Calibration of Stochastic Volatility Models: An Optimal Control

Calibration of Stochastic Volatility Models: An Optimal Control

Approach∗

Min Dai† Ling Tang‡ Xingye Yue§

First version: Apr 2012; This version: Oct 2012

Abstract

We aim to calibrate stochastic volatility models from option prices. We develop an optimalcontrol approach to recover the risk neutral drift term of stochastic volatility. An efficient nu-merical algorithm is given. Numerical results and empirical studies are presented to demonstrateour algorithm. In contrast to existing literature, we do not assume that the stochastic volatilitymodel has special structure, so our algorithm applies to calibration of general stochastic volatil-ity models. In addition, our empirical results reveal that the risk neutral process of volatilityrecovered from market prices of options on S&P 500 index is indeed linearly mean-reverting.

Subject classifications: Finance: asset pricing; Dynamic programmingoptimal control: ap-plications.

Area of review: Financial engineering.

1 Introduction

Black and Scholes (1973) develop the celebrated option pricing model under the assumption that

the underlying stock price follows a log-normal distribution with constant volatility. The model

possesses some appealing properties, including the uniqueness of prices and the perfect replication

of options. Unfortunately, the constant volatility assumption apparently conflicts with the volatility

smile phenomenon observed in options markets.

As a remedy, Dupire (1994) considers the local volatility model, where volatility is a deter-

ministic function of the underlying stock price and time. He derives an insightful equation, known

as Dupire’s equation, revealing that there is a unique diffusion process consistent with the risk neu-

tral densities derived from the market prices of options. In spite that a limited number of market∗This work is partially supported by Singapore MOE AcRF grant (No. R-146-000-138-112) and GAI-LCF grant

(R-146-000-160-646). We thank Baojun Bian, Lishang Jiang, Steven Kou, Lixin Wu and seminar participants at 2012IMS Workshop on Probability and Statistics in Finance, The 2012 Informs International Conference, Informs AnnualMeeting 2012, and Workshop 2012 in memory of Professor Xunjing Li for helpful discussions and comments.

†Dept of Math and Centre for Quantitative Finance, National University of Singapore.‡Dept of Math, National University of Singapore.§School of Mathematical Sciences and Financial Engineering Centre, Soochow University, Suzhou 215006, China.

1

prices may be insufficient to recover the entire volatility curve, Dupire’s equation provides very

useful insights into the inverse problem of calibrating the local volatility model which has been

extensively studied (e.g., Bouchouev and Isakov (1997,1999), Jiang et al. (2003), Egger and Engl

(2005), Jiang and Bian (2012)).

The local volatility model inherits some nice properties from the Black-Scholes model, such

as the law of unique price and perfect replication. However, Dumas et al. (1998) find that the out-

of-sample performance of deterministic volatility models is poor. Empirical evidences by Brockman

and Chowdhury (1997) and Buraschi and Jackwerth (2001) show that volatility is nondeterministic.

This motives us to instead consider stochastic volatility models.

There is an extensive literature devoted to stochastic volatility models1, among which there

are some named models, including Hull and White model (1987), Stein and Stein model (1991),

Heston model (1993), etc. All of these models assume special structure of the volatility (or variance)

process so as to obtain analytical pricing formulas for European vanilla options, which simplifies

model calibration.

In this paper we consider calibration of general stochastic volatility models and aim to identify,

through market prices of options, the risk neutral drift term of the volatility (or variance) process.

The reason we only calibrate the drift term is that the volatility of the volatility (or variance)

process does not change from the real world to the risk neutral world, whereas the drift term does.

Thus the volatility of the volatility (or variance) process can be estimated by historical underlying

stock prices whereas the risk-neutral drift term must be recovered from options market.

In contrast to standard literature, we assume that the risk neutral drift term of the volatility

(or variance) process is a deterministic function of instantaneous volatility (or variance) and time

and does not possess any special structure. As a consequence, analytical price formulas European

options are unavailable. We shall formulate the calibration problem as a standard inverse problem

of partial differential equations (PDEs) that can be attacked by an optimal control approach.

The contributions of the paper are summarized as follows. First, using the Dupire’s equation

associated with stochastic volatility models, we formulate the calibration problem as a standard

inverse problem of PDEs. Second, we solve the inverse problem in terms of an optimal control

approach with Tikhonov regularization. We derive a necessary condition that the optimal solution

satisfies. We further reduce the necessary condition, which plays a critical role in algorithm design.

Third, by the reduced necessary condition, we propose a gradient descent algorithm to numerically1Note that for the stochastic volatility model, the law of unique price loses effect and perfect replication is no

longer possible.

2

find the optimal solution. We highlight that our algorithm applies to general stochastic volatility

models. Last but not least, we conduct an extensive numerical analysis to demonstrate the efficiency

of our numerical algorithm. Moreover, by virtue of real market data, we reveal that the risk neutral

drift term of the variance process of S&P 500 index is indeed linearly mean-reverting.

The rest of the paper is organized as follows. In Section 2, we formulate the calibration

problem as a standard inverse problem of PDEs. In Section 3, we solve the inverse problem by

an optimal control approach with the Tikhnonov regularization technique. We derive a necessary

condition of the control problem by which we propose a numerical algorithm. Numerical and

empirical results are presented in Section 4. We conclude in Section 5.

2 Problem Formulation

2.1 Calibration Problem

Let Xt be the price process of the underlying stock. Under the risk-neutral world,

dXt

Xt= (r − q)dt + σtdW 0

t ,

σt = f(Yt),

dYt = b(Yt, t)dt + β(Yt)dW 1t ,

where r, q represent the riskfree rate and the dividend yield of the underlying, respectively, W 0t

and W 1t are standard 1-dimensional Brownian motions with constant correlation coefficient ρ, i.e.,

dW 0t · dW 1

t = ρdt. We assume that b(·, ·) and β(·) are deterministic functions, and Yt is either the

volatility (i.e. f(Yt) = Yt) of or the variance (i.e. f(Yt) =√

Yt) of the underlying.

Consider a European call option with strike price K and maturity T . The option price,

denoted by V (x, y, t;K, T ), satisfies the following equation:

∂tV + LV = 0 in x > 0, y > 0, t < T (2.1)

with the terminal condition

V |t=T = (x−K)+, (2.2)

where

LV ≡ 12x2f2(y)∂xxV + ρxf(y)β(y)∂xyV +

12β2(y)∂yyV

+(r − q)x∂xV + b(y, t)∂yV − rV.

Apparently V (x, y, t;K, T ) depends on b(y, t).

3

Denote the current time, current stock price, and current volatility by t∗, x∗, and σ∗, respec-

tively, and y∗ ≡ f−1(σ∗). Suppose we can observe the market prices of options with all maturities

and strike prices, denoted by V ∗(K,T ). We consider the following calibration problem.

Problem A: Find a pair of functions b(y, t) and V (x, y, t;K, T ) that satisfy (2.1)-(2.2) from market

prices of options

V (x∗, y∗, t∗;K, T ) = V ∗(K, T ) for all K, T.

Note that the observed information V ∗(·, ·) is two dimensional, so we require that b(·, ·), to

be recovered, be also of two dimension. Regarding β and ρ that are given, there is no restriction on

their functional forms. We use the present ones because it is easier to identify them by statistical

approach.

It should be pointed out that Problem A is not a standard inverse problem of PDEs because

the state variables of equation (2.1) are x, y, t, but the data V ∗(K, T ) observed is with respect to

K and T .2 To be consistent with the data, we need to derive a PDE with state variables K,T, as

Dupire (1994) does.

2.2 Formulation as an Inverse Problem

Define ψ(x, y, t;K, y, T ) as the fundamental solution to (2.1), that is,

∂tψ + Lψ = 0 in x > 0, y > 0, t < T

ψ|t=T = δ(x−K)δ(y − y).

From the well known property of fundamental solution, ψ(x∗, y∗, t∗;K, y, T ), as a function of K, y,

and T , satisfies the adjoint equation of (2.1):

∂T ψ − L∗ψ = 0 in K > 0, y > 0, T > t∗ (2.3)

ψ|T=t∗ = δ(x∗ −K)δ(y∗ − y), (2.4)

where

L∗ψ ≡ 12∂KK(K2f2(y)ψ) + ∂Ky(ρKf(y)β(y)ψ)

+12∂yy(β2(y)ψ)− (r − q)∂K(Kψ)− ∂y(b(y, T )ψ)− rψ.

2Lagnado and Osher (1997) directly solve a non-standard inverse problem arising from calibration of the localvolatility model. However their algorithm is rather complicated, as compared with that of Bouchouev and Isakov(1997) who use the Dupire’s equation.

4

In what follows, we suppress the dependence on x∗, y∗, t∗ for notational simplicity, namely,

ψ(K, y, T ) ≡ ψ(x∗, y∗, t∗;K, y, T ),

V (K, T ) ≡ V (x∗, y∗, t∗;K, T ).

The following proposition plays a critical role in formulating Problem A as a standard inverse

problem.

Proposition 1. Let ψ be the fundamental solution as define above. Then V (K, T ) satisfies

∂T V + (r − q)K∂KV + qV =12K2

∫ ∞

0f2(y)ψ(K, y, T )dy in K > 0, T > t∗ (2.5)

with the initial condition

V |T=t∗ = (x∗ −K)+. (2.6)

Furthermore, (2.5)-(2.6) permits an analytic solution:

V (K,T ) = e−q(T−t∗)(x∗ − e−(r−q)(T−t∗)K

)+

+K2

2

∫ T

t∗

∫ ∞

0e(−2r+q)(T−τ)f2(y)ψ(Ke−(r−q)(T−τ), y, τ)dydτ. (2.7)

A proof of the proposition is in Appendix A.

We call Eq. (2.5) the modified Dupire’s equation associated with stochastic volatility models.

We would like to relate the equation to the following result obtained by Derman and Kani (1998):

σ2loc(K, T ) = E

[σ2

T |XT = K], (2.8)

where

σ2loc(K, T ) =

∂T V + (r − q) K∂KV + qV12K2∂KKV

. (2.9)

Indeed, combining (2.8) with (2.9), we have

∂T V + (r − q) K∂KV + qV =12K2∂KKV E

[σ2

T |XT = K]. (2.10)

By definition,

E[σ2

T |XT = K]

=∫ ∞

0f2 (y)

ψ (K, y, T )∫∞0 ψ(K, y, T )dy

dy.

It is worthwhile pointing out [see (A.2) in Appendix]

∂KKV =∫ ∞

0ψ(K, y, T )dy. (2.11)

5

Combination of (2.10)-(2.11) yields the modified Dupire’s equation (2.5).

Clearly ψ(·, ·, ·) depends on b(·, ·), so does V (·, ·) via ψ(·, ·, ·). To emphasize the dependence on

b(·, ·), we use the notations ψ = ψ(·, ·, ·; b) and V = V (·, ·; b). By Proposition 1, we can reformulate

Problem A as follows.

Problem B: Find a triple of functions b(·, ·), ψ(K, y, T ; b), and V (K, T ; b) that satisfy (2.3)-(2.4)

and (2.7) from market prices of options

V (K,T ; b) = V ∗(K, T ) for all K, T.

3 An Optimal Control Approach

In this section, we use an optimal control approach with the Tikhonov regularization technique [cf.

Tikhonov et al. (1995)] to solve Problem B.

Let t∗ = T0 < T1 < T2 < · · · < TN with Tn = T0 + nh and h = (TN − T0)/N .3 Given

b0(·) ≡ b(·, T0) ∈ B = H1(R+) ≡ v ∈ L2(R+) : ∂yv ∈ L2(R+), we inductively construct two

sequences of cost functional Jn and optimal solution bn(·), n = 1, 2, · · · , N . That is, for each n,

given b0(·), b1(·), · · · , bn−1(·) ∈ B, we introduce the cost functional

Jn(b) =ε12h‖b− bn−1‖2

L2(R+) +ε22‖∂y b(y)‖2

L2(R+) +1

2h2‖Vh(·, Tn; bh)− V ∗(·, Tn)‖2

L2(R+) (3.1)

for b ∈ B, where ε1, ε2 > 0 are two regularization parameters, and Vh(·, ·; bh) is as given by (2.7)

with the coefficient

bh(y, T ) ≡

T−Tn−1

h b(y) + Tn−Th bn−1(y), Tn−1 ≤ T ≤ Tn

T−Tk−1

h bk(y) + Tk−Th bk−1(y), Tk−1 ≤ T ≤ Tk 1 ≤ k ≤ n− 1.

(3.2)

We aim to solve the following control problem: given b0(·), b1(·), · · · , bn−1(·) ∈ B, find bn(·) ∈ B

such that

Jn(bn) = infb∈B

Jn(b). (3.3)

The first two terms on the right hand side of (3.1) are the so-called Tikhonov regularization

which are widely utilized to handle ill posed problems. It is worthwhile pointing out that the termε12h‖b− bn−1‖2

L2(R+), first introduced by Jiang and Bian (2012) to recover the time-dependent local

volatility, enables us to achieve certain regularity of b(·, ·) in T which is needed for convergence

analysis. However, for numerical experiments presented in Section 4, this term is actually removed

and we find that it does not cause any oscillation.3For ease of presentation, we assume a regular partition in [t∗, TN ].

6

3.1 A Necessary Condition

Let us derive the necessary condition for optimality of problem (3.3). Assume that bn(·) is the

optimal solution of (3.3). For any ω ∈ B, we denote bλ = bn + λω ∈ B, λ ∈ R, and define

j(λ) = Jn(bλ), λ ∈ R.

Note that j(λ) reaches its minimum at λ = 0. It follows

j′(0) =d

dλJn(bλ)|λ=0 = 0, (3.4)

from which we can derive the following necessary condition.

Proposition 2. Let bn be the minimizer of problem (3.3). Let bh(·, ·) be as given in (3.2) with

bn in place of b, and let ψh(K, y, T ) and Vh(K, T ) be the solution to (2.3)-(2.4) and as given by

(2.7), respectively, where the coefficient b(·, ·) is replaced by bh(·, ·). Then bn satisfies the following

variational problem∫ ∞

0

(ε1

bn − bn−1

h· ω + ε2∂ybn · ∂yω

− 1h2

∫ Tn

Tn−1

∫ ∞

0

τ − Tn−1

hgh(K, y, τ)∂y (ψh(K, y, τ) · ω) dτdK

)dy = 0, (3.5)

for any ω ∈ B, where gh(K, y, τ) satisfies

∂τgh + Lhgh = −K2

2

[Vh(Ke(r−q)(Tn−τ), Tn)− V ∗(Ke(r−q)(Tn−τ), Tn)

]e(r−2q)(Tn−τ)f2(y) (3.6)

K > 0, y > 0, τ ∈ [Tn−1, Tn),

gh(K, y, Tn) = 0,

where Lh is the operator L with coefficient bh(·, ·) in place of b(·, ·).

The proof is in Appendix B.

It is somewhat time consuming to find numerical solutions of (3.5)-(3.6) because (3.6) is a

two dimensional time-dependent problem. To simplify computations, we send h → 0 to obtain the

following limiting equation of (3.5):∫ T

t∗dτ

∫ ∞

0

ε1∂τ b · ω + ε2∂yb · ∂yω

+12f(y)f ′(y)

∫ ∞

0K2[V (K, τ ; b)− V ∗(K, τ)]ψ (K, y, τ ; b) dK · ω

dy = 0 (3.7)

7

for any ω(y, τ) ∈ C∞0 ((0,+∞)× (t∗,+∞)), where b = b (y, t) , ψ(K, y, T ; b) and V (K, T ; b) are the

solution to (2.3)-(2.4) and as given by (2.7), respectively. The derivation of (3.7) is in Appendix C.

(3.7) can be regarded as the necessary condition of the optimal control problem in continuous

time. It is apparent that (3.7) is the weak formulation of the following equation:

ε1∂T b− ε2∂yyb +12f(y)f ′(y)

∫ ∞

0K2[V (K, T )− V ∗(K, T )]ψ (K, y, T ; b) dK = 0 (3.8)

in y > 0, T > t∗. We will design an algorithm based on a discretization of (3.8) to find the optimal

solution.4

3.2 Numerical Algorithm

We have seen that the optimal solution is determined by a system of equations (2.3)-(2.4), (2.7),

and (3.8) with certain initial/boundary conditions. This requires the numerical solution of the

system which can be accomplished using a finite difference method (see e.g., Quarteroni and Valli

(1994)). Due to stability concern, we use the implicit finite difference method which requires that

the solution domain be truncated into a bounded domain:

K ∈ [Kmin,Kmax], y ∈ [0, ymax], t ∈ [t∗, TN ].

Appropriate boundary conditions will be prescribed.

To describe our algorithm, we use semi-discretization schemes. Let us choose h > 0 as the

step size of time and let t∗ = T0 < T1 < T2 < · · · < TN be the partition of [t∗, TN ] with Tn = T0+nh.

Denote

bn(·) ≡ b(·, Tn), ψn (·, ·) = ψ (·, ·, Tn) , Vn (·) = V (·, Tn) .

Suppose b0(·) is given with ψ0(K, y) = δ(x∗ −K)δ(y∗ − y), V0(K) = (x∗ −K)+. We inductively

solve for bn(·) as well as ψn(·, ·) and Vn(·), n = 1, 2, · · · , N through an implicit discretization of the

system. Because (3.8) is nonlinear, an iterative procedure is required at each time step. Let bn,k,

ψn,k, and Vn,k be the values at the k-th iteration of the n-th time step. The iteration procedure for

the n-th time step is given as follow.

Iteration procedure for the n-th time step:

1. Choose a tolerance ε > 0 and an initial guess

bn,0(·) = bn−1(·), ψn,0(·, ·) = ψn−1(·, ·), Vn,0(·) = Vn−1(·),4We also tried solving (3.5)-(3.6) and obtained almost the same numerical results, though it was relatively time-

consuming.

8

and set k = 1.

2. Solve for bn,k in terms of an implicit discretization of (3.8):

ε1bn,k − bn−1

h− ε2∂yybn,k + F (y, ψn,k−1, Vn,k−1) = 0 for y ∈ (0, ymax) , (3.9)

where

F (y, ψn,k−1, Vn,k−1) =12f(y)f ′(y)

∫ Kmax

Kmin

K2(Vn,k−1(K)− V ∗(K, Tn))ψn,k−1(K, y)dK,

and either Dirichlet or Neumann boundary conditions are given at y = 0, Ymax and are

elaborated in the next section.

3. Use the updated bn,k to solve for ψn,k(·, ·) in terms of an implicit discretization of (2.3):

ψn,k − ψn−1

h− L∗ψn,k = 0 for K ∈ (Kmin,Kmax) , y ∈ (0, ymax)

with boundary conditions

ψn,k(Kmin, ·) = ψn,k(Kmax, ·) = 0,

ψn,k(·, 0) = ψn,k(·, ymax) = 0,

where the risk-neutral drift term of volatility process appearing in L∗ takes the form of linear

interpolation5

b(·, T ) ≡ T − Tn−1

hbn,k(·) +

Tn − T

hbn−1(·), Tn−1 ≤ T ≤ Tn.

4. Find Vn,k(·) by the updated ψn,k(·) and a variation of (2.7)

Vn,k(K) = e−qhVn−1(e−(r−q)hK)

+K2

2

∫ Tn

Tn−1

e(−2r+q)(Tn−τ)

∫ ∞

0f2(y)ψ(Ke−(r−q)(Tn−τ), y, τ)dydτ.

Using a simple numerical integration, we have

Vn,k(K) = e−qhVn−1(e−(r−q)hK)

+K2

2

∫ Ymax

0

h

2f2(y)

[e(−2r+q)hψn−1(Ke−(r−q)h, y) + ψn,k(K, y)

]dy.

5. Let bn = bn,k, ψn = ψn,k, Vn = Vn,k and stop if ‖bn,k(·) − bn,k−1(·)‖ < ε. Otherwise set

k = k + 1 and go to Step 2.5Here we use the fully implicit scheme, so only bn,k is actually used. The form of interpolation will be needed

when the Crank-Nicolson scheme is adopted.

9

To make the above iteration more smoothly, we introduce a ‘relaxed’ time scale κ at Step 2

of the above procedure. That is, we replace (3.9) by

bn,k − bn,k−1

κ+ ε1

bn,k − bn−1

h− ε2∂yybn,k + F (y, ψn,k−1, Vn,k−1) = 0. (3.10)

Note that (3.10) can be rewritten as

bn,k = bn,k−1 − κ

[ε1

bn,k − bn−1

h− ε2∂yybn,k + F (y, ψn,k−1, Vn,k−1)

]. (3.11)

Recalling the derivation procedure of the necessary condition, we can see that the last term [·]on the right hand side of (3.11) is essentially the Frechet derivative of the cost functional Jn(·).Hence, this is equivalent to employing the steepest descent method to find the minimizer of the

optimization problem (3.3). To ensure stability, we still use an implicit scheme to solve (3.10).

A natural question is how to choose the values of regularity parameters ε1, ε2. Theoretically

the smaller ε1, ε2, the more accurate the solutions. However decreasing the values of ε1, ε2 may

cause numerical oscillation. Luckily our numerical experiments reveal that we can always choose

ε1 = 0 which nonetheless does not incur any oscillation. The value of ε2 varies for different examples

but is also very small.

The remaining problem is how to choose the initial guess b0(·). By numerical experiments,

we find that owing to ε1 = 0, the recovered b(·, ·) is insensitive to b0(·). Hence, for all experiments

presented in the next section, we always choose b0(·) ≡ 0.

4 Numerical Experiments and Market Tests

To test our algorithm, we consider an extended Hull-White model:6

dXt

Xt= (r − q)dt + σtdW 0

t ,

σt =√

Yt,

dYt = b(Yt, t)dt + γYtdW 1t

with dW 0t dW 1

t = ρdt. That is,

f(y) =√

y, β(y) = γy.

Here ρ and γ are constants that can be estimated from historical data of the underlying. We

implement the algorithm presented in Section 3.2 to recover b(·, ·).6The reason we are interested in the Hull-White model is that it is consistent with the GARCH(1,1) model. In the

extended Hull-White model considered here, the risk-neutral drift term of the variance process is a general functionof instantaneous variance and time. We highlight that our approach applies to other stochastic volatility models.

10

We use a uniform mesh with NK subintervals in [Kmin,Kmax] and Ny subintervals in [0, ymax] ,

that is,

Ki = Kmin + iKmax −Kmin

NK, i = 0, 1, ..., NK ,

yj = jymax

Ny, j = 0, 1, ..., Ny.

4.1 Numerical Experiments

To demonstrate the efficiency of our algorithm, we first carry out tests with exact solutions. Suppose

the exact solution is b∗(·, ·) with which we solve (2.1)-(2.2) to get V (x∗, y∗, t∗;Ki, Tn) for i =

0, 1, ..., NK ; n = 0, 1, ..., N . We then treat V ∗(Ki, Tj) ≡ V (x∗, y∗, t∗;Ki, Tj) as the market prices of

options which are used to recover b(·, ·) through the algorithm presented in Section 3.2.

At Step 2 of the iteration procedure, the boundary conditions for bn,k are needed. We then

use the exact boundary conditions. Both Dirichlet conditions and Neumann conditions are tested

respectively, that is,

Dirichlet conditions: bn,k(0) = b∗(0, Tn), bn,k(ymax) = b∗(ymax, Tn);

Neumann conditions: ∂ybn,k(0) = ∂yb∗(0, Tn), ∂ybn,k(ymax) = ∂yb

∗(ymax, Tn).

The default parameter values are given as follows:

x∗ = 1, y∗ = 0.02, t∗ = 0, Kmin = 0.5, Kmax = 1.5, Ymax = 0.1, NK = 50, Ny = 50,

r = 0.02, q = 0, γ = 0.1, ρ = 0.5, ε = 2 ∗ 10−9, ε1 = 0, ε2 = 4 ∗ 10−5, κ = 2 ∗ 10−6.

Test 1. b∗ = b∗(y).

We first consider a simple case in which b∗ is independent of time. Then we only need the prices

of options with one maturity T1 to do calibration. So, N = 1, and we take T1 = 1/12.

We test three kinds of profile for b, corresponding constant, linear, and cubic dependence on

y, respectively.

1. b∗ (y) = 0.2;

2. b∗(y) = 2(0.08− y);

3. b∗(y) = −(y − 0.05)3.

For all three cases we use b0 (·) ≡ 0 as an initial guess. The results are presented in Figure

1, Figure 2, and Figure 3, respectively. It can be observed that the recovered values coincide with

the exact values perfectly.

11

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

y

b

Initial guessRecovered valueTrue value

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

y

b


Figure 1: Recovery of constant drift term. The left figure is with Dirichlet conditions while theright is with Neumann conditions. The dotted line is the initial guess b0(·) ≡ 0, the solid line is theexact drift term b∗(·) = 0.2, and the circles are the drift term b(·) recovered by our algorithm fromoption prices that are generated with the exact drift term.

Test 2. b∗ = b∗(y, T )

Now we examine the time-dependent case in which only Neumann conditions are given for

(3.10). We need to use three months data, so we take N = 3, T1 = 112 , T2 = 2

12 , T3 = 312 .

The exact value b∗(y, T ) = −(y − 0.05)3/(1 + 2T ) with which we generate the option prices

as the market prices. We again use the initial value b0 (·) ≡ 0 and Neumann boundary conditions,

and the results are shown in Figure 4. We can see that our algorithm still performs very well.

Test 3. Stability with data

To investigate the stability of our algorithm, we add a white noise to the ‘observed’ data. The

whole procedure is elaborated below. We consider the case in Test 1 with the exact risk-neutral

drift term b∗(y) = 2(0.08− y). As before we solve (2.1)-(2.2) with b∗ (·) to get V (x∗, y∗, t∗;Ki, Tn).

For any i = 0, 1, ..., NK , n = 0, 1, ..., N , we let

V ∗ (Ki, Tn) =(

1 +ξi,n

50

)V (x∗, y∗, t∗;Ki, Tn),

where ξi,n is a random variable drawn from the standardized normal distribution. Then we use

V ∗ (Ki, Tn) to do calibration. The results, presented in Figures 5, verify the stability of our algo-

rithm.

12

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−0.05

0

0.05

0.1

0.15

0.2

y

b


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0

0.05

0.1

0.15

0.2

y

b


Figure 2: Recovery of mean-reverting drift term. The left figure is with Dirichlet conditions whilethe right is with Neumann conditions. The dotted line is the initial guess b0(·) ≡ 0, the solid lineis the exact drift term b∗(y) = 2(0.08− y), and the circles are the drift term b(·) recovered by ouralgorithm from option prices that are generated with the exact drift term.

4.2 Market Tests

We now test the performance of our algorithm with real market data. Here the underlying asset is

SPY, an exchange traded fund (ETF) on SP 500 index. To calibrate the extended Hull-White model,

we use the quoted prices of call options on SPY with maturities T1 = 28/09/2012, T2 = 20/10/2012,

T3 = 17/11/2012, which are documented in Appendix D.7 All quoted prices are of t∗ = 30/08/2012,

and the SPY price on the day is x∗ = 140.49. The default parameter values are given as follows.

y∗ = 0.024, Kmin = 0.4x∗, Kmax = 1.4x∗, ymax = 0.1,

NK = 100, Ny = 50, r = 0.005, q = 0.03, γ = 0.1, ρ = −0.5,

ε1 = 0, ε2 = 5 ∗ 10−4x∗2, ε = 10−4, κ = 10−2.

We first use the prices of the options with maturity T1 to recover the time-independent b(·),where one time step is needed, i.e., N = 1. As before we need to prescribe the boundary conditions

for bn,k in Step 2 of the iteration procedure. To examine the effect of boundary conditions on

calibrations, we consider the following three (Neumann) boundary conditions:

1. ∂yb1,k(0) = −2, ∂yb1,k(ymax) = −2;

2. ∂yb1,k(0) = −3, ∂yb1,k(ymax) = −3;

7Bid and ask prices are recorded, and we use their average as market prices for calibration.

13

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

y

b


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

y

b

Initial valueRecovered valueTrue value

Figure 3: Recovery of cubic drift term. The left figure is with Dirichlet conditions while the right iswith Neumann conditions. The dotted line is the initial guess b0(·) ≡ 0, the solid line is the exactdrift term b∗(y) = −(y − 0.05)3, and the circles are the drift term b(·) recovered by our algorithmfrom option prices that are generated with the exact drift term.

3. ∂yb1,k(0) = −3, ∂yb1,k(ymax) = −2.

For all cases we use the initial guess b0(·) ≡ 0. The recovered results are reported in Figures

6-8. In each figure, the left shows the recovered b(·) while the right presents the market prices of

options used for calibration (dots) and the option prices generated from the recovered b(·) (solid

line). We observe that the boundary conditions significantly affect the recovered b(·). However,

for all cases, the option prices generated from the recovered b(·) coincide with the market prices

very well. This implies that the solution to the calibration problem is not unique. Moreover, using

a least-square fitting, we can fit the recovered b(·) by a straight line α(θ − y), where (α, θ) =

(1.997, 0.007) , (2.998, 0.012) , (2.308, 0.105) for the above three boundary conditions, respectively.

This indicates that the calibrated risk-neutral variance process is linearly mean-reverting.

Now we further use the real market data on T2 and T3 to calibrate b(·, ·), with boundary

conditions ∂ybn,k(0) = −3, ∂ybn,k(ymax) = −3. The calibration results are shown in Figure 9, where

the left presents calibrated b(·, Ti), i = 1, 2, 3, and the right presents the market option prices (dots)

and the option prices generated from the recovered b(·, ·) (solid line). Again we can observe that

our algorithm performs very well. The recovered b(·, Ti) can be fitted by straight lines αi(θi − y)

14

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

y

b

T=1/12: True bT=2/12: True bT=3/12: True bT=1/12: Recovered bT=1/12: Recovered bT=3/12: Recovered b

Figure 4: Recovery of time-varying drift term. Neumann conditions are given, and the initial guessb0(·) ≡ 0. The dotted, solid, and dashed lines are the exact drift term b∗(y, T ) = −(y− 0.05)3/(1+2T ) with T = 1/12, 2/12, 3/12, respectively, and the circles, dots, and plus are the drift termsb(·, 1/12), b(·, 2/12), b(·, 3/12) recovered by our algorithm from option prices generated with theexact drift term.

with

α1 = 2.998, θ1 = 0.012,

α2 = 2.990, θ2 = 0.093,

α3 = 2.978, θ3 = 0.091,

respectively.

5 Conclusion

In this article, we propose an optimal control approach to recover, from the options market, the risk-

neutral drift term of the volatility or variance process in the stochastic volatility model. In contrast

to the existing literature, the drift term does not possess any special structure and analytical pricing

formulas for European options are unavailable.

We first present a modified Dupire’s equation associated with stochastic volatility models,

which allows us to formulate the calibration problem as a standard inverse problem of partial

differential equations. We then use an optimal control approach with Tikhonov regularization to

solve the inverse problem. A necessary condition that the optimal solution satisfies is derived.

15

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−0.05

0

0.05

0.1

0.15

0.2

y

b


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

K

V

Option prices generated from exact b*

Option prices with white noiseOption prices generated from the recovered b

Figure 5: Stability analysis of the algorithm. The exact drift term is b∗(y) = 2(0.08− y). The leftfigure shows the calibration results, where the dotted line is the initial guess b0(·) ≡ 0, the solidline is the exact drift term, and the circles are the drift term b(·) recovered by our algorithm fromoption prices generated with the exact drift term and white ‘noise’. The right figure presents theoption prices generated with the exact drift term (solid line), the prices with white ‘noise’ (dots),and the prices generated with the recovered drift term (circles).

We further simplify the necessary condition and then propose a gradient descent algorithm to

numerically find the optimal solution. Our algorithm can be applied to calibrate general stochastic

volatility models. An extensive numerical analysis is presented to demonstrate the efficiency of our

numerical algorithm. It is worthwhile pointing out that the boundary conditions required in the

algorithm significantly affect the calibration results, but in all cases the option prices generated

from the stochastic volatility model with the recovered drift term coincide very well with the market

prices of options. Moreover, we find that the variance process of S&P 500 index recovered from the

options market is indeed linearly mean-reverting.

Appendix A: Proof of Proposition 1

Let us first prove (2.5). The original idea stems from Dai and Wu (2002). Denote

u = ∂KKV. (A.1)

Differentiating (2.1)-(2.2) w.r.t. K twice, we obtain

∂tu + Lu = 0 in x > 0, y > 0, t ∈ [0, T )

u|t=T = δ(x−K).

Here δ(x −K) is the Dirac’s delta function concentrated at x = K. Note that δ(x −K) is a line

16

0 0.02 0.04 0.06 0.08 0.1−0.2

−0.15

−0.1

−0.05

0

0.05

y

b

Recovered b

90 100 110 120 130 140 150 160 170 180−5

0

5

10

15

20

25

30

35

40

45

50

K

V

Option prices generated from the recovered bReal market prices

Figure 6: Market test with boundary conditions ∂yb1,k(0) = −2, ∂yb1,k(ymax) = −2. The quotedprices of call options on SPY with maturity T1 = 28/09/2012 are used. The current time t∗ =30/08/2012, and the SPY price on the day is 140.49. The left figure presents the calibrated b(·)while the right figure presents the market prices of options used for calibration (circles) and theoption prices generated from the recovered b(·) (solid line). The recovered b(·) can be fitted by astraight line α(θ − y) with α = 1.997 and θ = 0.007.

source, instead of a point source, since the space is two dimensional. Therefore,

u(x, y, t;K, T ) =∫ ∞

0ψ(x, y, t;K, y, T )dy (A.2)

by which, we integrate (2.3) with respect to y to get

∂T u =12∂KK

(K2

∫ ∞

0f2(y)ψ(x, y, t;K, y, T )dy

)+ ∂K [ρKf (y) β (y) ψ(x, y, t;K, y, T )]|∞0

+12∂y[β2(y)ψ(x, y, t;K, y, T )]|∞0 − (r − q)∂K(Ku)

−(b(y, T )ψ(x, y, t;K, y, T ))|∞0 − ru in K > 0, T > t.

Since ψ decays fast enough at y = 0,∞, we deduce

∂T u =12∂KK

(K2

∫ ∞

0f2(y)ψ(x, y, t;K, y, T )dy

)− (r − q)∂K(Ku)− ru in K > 0, T > t.

Integrating this equation twice w.r.t. K and restricting attention to x∗, y∗, t∗, we get (2.5).

Given ψ, (2.5) is a first-order linear equation. Now let us find the analytical solution of (2.5)

subject to terminal condition (2.6). By transformation

H (Z, T ) = eqT V (K, T ) , Z = log K,

we have∂H

∂T+ (r − q)

∂H

∂Z=

12eqT e2Z

∫ ∞

0f2(y)ψ(eZ , y, T )dy

17

0 0.02 0.04 0.06 0.08 0.1−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

y

b

Recovered b

90 100 110 120 130 140 150 160 170 180−5

0

5

10

15

20

25

30

35

40

45

50

K

V



in Z ∈ (−∞,∞) , T > t∗. Consider the characteristic line Z(T ) = Z(t∗) + (r− q)T along which we

havedH(Z(T ), T )

dT=

12eqT e2Z(T )

∫ ∞

0f2(y)ψ(eZ(T ), y, T )dy.

It follows

H(Z(T ), T ) = H(Z(t∗), t∗) +12

∫ T

t∗eqτe2Z(τ)

∫ ∞

0f2(y)ψ(eZ(τ), y, τ)dydτ

Since Z(τ) = Z(T )− (r − q)(T − τ), we have

H(Z(T ), T ) = H(Z(T )− (r − q)(T − t∗), t∗) +12

∫ T

t∗eqτe2(Z(T )−(r−q)(T−τ))

∫ ∞

0f2(y)ψ(eZ(T )−(r−q)(T−τ), y, τ)dydτ.

Hence,

H(Z, T ) = H(Z−(r−q)(T−t∗), t∗)+12

∫ T

t∗eqτe2(Z−(r−q)(T−τ))

∫ ∞

0f2(y)ψ(eZ−(r−q)(T−τ), y, τ)dydτ

Changing to the original variables gives the desired result.

Appendix B: Proof of Proposition 2

18

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

y

b

Recovered b

90 100 110 120 130 140 150 160 170 180−5

0

5

10

15

20

25

30

35

40

45

50

K

V



(3.4) is equivalent to

d

dλ

(∫ ∞

0

ε12h|bλ − bn−1|2 +

ε22|∂ybλ(y)|2dy +

12h2

∫ ∞

0|Vh(K,Tn; bh

λ)− V ∗(K, Tn)|2dK

)∣∣∣∣λ=0

= 0,

(B.1)

where Vh(K,Tn; bhλ) is as given by (2.7) with the coefficient bh

λ(·, ·) ≡ bh(·, ·) + λω(·). Setting

ξ(K, T ) = dVh(K,T ;bhλ)

dλ |λ=0, we infer from (B.1) that∫ ∞

0ε1

bn − bn−1

h·ω+ε2∂ybn(y)∂yω(y)dy+

1h2

∫ ∞

0(Vh(K, Tn)−V ∗(K, Tn))ξ(K, Tn)dK = 0 (B.2)

for any ω ∈ B and Tn−1 ≤ T ≤ Tn. By (2.5), ξ satisfies

∂T ξ + (r − q)K∂Kξ + qξ = K2

2

∫∞0 f2(y)η(K, y, T )dy, Tn−1 ≤ T ≤ Tn,

ξ(K, Tn−1) = 0,

with η(K, y, T ) = dψh(K,y,T ;bhλ)

dλ |λ=0, where ψh(K, y, T ; bhλ) is the solution to (2.3)-(2.4) with the

coefficient bhλ. Similar to (2.7), we have an analytical expression form for ξ,

ξ(K, T ) =∫ T

Tn−1

K2

2e−(2r−q)(T−τ)

∫ ∞

0f2(y)η(Ke−(r−q)(T−τ), y, τ)dydτ. (B.3)

Note that η(K, y, T ) is the solution to the following problem:

∂T η − L∗hη = −∂y(ψhω)T−Tn−1

h , Tn−1 < T ≤ Tn,

η(K, y, Tn−1) = 0.

19

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

y

b

T=T

1: Recovered b

T=T2: Recovered b

T=T3: Recovered b

90 100 110 120 130 140 150 160 170 180−5

0

5

10

15

20

25

30

35

40

45

50

K

V

T1: Option prices generated from the recovered b



T1: Real market prices



Figure 9: Market test with boundary conditions ∂ybn,k(0) = −3, ∂ybn,k(ymax) = −3, n = 1, 2, 3.The quoted prices of call options on SPY with maturities T1 = 28/09/2012, T2 = 20/10/2012, andT3 = 17/11/2012 are used. The current time t∗ = 30/08/2012, and the SPY price on the day is140.49. The left figure presents the calibrated b(·, Ti), i = 1, 2, 3 while the right figure presents themarket prices of options used for calibration and the option prices generated from the recoveredb(·). The recovered b(·, Ti) can be fitted by a straight line αi(θi − y) with (α1, θ1) = (2.998, 0.012),(α2, θ2) = (2.990, 0.093), (α3, θ3) = (2.978, 0.091), respectively.

Let Gh be the Green function associated with operator L∗h. By Green’s formula,

η(K, y, T ) =∫ T

Tn−1

dT

∫ ∞

0dK

∫ ∞

0Gh(K, y, T ; K, y, T )∂y(−ψhω)

T − Tn−1

hdy. (B.4)

We now compute the third term on the left-hand side of (B.2):

(∗) =∫ ∞

0(Vh(K, Tn)− V ∗(K, Tn))ξ(K, Tn)dK. (B.5)

Plugging (B.3) into (B.5), we have

(∗) =∫ ∞

0(Vh(K,Tn; bh)− V ∗(K, Tn))

[∫ Tn

Tn−1

K2

2e(−2r+q)(Tn−τ)

∫ ∞

0f2(y)

η(Ke−(r−q)(Tn−τ), y, τ)dydτ]dK

=∫ ∞

0dK

∫ Tn

Tn−1

dτ

∫ ∞

0dy

[(Vh(Ke(r−q)(Tn−τ), Tn)− V ∗(Ke(r−q)(Tn−τ), Tn)

)

K2

2e(r−2q)(Tn−τ)f2(y)η(K, y, τ)

],

where we have used a change of variable on K in the second equality. Substituting (B.4) into the

20

above equation, we obtain

(∗) =∫ ∞

0dK

∫ Tn

Tn−1

dτ

∫ ∞

0dy

[(Vh(Ke(r−q)(Tn−τ), Tn)− V ∗(Ke(r−q)(Tn−τ), Tn)

) K2

2e(r−2q)(Tn−τ)

f2(y)∫ τ

Tn−1

dT

∫ ∞

0dK

∫ ∞

0dyGh(K, y, T ; K, y, T )∂y(−ψhω)

T − Tn−1

h

]

=∫ Tn

Tn−1

dT

∫ ∞

0dK

∫ ∞

0dy(−ψhω)y

T − Tn−1

hgh(K, y, T ),

where the order of integrations is changed in the second equality and

gh(K, y, T ) =∫ Tn

Tdτ

∫ ∞

0dK

∫ ∞

0dyGh(K, y, T ; K, y, T )

(Vh(Ke(r−q)(Tn−τ), Tn)− V ∗(Ke(r−q)(Tn−τ), Tn)

) K2

2e(r−2q)(Tn−τ)f2(y).

It is easy to see gh(K, y, T ) satisfies (3.6). Substituting the term (∗) into (B.2) yields the desired

result. The proof is completed.

Appendix C: Derivation of the necessary condition (3.7) in continuous time

We restrict our attention to a bounded domain Q ≡ I × Ω × (Tmin, TN ) ≡ (0,Kmax) ×(0, Ymax) × (Tmin, TN ), with any fixed Tmin > t∗.8 Without loss of generality, we always impose

homogeneous boundary conditions on the boundary of Q for illustration.9 We still use the previous

notations except T0 = Tmin. Problem (3.5) is restated as follows:

Given b0, · · · , bn−1 ∈ H10 (Ω), find bn ∈ H1

0 (Ω) such that for any ω ∈ H10 (Ω),

∫

Ω

(ε1

bn − bn−1

h· ω + ε2∂ybn · ∂yω

− 1h2

∫ Tn

Tn−1

∫

I

τ − Tn−1

hgh(K, y, τ ; bh)

(ψh(K, y, τ ; bh) · ω

)ydKdτ

)dy = 0,

where gh and ψh are both given homogeneous boundary conditions on Q, and Vh(0, t) = e−q(T−t)x∗,

Vh (Kmax, t) = 0.

As before, bh (·, ·) is constructed from bn (·) , n = 0, ..., TN by piecewise linear interpolation

in time. We aim to show that as h → 0, there is a subsequence of bh (·, ·) that weakly converges to

the solution to the following problem:8The reason we require Tmin > t∗ is that we want to avoid the non-smooth initial conditions at t∗ for ψ and V.

Note that we are only interested in Eq. (3.8) for T > t∗ and the test function ω in (3.7) is C∞0 ((0, +∞)× (t∗, +∞)) .9Other boundary conditions can be treated in a similar way. We point out that a rigorous proof relies on formulating

the original problem in a bounded region with appropriate boundary conditions.

21

For any T ≥ Tmin and any ω(y, T ) ∈ S ≡ H1(Ω× (Tmin, TN )) ∩ L∞((Tmin, TN );H10 (Ω)),

∫ T

Tmin

dτ

∫

Ω

(ε1

∂b

∂τ· ω + ε2by · ωy+

+12f(y)f ′(y)

∫

IK2(V (K, τ)− V ∗(K, τ))ψ (K, y, τ ; b) ω

)dKdy = 0,

where the boundary conditions for ψ and V are similar to those of ψh and Vh, respectively.

We now give a sketch of the proof under the assumption that bh is uniformly bounded in

H1(Ω × (Tmin, TN )). We can show that there exist some α ∈ (0, 1) and some positive constant C

independent of h and n, such that

‖bh‖Cα(Ω×(Tmin,TN )) ≤ C,

where Cα(Ω × (Tmin, TN )) is a space of all H··older continuous functions. Then there exists a

subsequence of bh, still denoted by bh, weakly converges to a function denoted by b(y, T ) in

H1(Ω× (Tmin, TN )) and uniformly in C0(Ω× (Tmin, TN )) by Rellich’s Theorem and Arzela-Ascoli’s

Theorem. Hence

n∑

k=1

h

∫

Ω

(ε1

bk − bk−1

hω(·, tk) + ε2∂ybk ∂yω(·, tk)

)dy

∫ Tn

Tmin

dτ

∫

Ω

(ε1

∂b

∂τω + ε2by ωy

)dy

for any ω(y, T ) ∈ S.

It remains to show

−n∑

k=1

1h

∫

Ω

∫ Tk

Tk−1

∫

I

τ − Tk−1

hgh(K, y, τ ; bh)

(ψh(K, y, τ ; bh)ω

)ydKdτdy

∫ Tn

Tmin

dτ

∫

Ω

12f(y)f ′(y)

∫

IK2 [V (K, τ)− V ∗(K, τ)]ψ (K, y, τ ; b) ωdKdy. (C.1)

The following result plays a critical role: Let φ1(K, y, τ) and φ2(K, y, τ) be the solutions of the

problems

∂τφ1 + Lhφ1 = −T (K, y, τ)h

, τ ∈ (Tk−1, Tk), (K, y) ∈ I × Ω

φ1(K, y, Tk) = 0,

and

∂τφ2 + Lhφ2 = 0, τ ∈ (Tk−1, Tk), (K, y) ∈ I × Ω

φ2(K, y, Tk) = T (K, y, Tk),

22

respectively, and both are subject to homogeneous boundary conditions. If T is smooth, then there

exists some α1 ∈ (0, 1) such that

||φ1 − φ2||C0(I×Ω×[Tk−1,Tk]) ≤ Chα1 ,

where C > 0 is independent of h. Using this result and continuity, we can rewrite the left hand

side of (C.1) as

−n∑

k=1

∫

Ω

∫ Tk

Tk−1

∫

I

τ − Tk−1

h

K2

2[Vh (K, Tk)− V ∗ (K, Tk)] f2 (y)

(ψh(K, y, τ ; bh) · ω

)ydKdτdy

= −n∑

k=1

h

2

∫

Ω

∫

I

K2

2[Vh (K,Tk)− V ∗ (K,Tk)] f2 (y)

(ψh(K, y, Tk; bh) · ω

)ydKdy (C.2)

where we have neglected a term of order hα1 . We can show Vh(K, T ; bh), ψh(K, y, T ; bh) and

∂yψh(K, y, T ; bh) converge to V (K,T ; b), ψ(K, y, T ; b) and ∂yψ(K, y, T ; b) uniformly in C0(I ×Ω×(Tmin, TN )), respectively. Taking the limit on the right hand side of (C.2) yields

−12

∫ Tn

T ∗dτ

∫

Ω

∫

I

K2

2[V (K, τ)− V ∗ (K, τ)] f2 (y) (ψ(K, y, τ ; b) · ω)y dKdy

=∫ Tn

T ∗dτ

∫

Ω

∫

I

K2

2[V (K, τ)− V ∗ (K, τ)] f (y) f ′ (y) ψ(K, y, τ ; b) · ωdKdy,

where the integration by parts with respect to y is used in the last equality. This gives the desired

result.

Appendix D: Market Data

Strike Bid Ask Strike Bid Ask Strike Bid Ask

70 70.24 70.45 120 20.29 20.51 142 1.38 1.4190 50.24 50.45 121 19.30 19.52 143 0.99 1.00100 40.24 40.45 122 18.31 18.54 144 0.67 0.69101 39.24 39.45 123 17.32 17.55 145 0.43 0.45102 38.24 38.45 124 16.34 16.57 146 0.28 0.29103 37.24 37.46 125 15.36 15.59 147 0.17 0.18104 36.24 36.46 126 14.39 14.61 148 0.10 0.11105 35.24 35.46 127 13.42 13.64 149 0.06 0.07106 34.24 34.46 128 12.45 12.68 150 0.04 0.05107 33.24 33.46 129 11.52 11.72 151 0.03 0.04108 32.24 32.46 130 10.57 10.78 152 0.02 0.03109 31.24 31.46 131 9.63 9.84 153 0.01 0.02110 30.25 30.46 132 8.70 8.92 154 0.01 0.02111 29.25 29.47 133 7.81 8.01 155 0.00 0.02112 28.25 28.47 134 6.96 7.10 156 0.00 0.01113 27.25 27.47 135 6.09 6.26 157 0.00 0.01114 26.26 26.48 136 5.27 5.43 158 0.00 0.01115 25.26 25.48 137 4.50 4.61 159 0.00 0.01116 24.26 24.48 138 3.76 3.85 160 0.00 0.01117 23.27 23.49 139 3.07 3.12 161 0.00 0.01118 22.27 22.50 140 2.44 2.47 162 0.00 0.01119 21.28 21.50 141 1.87 1.89 163 0.00 0.01

Table 1: Quoted prices of call options on SPY recorded on 30/08/2012: Expire at close Friday, 28/09/2012.

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80 60.24 60.45 125 15.54 15.75 153 0.06 0.0793 47.24 47.45 126 14.59 14.80 154 0.04 0.0594 46.24 46.45 127 13.65 13.86 155 0.03 0.04100 40.24 40.46 128 12.71 12.93 156 0.02 0.03101 39.24 39.46 129 11.79 12.01 157 0.02 0.03102 38.24 38.46 130 10.90 11.10 158 0.01 0.02103 37.24 37.46 131 10.03 10.17 159 0.01 0.02104 36.24 36.46 132 9.16 9.33 160 0.01 0.02105 35.24 35.46 133 8.32 8.48 161 0.00 0.01106 34.25 34.46 134 7.47 7.64 162 0.00 0.01107 33.25 33.47 135 6.67 6.81 163 0.00 0.01108 32.25 32.47 136 5.91 6.05 164 0.00 0.01109 31.26 31.47 137 5.19 5.28 165 0.00 0.01110 30.26 30.48 138 4.49 4.57 166 0.00 0.01111 29.26 29.48 139 3.85 3.90 167 0.00 0.01112 28.27 28.49 140 3.24 3.27 168 0.00 0.01113 27.27 27.49 141 2.67 2.70 169 0.00 0.01114 26.28 26.50 142 2.17 2.19 170 0.00 0.01115 25.29 25.51 143 1.72 1.74 171 0.00 0.01116 24.30 24.52 144 1.33 1.35 172 0.00 0.01117 23.31 23.53 145 1.00 1.02 173 0.00 0.01118 22.33 22.55 146 0.73 0.75 174 0.00 0.01119 21.34 21.56 147 0.52 0.54 175 0.00 0.01120 20.36 20.58 148 0.37 0.38 176 0.00 0.01121 19.40 19.61 149 0.25 0.27 177 0.00 0.01122 18.41 18.63 150 0.17 0.18 178 0.00 0.01123 17.44 17.67 151 0.12 0.13 179 0.00 0.01124 16.50 16.70 152 0.08 0.09 180 0.00 0.01



99 41.25 41.47 127 14.14 14.36 150 0.61 0.62100 40.25 40.47 128 13.26 13.49 151 0.46 0.48101 39.25 39.47 129 12.39 12.61 152 0.34 0.36105 35.27 35.49 130 11.57 11.75 153 0.26 0.27108 32.30 32.51 131 10.75 10.90 154 0.19 0.20109 31.31 31.52 132 9.93 10.11 155 0.14 0.15110 30.32 30.54 133 9.11 9.30 156 0.11 0.12111 29.33 29.55 134 8.38 8.49 157 0.08 0.09112 28.34 28.57 135 7.64 7.75 158 0.06 0.07113 27.38 27.59 136 6.90 7.01 159 0.05 0.06114 26.39 26.61 137 6.23 6.30 160 0.04 0.05115 25.41 25.63 138 5.59 5.64 161 0.03 0.04116 24.46 24.66 139 4.96 5.00 162 0.02 0.03117 23.46 23.69 140 4.36 4.39 163 0.02 0.03118 22.51 22.73 141 3.79 3.82 164 0.02 0.03119 21.54 21.77 142 3.26 3.29 165 0.01 0.02120 20.59 20.82 143 2.78 2.80 166 0.01 0.02121 19.66 19.87 144 2.33 2.35 167 0.01 0.02122 18.70 18.93 145 1.93 1.95 168 0.01 0.02123 17.77 18.00 146 1.58 1.60 169 0.01 0.02124 16.85 17.07 147 1.27 1.28 170 0.01 0.02125 15.95 16.16 148 1.01 1.03 171 0.01 0.02126 15.03 15.25 149 0.78 0.80 172 0.00 0.02


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Calibration of Stochastic Volatility Models: An Optimal Control

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