ω, so the real market data are in need of a proper interpolation, minimizing the size of second derivatives of F . A choice of an appropriate smoothing interpolation and an intensive numerical testing will be a subject of future work. Our immediate goal was to develop some mathematical theory of the inverse option pricing problem, to suggest a numerical algorithm, and to test it on simulated data. The next step is an additional testing on simulated and real data. For simplicity we considered only European options. We hope to adjust the linearization technique to American and more complicated options, which are in particular described by free boundary problems. So far there are actually no results in this very important practical case.

The work of Victor Isakov was in part supported by the NSF grants DMS 98-03397 and DMS 01-04029.

References [1] Avellaneda M, Friedman C, Holmes R and Sampieri L 1997 Calibrating volatility surfaces via relative entropy minimization Appl. Math. Finance 4 37–64 [2] Berestycki H, Busca J and Florent I 2000 An inverse parabolic problem arising in finance C. R. Acad. Sci. Paris 331 965–9 [3] Bouchouev I and Isakov V 1997 The inverse problem of option pricing Inverse Problems 13 L1–7 [4] Bouchouev I and Isakov V 1999 Uniqueness, stability, and numerical methods for the inverse problem that arises in financial markets Inverse Problems 15 R95–116 [5] Bodurtha J N and Jermakyan M 1999 Non-parametric estimation of an implied volatility surface J. Comput. Finance 2 29–61 [6] Black F and Scholes M 1973 The pricing of options and corporate liabilities J. Political Econ. 81 637–59 [7] Dupire B 1994 Pricing with a smile RISK 7 18–20 [8] Friedman A 1964 Partial Differential Equations of Parabolic Type (Englewood Cliffs, NJ: Prentice-Hall) [9] Hull J 1997 Options, Futures, and Other Derivatives (Englewood Cliffs, NJ: Prentice-Hall) [10] Isakov V 1990 Inverse Source Problems (Providence, RI: American Mathematical Society) [11] Isakov V 1997 Inverse Problems for PDE (New York: Springer) [12] Ladyzenskaja O A, Solonnikov V A and Uraltseva N N 1969 Linear and Quasilinear Equations of Parabolic Type (New York: Academic) [13] Lagnado R and Osher S 1997 A technique for calibrating derivation of the security pricing models: numerical solution of the inverse problem J. Comput. Finance 1 13–25 [14] Lavrentiev M A and Shabat B A 1965 Methods of Theory of Functions of One Complex Variable (Moscow: Nauka)

263