Table Of ContentA simple time-consistent model for the forward density
process
3 Henrik Hult, Filip Lindskog, and Johan Nykvist*
1
0
2 Abstract. In this paper a simple model for the evolution of the forward
density of the future value of an asset is proposed. The model allows for
n
a straightforward initial calibration to option prices and has dynamics that
a
are consistent with empirical findings from option price data. The model is
J
constructed with the aim of being both simple and realistic, and avoid the
1 needforfrequentre-calibration. Themodelpricesofnoptions andaforward
2 contract are expressed as time-varying functions of an (n+1)-dimensional
Brownian motion and it is investigated how the Brownian trajectory can be
] determined from the trajectories of the price processes. An approach based
R
on particle filtering is presented for determining the location of the driving
P Brownian motion from option prices observed in discrete time. A simulation
. studyandanempiricalstudyofcalloptionsontheS&P500indexillustrates
n
thatthemodelprovidesagoodfittooptionpricedata.
i
f
-
q
[
1 1. Introduction
v
9 Considera financialmarketconsistingofa collectionofEuropeanoptionswith
6 maturity T > 0, written on the value S of an asset at time T. Suppose that the
T
8 optionprices atany time t [0,T]canbe expressedas discountedexpected option
4 ∈
payoffs, where the expectations are computed with respect to a density f of S .
. t T
1 The density f is often called the forward density of S . This paper addresses the
t T
0 modeling of the initial density f and the evolution of the density f over time,
0 t
3
1 {ft}t∈[0,T].
: The model is constructed on a filtered probability space (Ω, , t t∈[0,T],P)
v F {F }
with expectation operator E. For each t [0,T], the forward price of a derivative
i ∈
X payoff g(S ) is the expected payoff computed with respect to the density f :
T t
r
a (1) E[g(S ) ]= g(x)f (x)dx.
T |Ft Z t
In statements such as the above, to avoid technicalities, the functions mentioned
are assumed to satisfy measurability and integrability conditions necessary for the
statements to be meaningful. The market is assumed to consist of n+1 forward
contracts on European call options with payoffs (S K ) for 0 = K < K <
T j + 0 1
−
<K . If the originalmarketconsists of a mix of Europeanputs and calls, then
n
···
the put-call parity may be used to define an equivalent market consisting entirely
2000 Mathematics Subject Classification. 91B24,91B70(primary);60G44(secondary).
Key words and phrases. Optionpricing,mixturemodels.
JEL classification. C60,C63,G12,G13.
H.Hultacknowledges supportfromtheG¨oranGustafssonFoundation.
*Correspondingauthor.
1
2 H.HULT,F.LINDSKOG,ANDJ.NYKVIST
of forward contracts on call option payoffs. From (1) it follows that the forward
price processes {Gjt}t∈[0,T] are martingales satisfying the initial condition
(2) Gj = (x K ) f (x)dx for j =0,1,...,n.
0 Z − j + 0
A parametric form for f will be selected that allows its parameters to be set in a
0
straightforward manner from the n+1 equations in (2) and internally consistent
forward prices G0,G1,...,Gn.
0 0 0
The filtration t t∈[0,T] is assumed to be generated by a standard (n+1)-
{F }
dimensional Brownian motion (V1,V2,...,Vn+1) and we take, for all t, f to be
t
a function with parameters t,V1,V2,...,Vn+1 that vary over time and other pa-
t t t
rameters that are set in the initial calibration of f to the current price data. The
0
choice of f allows the Rn+1-valued forward price process (G0,G1,...,Gn) to be
t
expressed in terms of the Brownian motion (V1,V2,...,Vn+1) as
(G0,G1,...,Gn)=h (V1,V2,...,Vn+1)
t t t t t t t
for functions h : Rn+1 Rn+1, t [0,T]. It is desirable that the functions h
t t
→ ∈
are locally invertible so that the filtration t t∈[0,T] generated by the prices, the
{G }
filtration with an economic interpretation, equals the Brownian filtration. For the
model to be relevant the functions h must give rise to price processes with joint
t
dynamics that are in line with empirically observed stylized facts for option price
processes. Moreover, the range of option prices that the model can produce must
be largeenoughtocapturethe fluctuationsofobservedoptionprices andavoidthe
need for frequent recalibration. Frequent recalibration of a model’s parameters is
unattractive from a theoretical point of view and limits its practical utility.
The model for ft t∈[0,T] set up at time 0 is intended to be relevant also at
{ }
time t>0. Therefore, it makes sense to require that the realized forwardprices at
time t>0 should be possible realizations of the model prices G0,G1,...,Gn. The
t t t
following example illustrates that a simple model such as Black’s model does not
satisfy this requirement.
Example 1 (Black’s model). Consider the case n = 1 (one forward contract
and one call option on ST) and let (Wt)t∈[0,T] be standard Brownian motion with
respect to P. Black’s model, see [1], says that
σ2
S =G0exp σ W 0T
T 0 n 0 T − 2 o
which implies that the forward price G1 for the call option payoff (S K) is
0 T − +
given by
G1 =G0Φ(d ) KΦ(d ),
0 0 1 − 2
log(G0/K) σ √T
d = 0 + 0 , d =d σ √T.
1 2 1 0
σ √T 2 −
0
The parameter σ solving this equation is the option’s implied volatility (implied
0
from G0 and G1). Writing
0 0
σ2 σ2
S =G0exp σ W 0t exp σ (W W ) 0(T t)
T 0 n 0 t− 2 o n 0 T − t − 2 − o
and t = σ( Ws s∈[0,t]) we notice that the model allows stochastic fluctuations in
F { }
the forward price of S :
T
σ2
G0 =G0exp σ W 0t .
t 0 n 0 t− 2 o
A MODEL FOR THE FORWARD DENSITY PROCESS 3
However, the option’s implied volatility is required to stay constant over time. In
particular, the future realized prices are practically guaranteed to violate the model
which therefore has to be frequently recalibrated to fit the price data.
OnereasonfortheinabilityofthedynamicversionofBlack’smodelinExample
1to generatefuture optionpricesis thatthe filtration t t∈[0,T] is generatedbya
{F }
one-dimensionalBrownianmotion. Aftertheinitialcalibrationtherangeofpossible
forwardpricesthatthemodelproducesisverylimited: itislikelythat,afterashort
periodoftime,theobservedoptionpriceslieoutsidetherangeofthemodel. Similar
problemsoccurforinstanceforthelocalvolatilitymodelbyDupire[9]andformany
stochastic volatility models. In addition, the initial calibration for these models is
non-trivial.
Inthemodelwewillconsiderbelowwewant t t∈[0,T] tobeequivalenttothe
{F }
filtrationgeneratedbythe priceprocessesandconsiderthesituationwhennoprice
process can be determined from the other price processes.
We do not consider the spot price process for the underlying asset, only its
value at time T and forward and other derivative contracts written on that value.
If the asset is a non-dividend paying stock, then the spot price must equal the
discounted forward price in order to rule out arbitrage opportunities.
We do not pay attention to the subjective probability views of market partici-
pants. Thereforeitdoesnotmakemuchsenseheretodiscussequivalentmartingale
measures. However, by requiring that the conditional density process is a martin-
galeandthatitproducesrealisticdynamicsforthepriceprocessesweareimplicitly
saying that the model could be a natural candidate for an equivalent martingale
measure for informed market participants.
The paper [17] has a similar objective as ours. However, whereas in [17] the
authorssetupa systemofstochasticdifferentialequations(diffusion processes)for
the evolution of the spot price and the implied volatilities and address the diffi-
cult mathematical problem of determining conditions for the absence of arbitrage
opportunities, we consider a more explicit but less general class of models for the
conditional density process ft t∈[0,T]. A more general problem is investigated in
{ }
[5] and [13], where characterizations are provided of arbitrage-free dynamics for
markets with call options available for all strikes and all maturities. Conditional
density models, which are studied in this paper, are also studied in [10], where
the authors characterize the “volatility processes” {σtf(x)}t∈[0,T] in the stochastic
exponential representation
t
f (x)=f (x)+ σf(x)f (x)dV
t 0 Z s s s
0
thatgenerateproperconditionaldensityprocesses. Incontrast,wetakeaparticular
model for ft t∈[0,T] as the starting point whereas in [10] the conditional density
{ }
model is implied from the model for {σtf(x)}t∈[0,T].
In [4], [8], [7], and [14] the authors consider a setting with a finite number of
tradedoptionsforafinitesetofmaturitiesononeunderlyingasset,andcharacterize
absence of arbitrage in this setting. Both static arbitrage and arbitrage when
dynamic trading in the options is allowed are considered. In [4], [7], and [14]
explicit Markov martingales are constructed that give perfect initial calibration to
the observed option prices.
The outline of this paper is as follows. In Section 2 we consider a rather naive
model for f , a distribution of S that reproduces the given option prices, and
0 T
present a straightforward calibration procedure for the model parameters. The
model is the starting point for the conditional density model for ft t∈[0,T] that is
{ }
presentedinSection3. The theoreticalpropertiesofthe modelandadiscussionon
4 H.HULT,F.LINDSKOG,ANDJ.NYKVIST
howthemodelcanbesetuptomeetthenaturalrequirementsforagoodderivative
pricing model are also included in Section 3. Section 4 contains further theoretical
andnumericalinvestigationsofthe propertiesoftheconditionaldensitymodeland
it is evaluated on S&P 500 index option data and through simulation studies.
Our contributions can be summarized as follows. We propose a simple model
for the evolution of the forward density. At each time the forward density is a
mixture of lognormal distributions which makes it easy to make the initial cali-
bration of its parameters and price European type derivatives. On a market with
n liquidly traded call options and a forward contract, the model is driven by an
(n+1)-dimensionalBrownianmotion,makingitflexibleenoughtocapturerealized
option price fluctuations in a satisfactory way and avoids the need for frequent re-
calibration. The model is set up so that the filtration generated by the n+1 price
processes is essentially, see Section 3.1 for details, equal to the (n+1)-dimensional
Brownian filtration. Moreover, the model can easily be set up to capture stylized
features ofoptionprices, suchas a negativecorrelationbetweenchangesinthe for-
ward price and changes in implied volatility. A simulation study and an empirical
study of call options on the S&P 500 index illustrates that the model provides a
good fit to option data.
2. The spot price at maturity
We start by investigating a very simple model for S , which will be refined
T
later, that reproduces the n+1 observed forward prices. The random variable S
T
is assumed to be discrete and takes one of the values 0 x < < x < .
1 n+2
≤ ··· ∞
Let pk be the forward probability of the event S = x . The initial calibration
0 { T k}
requires solving a linear system of equations of the form Ap = b, where p is the
vector of forward probabilities of the events S =x :
T k
{ }
(3)
1 1 ... 1 1
p1
x x ... x 0 G0
1 2 n+2 p2 0
(x K ) (x K ) ... (x K ) 0 G1
... 1− 1 + 2− 1 + n+2− 1 + ... = ... 0 .
pn+2
(x K ) (x K ) ... (x K ) 0 Gn
1− n + 2− n + n+2− n + 0
If further x2 K1, xn+2 > Kn, and xk (Kk−2,Kk−1] for k = 3,...,n + 1,
≤ ∈
then the matrix on the left-hand side in (3) is one row operation away from an
invertible triangular matrix. In particular, the matrix equation Ap = b can be
solved explicitly for p by backwardsubstitution and then it only remains to verify
that p is a probability vector. In order to ensure the existence of a probability
vector solving (3) it must be assumed that
Gj−1 Gj
(4) 0 − 0 [0,1], j 1,
Kj Kj−1 ∈ ≥
−
where we set K =0, and
0
(5) Gj−1 Kj+1−Kj−1Gj + Kj −Kj−1Gj+1 0, j 1.
0 − K K 0 K K 0 ≥ ≥
j+1 j j+1 j
− −
The conditions (4) and (5) were considered in [6] and ensure that the market of
linear combinations of forward contracts together with a linear pricing rule is free
of static arbitrage opportunities. The following result, which is proved at the end
of the paper, is used as a starting point in the initial calibration of the model for
the forward price processes presented in Section 3. The result gives (necessary
and) sufficient conditions for the existence of a discrete distribution of S that is
T
A MODEL FOR THE FORWARD DENSITY PROCESS 5
consistent with the forward prices on S . The statement of Proposition 1 below is
T
a slight generalization of Proposition 3.1 in [4].
Proposition1. Supposethatthenon-negativeforwardpricesG0andG1,...,Gn
0 0 0
on the values S and (S K ) , for j = 1,...,n, at time T > 0, are ordered so
T T j +
−
that K1 < <Kn and satisfy (4) and (5). If xk =Kk−1 for k =2,...,n+1,
···
G0(K K )+G2K G1K
(6) x 0 2− 1 0 1− 0 2, and
1 ≤ (K K ) (G1 G2)
2− 1 − 0− 0
(7) x G0n−1Kn−Gn0Kn−1,
n+2 ≥ Gn−1 Gn
0 − 0
then there exist a unique probability vector (p ,...,p ) such that
1 n+2
n+2 n+2
G00 = pkxk and Gj0 = pk(Kk−1−Kj) for j =1,...,n.
X X
k=1 k=j+2
The p s are given by
k
K +G1 G0
p = 1 0− 0,
1
K x
1 1
−
x [G1 G2 (K K )]+G0(K K ) G1K +G2K
p = 1 0− 0− 2− 1 0 2− 1 − 0 2 0 1,
2
(K x )(K K )
1 1 2 1
− −
(8) p = G0k−2 G0k−1(Kk−Kk−2) + Gk0 ,
k
Kk−1 Kk−2 − (Kk−1 Kk−2)(Kk Kk−1) Kk Kk−1
− − − −
for k =3,...,n,
p = G0n−1 Gn0(xn+2−Kn−1) ,
n+1
Kn Kn−1 − (Kn Kn−1)(xn+2 Kn)
− − −
Gn
p = 0 .
n+2
x K
n+2 n
−
Remark 1. Notice that (4) and (6) imply that x < K and that (7) implies
1 1
that x > K . Notice also that Proposition 1 says that there exist indicators
n+2 n
I 0,1 satisfying I + +I =1 and p =E[I ] such that
k 1 n+2 k k
∈{ } ···
G0 =E I x and Gj =E I x K for j =1,...,n.
0 hX k ki 0 h(cid:16)X k k− j(cid:17)+i
k k
The conditions (4)-(7) are sharp: it can be seen from the proof that if any of them
is violated, then the conclusion of Proposition 1 does not hold.
Although the model in Proposition 1 for S under the forward probability
T
provides explicit expressions for the model parameters in terms of the prices and
reproduces any set of observedprices satisfying (4) and (5) it is not a good model.
If we want to use the model for pricing new derivative contracts, then we should
feel uncomfortable with having a finite grid of points as the only possible values
for S . For instance, the contract that pays 1 if S takes a value other than one
T T
of the grid points would be assigned a zero price and this would be viewed as an
arbitrage opportunity by most (all) market participants.
A simple extension is to model S as the random variable
T
n+2
(9) S = I x Z
T k k k
X
k=1
which corresponds to replacing the fixed values x ,...,x by random values
1 n+2
x Z ,...,x Z for some suitably chosen random variables Z ,...,Z that
1 1 n+2 n+2 1 n+2
6 H.HULT,F.LINDSKOG,ANDJ.NYKVIST
are independent of I ,...,I . Take
1 n+2
σ2
(10) Z =exp kT +σ B ,
k k T
n− 2 o
B
whereB isN(0,T)-distributed. ThenE[(x Z K ) ]=G (x ,σ ,K ,T),where
T k k j + k k j
−
B
(11) G (x,σ,K,T)=xΦ(d ) KΦ(d ),
1 2
−
log(x/K) σ√T
d = + , d =d σ√T,
1 2 1
σ√T 2 −
is Black’sformulafor the forwardprice ofaEuropeancalloptionmaturingattime
T, where x is the forwardprice of S , σ is the volatility, and K is the strike price.
T
The initial calibration problem for the modified model amounts to finding a
probabilityvectorp solvingthe linearequationAp=b,whereAis a squarematrix
with n+2 rows and columns with
B
A1,k =1 for all k and Aj,k =G (xk,σk,Kj−2,T) for j 2 and all k,
≥
and where b = (1,G0,...,Gn)T. The solution p to Ap = b can, as before, be
0 0
expressedasp=A−1baslongaswespecify the x sandσ ssothatAisinvertible.
k k
In general A will not be close to a diagonal matrix and therefore p= A−1b has to
becomputednumerically. Noticethatforavectorbofinternallyconsistentforward
pricesandaninvertiblematrixAwemayfindthatA−1bhasnegativecomponents.
Inthatcasethepricevectorbisoutsidetherangeofpricevectorsthatthemodelcan
generate. Fortunatelyitisnothardtodeterminetherangeofforwardpricevectors
that the model can produce. The simplex = p Rn+2 :p 0,1Tp=1 , where
S { ∈ ≥ }
1T = (1,...,1), is a convex set and a linear transformation A of a convex set is a
convexset. Moreover,theextremepointsof aremappedtotheextremepointsof
S
A . Thereforeitissufficienttodeterminethe points b =Ae fork =1,...,n+2,
k k
S
wheree isthekthbasisvectorinthe standardbasisforRn+2,andinvestigatethe
k
convex hull of b ,...,b . This is the set of price vectors that the model can
1 n+2
{ }
produce.
3. The forward price processes
A choice of the initial forward distribution, F (x) = P(S x) and forward
0 T
density f (x) = F′(x) has been proposed implicitly from (9) a≤nd (10). In this
0 0
section, the evolution of the forward distribution and density will be treated as a
stochastic process {ft}t∈[0,T], where Ft(x) = P(ST ≤ x | Ft) and ft(x) = Ft′(x).
Thefiltration t t∈[0,T]istakentobegeneratedbyan(n+1)-dimensionalstandard
{F }
Brownianmotion. Then-dimensionalBrownianmotioncorrespondingtothefirstn
components is denotedby W (Wk =Vk for k =1,...,n)and is used to modelthe
indicatorsI ,...,I , whereasthe 1-dimensionalBrownianmotioncorresponding
1 n+2
to the last component is denoted by B (B = Vn+1) and is used to model the
variables Z ,...,Z as in (10).
1 n+2
The forwardprice attime t ofa derivativecontractonS withpayofffunction
T
g is given by
n+2
E[g(S ) ]=E g I x Z
T t k k k t
|F h (cid:16)X (cid:17)|F i
k=1
n+2
= P(I =1 )E[g(x Z ) ].
k t k k t
|F |F
X
k=1
A MODEL FOR THE FORWARD DENSITY PROCESS 7
We consider a partition D ,...,D of Rn and set I = I W D . The
1 n+2 k T k
{ } { ∈ }
factors P(I =1 ) and E[g(x Z ) ] can be computed as follows:
k t k k t
|F |F
P(I =1 )=P(W D W ),
k t T k t
|F ∈ |
σ2
E[g(x Z ) ]=E g x exp kT +σ B B .
k k t k k T t
|F h (cid:16) n− 2 o(cid:17)| i
We write
D W
pk =P(I =1 )=P(W +W W D W )=Φ k− t ,
t k |Ft t T − t ∈ k | t n(cid:16) √T t (cid:17)
−
where Φ is the standard Gaussian distribution in Rn. Note that the stochastic
n
process {pt}t∈[0,T], where pt = (p1t,...,pnt+2), is a martingale on the simplex S =
p Rn+2 : p 0,1Tp = 1 with the property that p e ,...,e , where
T 1 n+2
{ ∈ ≥ } ∈ { }
the e s arethe basisvectorsofthe standardEuclideanbasisinRn+2. The forward
k
prices at time t are given by
B
G0 = pkxk and Gj = pkG (xk,σ ,K ,T t) for j =1,...,n,
t t t t t t k j −
X X
k k
B
where G denotes Black’s formula (11) for the forward price of a European call
option and
σ2 σ2
xk =x E exp kT +σ B B =x exp kt+σ B .
t k h n− 2 k To| ti k n− 2 k to
3.1. TrackingtheBrownianparticleincontinuoustime. Inordertouse
the model at time t (0,T) for pricing a Europeanderivative with payofffunction
∈
g, it is necessary to know the location of the Brownian particle (W1,...,Wn,B ).
t t t
That is, given the observed forward prices (G0,...,Gn) we need to infer the lo-
t t
cation of (W1,...,Wn,B ). We may express (G0,G1,...,Gn) as the value of a
t t t t t t
function ht evaluated at (Wt1,...,Wtn,Bt). The filtration {Gt}t∈[0,T] generated by
the vector (G0,G1,...,Gn) of price processes is therefore smaller than or equal to
the Brownian filtration t t∈[0,T] generated by (W1,...,Wn,B). We now inves-
{F }
tigatethe functions h inordertocomparethe twofiltrationsandtodeterminethe
t
dynamics of the price processes.
The mixture probabilities can be written as pk =pk(W ), where
t t t
(x w)T(x w)
(12) pk(w)= (2π(T t))−n/2exp − − dx.
t ZDk − n− 2(T −t) o
Write h = (h0,...,hn). Then the forward price G0 can be expressed as G0 =
t t t t t
h0(W ,B ), where
t t t
n+2 σ2
h0(w,b)= pk(w)x exp kt+σ b .
t X t k n− 2 k o
k=1
Similarly, Gj =hj(W ,B ), j =1,...,n, where
t t t t
n+2 B σ2
hj(w,b)= pk(w)G x exp kt+σ b ,σ ,K ,T t
t X t (cid:16) k n− 2 k o k j − (cid:17)
k=1
n+2 σ2
= pk(w) x exp kt+σ b Φ(d ) K Φ(d )
X t (cid:16) k n− 2 k o 1 − j 2 (cid:17)
k=1
8 H.HULT,F.LINDSKOG,ANDJ.NYKVIST
with d =d (j,k,b,T t) and d =d (j,k,b,T t) given by
1 1 2 2
− −
1 σ2 1
d = kt+σ b+log(x /K ) + σ √T t,
1 k k j k
σk√T t(cid:16)− 2 (cid:17) 2 −
−
d =d σ √T t.
2 1 k
− −
In particular,
(13) (G0,G1,...,Gn)=h (W ,B )
t t t t t t
=(h0(W ,B ),h1(W ,B ),...,hn(W ,B )).
t t t t t t t t t
If, for every t [0,T), h : Rn+1 Rn+1 is locally one-to-one everywhere, then
t
∈ →
an (n+1)-dimensional trajectory for the forward prices can be transformed into a
unique(n+1)-dimensionaltrajectoryforthe(n+1)-dimensionalstandardBrownian
motion(W,B). Fromtheinversefunctiontheorem(Theorem9.24in[16])weknow
that if the Jacobian matrix
∂h0t(w,b) ... ∂h0t (w,b) ∂h0t(w,b)
(14) h′(w,b)= ..∂w1 ∂wn ∂b ..
t . .
∂hnt (w,b) ... ∂hnt (w,b) ∂hnt (w,b)
∂w1 ∂wn ∂b
of the continuously differentiable function h is invertible at the point (w,b), then
t
h is one-to-one in a neighborhood of (w,b) and has a continuously differentiable
t
inverse in a neighborhood of h (w,b). The set
t
Γ = (w,b) Rn+1 :deth′(w,b)=0
t { ∈ t }
is the subset of Rn+1 where h is not locally one-to-one.
t
In order to investigate the sets Γ and in order to express the dynamics of the
t
price processes using Itoˆ’s formula the partial derivatives of the functions h must
t
be computed. We find that
∂h0 n+2 σ2
t(w,b)= pk(w)σ x exp kt+σ b
∂b X t k k n− 2 k o
k=1
and
∂hj n+2 σ2
t(w,b)= pk(w)Φ(d )σ x exp kt+σ b ,
∂b X t 1 k k n− 2 k o
k=1
where d =d (j,k) depends on j and k through K and σ . Similarly,
1 1 j k
∂h0 n+2 ∂pk σ2
t(w,b)= t (w)x exp kt+σ b
k k
∂wi X ∂wi n− 2 o
k=1
and
∂hj n+2 ∂pk B σ2
t(w,b)= t (w)G x exp kt+σ b ,σ ,K ,T t .
k k k j
∂wi X ∂wi (cid:16) n− 2 o − (cid:17)
k=1
Finally,
∂pk (x w ) (x w)T(x w)
t (w)= (2π(T t))−n/2 i− i exp − − dx.
∂wi ZDk − T −t n− 2(T −t) o
Numericalinvestigations,illustratedinFigure1,indicatethatΓ isasmoothsurface
t
of dimension n that varies continuously with t. If the function deth′ : Rn+1 R
has a nonzero gradient almost everywhere in Γ = (w,b) Rn+1 t: deth′ =→0 ,
t { ∈ t }
then the implicit function theorem(Theorem9.28in [16]) implies that Γ is a con-
t
tinuously differentiable hypersurface in Rn+1. Similarly, if the function (t,w,b)
7→
A MODEL FOR THE FORWARD DENSITY PROCESS 9
deth′(w,b) has a nonzero gradient almost everywhere in Γ = (t,w,b) Rn+2 :
deth′t =0 ,thenΓisacontinuouslydifferentiablehypersurfacein{Rn+2fr∈omwhich
t }
we conclude that the Γ s vary continuously with t. If the gradients are nonzero al-
t
most everywhere, then we conclude that P((W ,B ) Γ ) = 0 for all t but that
t t t
∈
P((W ,B ) Γ for some t)>0. In particular, if
t t t
∈
τ =inf t>0:(W ,B ) Γ ,
t t t
{ ∈ }
thefirsttimethatthe(n+1)-dimensionalBrownianmotion(W,B)arrivesatapoint
wherehτ isnotlocallyinvertible,thenthetrajectoryof (Wt,Bt) t∈[0,τ]isuniquely
{ }
determinedby thetrajectoryof ht(Wt,Bt) t∈[0,τ]. Therefore,τ isastoppingtime
{ }
with respect to t t∈[0,T] and t∧τ = t∧τ. However, whether the trajectory of
{G } G F
(Wt,Bt) t∈[0,T] is uniquely determined by the trajectory of ht(Wt,Bt) t∈[0,T] or
{ } { }
not depends on the function h in a neighborhood of (W ,B ) Γ .
τ τ τ τ
∈
In practice, only discrete observations of the forward prices are available, so it
will be impossible to track the Brownian motion exactly based on the discretely
observed forward prices. This issue is treated in some detail in Section 4 where
bothalocallinearapproximationandaparticlefilteringmethodisappliedtotrack
the location of the Brownian particle.
3 3
4000 10000
2 3500 2 8000
3000
1 1 6000
2500
4000
0 2000 0
1500 2000
−1 1000 −1 0
500 −2000
−2 0 −2
−4000
−500
−−33 −2 −1 0 1 2 3 −−33 −2 −1 0 1 2 3 −6000
3 x 104
8
2 6
4
1
2
0 0
−2
−1
−4
−2 −6
−8
−−33 −2 −1 0 1 2 3
Figure 1. Contour plots of the Jacobian determinant deth′, for
t
n=2 and b=0, as a function of (w ,w ) at times t =0, t =0.5,
1 2
and t=0.9. The zeros of the determinant are displayed along the
dotted curves. The functions h correspond to a forward density
t
process calibrated to S&P 500 option data presented in Section 4
and parameterized as in (18).
3.2. The forward price dynamics. Many popular models for derivative
pricingarebasedonmodelingthedynamicsoftheunderlyingspotpriceorforward
price directly. Examples are Black’s model, Dupire’s model, and stochastic volatil-
ity models. Our starting point is a model for the dynamics of the forward density.
Fromthe model for the forwarddensity process,the dynamics of the forwardprice
10 H.HULT,F.LINDSKOG,ANDJ.NYKVIST
process {G0t}t∈[0,T] are derived from the expressions for the partial derivatives of
h and Itoˆ’s formula (Theorem 33, p. 81, in [15]):
t
n t ∂h0 t ∂h0
G0 =G0+ s(W ,B )dWi+ s(W ,B )dB
t 0 Z ∂w s s s Z ∂b s s s
Xi=1 0 i 0
1 t n ∂2h0 ∂h0 ∂2h0
+ s(W ,B )+2 s(W ,B )+ s(W ,B ) ds
2Z0 (cid:16)Xi=1 ∂wi2 s s ∂s s s ∂b2 s s (cid:17)
n n+2 t ∂pk σ2
=G0+ s(W )x exp ks+σ B dWi
0 Xi=1Xk=1Z0 ∂wi s k n− 2 k so s
n+2 t σ2
+ pk(W )σ x exp ks+σ B dB
Xk=1Z0 s s k k n− 2 k so s
n+2 t 1 n ∂2pk ∂pk σ2
+ s(W )+ s(W ) x exp ks+σ B ds.
Xk=1Z0 (cid:16)2Xi=1 ∂wi2 s ∂s s (cid:17) k n− 2 k so
From e.g. the martingale representation theorem (Theorem 43, p. 186, in [15]) it
follows that the last sum above vanishes so that
n n+2 t ∂pk σ2
G0 =G0+ s(W )x exp ks+σ B dWi
t 0 Xi=1Xk=1Z0 ∂wi s k n− 2 k so s
n+2 t σ2
+ pk(W )σ x exp ks+σ B dB .
Xk=1Z0 s s k k n− 2 k so s
The derivatives computed so far can also be used to study the conditional
density process ft(x) t∈[0,T]. The conditional density
{ }
n+2
f (x)= pkfk(x)
t t t
X
k=1
isaconvexcombination,withrandomprobabilityweightspk asabove,oflognormal
t
densities fk(x), where
t
1 1 log(x/x )+σ2T/2 σ B 2
fk(x)= exp k k − k t .
t xσk 2π(T −t) n− 2(cid:16) σk√T −t (cid:17) o
p
Itoˆ’s formula and the martingale representation theorem yield, where the depen-
dence of f (x) on W through the pks and on B through the fks has been sup-
t t t t t
pressed,
n t ∂f (x) t ∂f (x)
f (x)=f (x)+ s dWi+ s dB
t 0 Z ∂w s Z ∂b s
Xi=1 0 i 0
1 t n ∂2f (x) ∂f (x) ∂2f (x)
s s s
+ +2 + ds
2Z0 (cid:16)Xi=1 ∂wi2 ∂s ∂b2 (cid:17)
n n+2 t ∂pk
=f (x)+ sfk(x)dWi
0 Z ∂w s s
Xi=1kX=1 0 i
n+2 t log(x/x )+σ2T/2 σ B
+ pkfk(x) k k − k s dB .
Xk=1Z0 s s (cid:16) σk(T −s) (cid:17) s
