Inverse Problems
LETTER TO THE EDITOR
The inverse problem of option pricing
To cite this article: Ilia Bouchouev and Victor Isakov 1997 Inverse Problems 13 L11
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Inverse Problems 13 (1997) L11–L17. Printed in the UK PII: S0266-5611(97)84870-8
LETTER TO THE EDITOR
The inverse problem of option pricing
Ilia Bouchouev and Victor Isakov
Department of Mathematics and Statistics, Wichita State University, Wichita, KS 67260-0033,
USA
Received 10 June 1997
Abstract. Valuation of options and other financial derivatives critically depends on the
underlying stochastic process specified for a particular market. An inverse problem of option
pricing is to determine the nature of this stochastic process, namely, the distribution of expected
asset returns implied by current market prices of options with different strikes. We give a
rigorous mathematical formulation of this inverse problem, establish uniqueness, and suggest
an efficient numerical solution. We apply the method to the S&P 500 Index and conclude that
the index is negatively skewed with a higher probability of the sudden decline of the US stock
market.
An option is a contract that gives the owner the right to buy or sell a specified amount of a
particular underlying asset at a fixed price within a fixed period of time. An option to buy
is a call option; an option to sell is a put option. A fixed price of the contract is an exercise
(strike) price. An expiration date of the option is called a maturity date. Options that can
be exercised at any time up to the maturity date are called American options, while options
that can only be exercised on the maturity date are European.
We consider the valuation of European options on stock (or stock index) but our results
are also applicable to certain other markets. The stock index such as the S&P 500 can be
considered as a single stock that pays a continuous dividend yield.
The option premium, u, is determined by a number of factors, including the underlying
stock price, x, current time, t , the maturity date, T , the exercise price, K , the risk-free
interest rate, R, the dividend yield on the stock, D, and local stock volatility, , that
measures the riskiness of the stock. Local volatility is defined as instantaneous time-
independent variance of expected stock returns. It is the only parameter that cannot be
observed directly. Moreover, it depends on the stock price, which, in turn changes over
time.
The call option premium u D u.x; tIK;R;D; .x// satisfies the Black–Scholes partial
differential equation
@u
@t
C 1
2
x2 2.x/
@2u
@x2
C .R −D/x @u
@x
− Ru D 0 .x; t/ 2 RC .t; T / (1)
derived in the seminal paper [1].
We assume that u.x; t/ 2 C.[0;1/ [t; T ]/ and 0 6 u.x; t/ 6 x. The last inequality
ensures that the price paid for the right to buy a stock cannot exceed the stock price itself. We
often write u D u.x; tIK/ and suppress for simplicity its dependence on other parameters.
0266-5611/97/050011+07$19.50 c© 1997 IOP Publishing Ltd L11
L12 Letter to the Editor
The boundary conditions are given by
u.0; tIK/ D 0 (2)
and
u.x; T IK/ D max.0; x −K/: (3)
The initial boundary value problem (1)–(3) has a closed-form solution only if volatility
is constant, or if it takes a rather special functional form. We refer to [7] and [15] for the
survey of analytic formulae and numerical methods applicable to the problem (1)–(3).
It was suggested in [3, 5, 14] that the local time-dependent volatility function can be
reconstructed by using market prices of options with all possible strikes and maturities.
Since in a real world we do not have enough market prices in the time direction, we
simplify the problem assuming that volatility is time-independent and using only option
prices with different strikes and a fixed maturity date. This assumption is realistic for
equity options but it is rather restrictive for foreign exchange and commodity markets.
Valuation of options with the local volatility function implied by the market became
known in finance as the Super Model of option pricing. We formulate this problem from
the mathematical standpoint as follows.
The inverse problem of option pricing (IPOP). To determine the pair of functions
u.x; tIK/ and .x/ that satisfy (1)–(3) from current market prices
u.x; tIK/ D u.K/ K 2 ! (4)
of options with different strikes.
Here, ! is an interval in RC, and .x/ is assumed to be known outside !. In particular,
! can be the whole RC. The subscript is used for current time and current market prices.
At present, a rigorous mathematical analysis of this inverse problem in the continuous-
time setting has not been given. Instead various recipes for reconstruction of the discrete
implied binomial tree have been suggested. An alternative optimization approach was
recently proposed in [11]. This algorithm gives a better accuracy but is highly demanding
from the computational standpoint.
In this letter we reduce (1)–(4) to the inverse parabolic problem with the final observation
and use the adjoint equation to establish a uniqueness result. Then we obtain a non-linear
Fredholm equation for volatility and solve it iteratively. As an example, we calculate the
local volatility function implied by options on the S&P 500 Index and conclude that the
distribution of stock market returns is negatively skewed with a higher probability of the
sudden decline of the US stock market.
For the general theory of parabolic equations and definition of appropriate norms in
the space of Holder continuous functions we refer to [6]. We differentiate (1) twice with
respect to K and introduce ’.x; tIK; T / as
@2u.x; tIK; T /
@K2
D ’.x; tIK; T /:
The existence of this derivative can be shown in a straightforward manner by using
the finite difference approximation and well known properties of solutions to the parabolic
equation.
The function ’.x; tIK; T / satisfies (1) with the terminal data
’.x; T IK; T / D .x −K/
where .x −K/ is the Dirac’s delta function concentrated at K .
Letter to the Editor L13
From the well known property of the Green’s function, ’.x; tIK; T / also satisfies the
adjoint equation:
@’
@T
D 1
2
@2
@K2
.K2 2.K/’/− .R −D/ @
@K
.K’/− R’:
We integrate this equation twice with respect to K , and using integration by parts arrive
at
@u
@
D 1
2
K2 2.K/
@2u
@K2
C .D − R/K @u
@K
−Du .K; / 2 RC .0; / (5)
where D T − t is the time remaining to maturity and D T − t. A simplified version
of this equation has been derived in a slightly different way in [3] and [5].
Adding natural boundary conditions for u D u.K; /:
u.0; / D e−Dx (6)
u.K; 0/ D max.0; x −K/ (7)
and additional market data
u.K; / D u.K/; (8)
we obtain the inverse coefficient problem in variables K and with the final observation
at D . Here, the current stock price is a known parameter.
Equations (5)–(7) can be considered as the initial boundary value problem for the put
option premium where the underlying price is interchanged with the strike, and the interest
rate is interchanged with the dividend yield. This duality is important for foreign-exchange
options where the dividend yield is just the risk-free interest rate in the foreign currency.
Theorem 1. Assume that .K/ is known on a certain interval  !. Then a pair of
functions .u.K; /; .K// that satisfies (5)–(8) and, therefore, (1)–(4) is unique.
If, in addition, ! is bounded, 0 62 N! and jj j.!/ 6 M , j D 1; 2; 0 < < 1, then the
solution is stable: there exists a constant C D C.M/ such that
j1 − 2j.!/ 6 Cju1.; /− u2.; /j2C.!/
where u1 and u2 are solutions corresponding to 1 and 2.
Proof. First we note that all classical results for parabolic equations are applicable to
(5) since the degeneration there can be removed if we use lnK as a new state variable.
Subtracting equations for two solution pairs .u1; 1/ and .u2; 2/ we obtain that u D u2−u1
satisfies:
@u
@
D 1
2
K2 22 .K/
@2u
@K2
C .D − R/K @u
@K
−DuC f .K/@
2u1
@K2
(9)
where
f .K/ D 12K2. 22 .K/− 21 .K//:
We consider this equation on  fg where f .K/ is equal to zero. Since u D 0 on
fg, (9) implies that @u@ D 0 on fg. We differentiate this equation with respect to
and repeating the same argument conclude that all derivatives of the function u.K; / with
respect to also vanish on fg. Since u.K; / is an analytic function of , the solution
is zero on  .0; /. Extending the equation (5) for 2 .; 2/, and using analyticity
again we show that u.K; / vanish on  .0; 2/. Finally, we consider the equation onQ .0; 2/, Q D RC n N with zero Cauchy’s data on the lateral boundary @ .0; 2/,
L14 Letter to the Editor
and, using one known result (see [8, Theorem 6.4.1]), we obtain that f .K/ D 0 on Q.
More details are given in [2].
From the Schauder-type estimate for the inverse source problem (see [9]), we have
jf j.!/ 6 C.ju.; /j2C.!/C juj0.RC .0; ///:
Since the solution is unique, we apply the argument of [8] and [9] and eliminate the
last term in the above estimate.
Now we write an integral representation for the solution to the problem (1)–(3) that will
be used for the numerical computation of unknown volatility.
It is convenient to make the change of variables
D T − t y D ln x
K
:
Let
U.y; IK/ D u.x; tIK/ a.y/ D .x/:
Then equations (1) and (3) can be written as
@U
@
D 1
2
a2.y/

@2U
@y2
− @U
@y

C .R −D/− RU .y; / 2 R .0; /
U.y; 0IK/ D K max.0; ey − 1/:
(10)
The function V D @U
@
also satisfies (10) with the initial data given by
V .y; 0IK/ D K. 12a2.y/.y/C RH.y/−DH.y/ey/
where .y/ is the Dirac’s delta function, and H.y/ is the Heaviside’s step function. Using
the integral representation of V .y; IK/, we obtain that
U.y; IK/ D U.y; 0IK/C
Z
0
V .y; IK/ d
D K max.0; ey − 1/C 12a2.0/K
Z
0
0.y; I 0; 0/ d
CRK
Z
0
Z 1
0
0.y; I ; 0/ d d
−DK
Z
0
Z 1
0
0.y; I ; 0/e d d (11)
where 0.y; I ; s/ is the fundamental solution to equation (10).
Theorem 2. Let .x/ be a strictly positive bounded Lipschitz continuous function. Then
u.x; tIK/ D max.0; x −K/C I .K/C
Z 1
0
I1.K; / d
C
Z 1
K
I2.K; / d C o.T − t/ (12)
where
I .K/ D Kp
2
Z .K/pT−t
0
e−.ln x=K/
2=2s2 ds
I1.K; / D K4.K/./
Z T−t
0
Z
0

2./− 2.K/
2&

.ln
K
/2
& 2.K/
− 1

Letter to the Editor L15
C

2./
2
− R CD
ln
K
&

1p
. − &/&
exp

− .ln
x

/2
2. − &/ 2./
!
exp

− .ln

K
/2
2& 2.K/
!
d& d
I2.K; / D 2p
2 2./

RK

−D
Z ./pT−t
0
e−.ln x=/
2=2s2 ds
lim
t!T
o.T − t/
T − t D 0:
Proof. Following [6], we construct the fundamental solution 0.y; I ; s/ in the form of
infinite series and explicitly evaluate terms that do not vanish as ! s. It gives
0.y; I ; s/ D Z.y; I ; s/C Z1.y; I ; s/C Z.y; I ; s/
where
Z.y; I ; s/ D 1p
2. − s/a./e
−.y−/2=2.−s/a2./
Z1.y; I ; s/ D
Z
s
Z 1
−1

a2./− a2./
2.& − s/a2./

. − /2
.& − s/a2./ − 1

C

a2./
2
− R CD

. − /
.& − s/a2./

Z.y; I ; &/Z.; &I ; s/ d d&
and Z.y; I ; s/! 0 as ! s. MoreoverZ 1
0
0.y; I ; s/ d D
Z 1
0
Z.y; I ; s/ d C Z.y; I s/
such that Z.y; I s/! 0 as ! s.
We substitute these expressions into (11), return to the original variables .x; t/, and
make appropriate substitutions to remove integrable singularities in I and I2 . We refer
to [2] for details.
We now let x D x in (12), drop terms of the order o.T − t/, and use market data
(4). Thus, we have obtained a nonlinear Fredholm integral equation of the second kind for
unknown volatility. This equation can be solved by using the following iterations:
I .mC1/.K/ D u.K/−max.0; x −K/−
Z 1
0
I1
.m/.K; / d −
Z 1
K
I2
.m/.K; / d:
We use standard one-dimensional Gaussian quadrature formulae to approximate I2 .
As we can see from the definition of I1 , the integration there is taken over a triangle in
the .; &/ plane. For the triangular regions a variety of Gaussian quadrature formulae are
readily available as well (see, for example, [12] and [13]), and we use them to approximate
I1 .
Let Ki D KminCh.i−1/, i D 1; : : : ; nC1, n D .Kmax−Kmin/=h, where Kmin and Kmax
are, respectively, the smallest and largest strikes available for trading on a particular day,
and h is the strike interval specified by an option contract. The lack of data for K > Kmax
and K < Kmin does not cause a major problem since integrands in (12) for this range of
K are exponentially small. If we cut off these integrals at K D Kmin and K D Kmax, the
approximation error is usually negligible. An alternative approach would be to extrapolate
data for a larger set of strike prices.
L16 Letter to the Editor
Figure 1. Local volatility versus implied Black–Scholes volatility (from S&P 500 Index,
18 October 1996).
Figure 2. Implied distribution versus normal distribution (from S&P 500 Index, 18 October
1996).
We use the extended trapezoidal rule to approximate integrals in (12). Then on each
iteration step we solve the following nonlinear equations for .Ki/, i D 1; : : : ; nC 1:
I .mC1/.Ki/ D u.Ki/−max.0; x −Ki/− h
nX
jD2
I1
.m/.Ki;Kj /− h2 I1
.m/.Ki;K1/
−h
2
I1
.m/.Ki;KnC1/− h
nX
jDiC1
I2
.m/.Ki;Kj /− h2 I2
.m/.Ki;Ki/
−h
2
I1
.m/.Ki;KnC1/: (13)
For i D nC 1, the last two terms should be dropped.
We apply our iterative method (13) to the S&P 500 Index using market prices of options
on 18 October 1996, and the following set of input parameters: x D 710:82, T − t D 43252(43 trading days until maturity), R D 0:05, D D 0:02. The average of bid–ask prices for
options has been used as input data u.K/. The local volatility is reconstructed on [650,
750]. Ten iterations were needed to achieve a convergence of the order 10−5.
It is a common practice to price options using implied Black–Scholes volatility obtained
by inverting the Black–Scholes formula for volatility successively for different strikes.
Letter to the Editor L17
Implied and local volatility curves have a similar shape but the latter decreases more rapidly.
Figure 1 confirms the important difference between two curves discussed in [4].
Using the local-volatility function, we then solve numerically the corresponding Fokker–
Planck equation by using the implicit Crank–Nicholson scheme and determine the implied
dispersion of expected returns. Figure 2 shows that the implied distribution differs
significantly from the normal distribution often assumed by option traders and analysts.
This skewness is typical for the S&P 500 Index, and it agrees with other empirical and
theoretical results in this area (see [3, 10]). We conclude the US stock market follows a
distribution with a fatter left tail, i.e. with higher probabilities of sharp downside movements.
The implied distribution of expected returns can be used for various purposes such as
risk management and hedging, arbitrage and volatility trading, and pricing exotic and more
complicated over-the-counter options.
VI was partially supported by NSF grant DMS 9501510.
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