Download Consistent Variance Curve Models

Consistent Variance Curve Models
Hans Buehler∗
Deutsche Bank AG London
Global Equities
1 Great Winchester Street
London EC2N 2EQ, United Kingdom
e-mail: [email protected]
Institut für Mathematik, MA 7-4
TU Berlin, Strasse des 17. Juni 136
10623 Berlin, Germany
e-mail: [email protected]
Version 3.5 June 7 2005 first draft June 2004
Abstract
We introduce a general approach to model a joint market of stock price
and a term structure of variance swaps in an HJM-type framework. In
such a model, volatility-dependent contracts in particular can be priced
and risk-managed in terms of the observed stock and variance swap prices.
To this end, we introduce equity forward variance term-structure models and derive the respective HJM-type arbitrage conditions. We then
discuss finite-dimensional Markovian representations of the fixed time-tomaturity forward variance swap curve and derive consistency results.
We then give a few examples of such variance curve functionals and
discuss briefly hedging in such models. As a further application, we show
that the speed of mean-reversion in some standard stochastic volatility
models should be kept constant when the model is recalibrated.
1
Introduction
Standard financial equity market models model exclusively the price process of
the stock prices. The prices of liquid derivatives, like plain vanilla calls, are
only used to calibrate the parameters of the model. A more natural approach
∗ Supported by the DFG Research Center Matheon “Mathematics for key technologies”
(FZT 86)
1
would be to model the evolution of stock price and some liquid instruments
simultaneously.
Various approaches, such as those by Brace et al. [BGKW01], Cont et al. [CFD02],
Fengler et al. [FHM03], or Haffner [H04], take on the problem by modeling the
stochastic evolution of implied volatility surfaces (and therefore European call
prices) according to certain stylized facts or statistical observations. The price
processes obtained from these so-called “stochastic implied volatility models”,
however, are not guaranteed to remain martingales. Schönbucher [S99] and
Schweizer/Wissel [SW05], on the other hand, focus on the term structure of
implied volatility for a single fixed strike, which is too restrictive for the applications we have in mind.
In this article, we consider variance swaps as liquid derivatives and derive
conditions such that the joint market of stock price and variance swap prices
is free of arbitrage. Such models can then be used to price exotic options and
allow the computation of hedges with respect to stock and variance swaps in
a consistent way. This approach is therefore particular suitable for options on
realized variance and other strongly volatility dependent products.
Variance Swap Markets
For the world’s equity stock indices, a fairly liquid market of variance swaps
has evolved in recent years. Given an index S, such a variance swap exchanges
the payment of realized variance of the log-returns against a previously agreed
strike price. The (zero mean) annualized realized variance for the period [0, T ]
with business days 0 = t0 < . . . < tn = T is usually defined as
µ
¶2
d X
S ti
log
,
n i=1,...,n
Sti−1
but contracts may vary. The constant d denotes the number of trading days per
year. A standard result (e.g. Protter [P04], p. 66) gives that
¶2
X µ
S ti
hlog SiT = lim
log
m↑∞
Sti−1
t ∈τ
i
m
m
where the limit is taken over refining subdivisions τm = (0 = tm
0 < · · · < tm =
T ), that is limm↑∞ supi=1,...,m |ti − ti−1 | = 0. We can focus without loss of
generality on variance swaps with a zero strike price.
We will assume in this article that the realized variance paid by a variance
swap is the realized quadratic variation of the logarithm of the index price, i.e.
hlog ST i. We also assume that the stock price process is continuous. In this
case, Neuberger [N92] has shown that the price of a variance swap is given in
terms of liquid option prices, if they are available (see also section 2.1).
Our starting point will be to assume that variance swaps are liquidly traded
for all maturities.
2
The aim of this article is to first model the variance swap prices and then
find an associated stock price process with compatible dynamics:
Problem (P1)
Given today’s variance swap prices V0 (T ) for all maturities T ∈ [0, ∞), we
want to model the price processes V (T ) = (Vt (T ))t∈[0,∞) and the stock price S
together, such that the joint market with all variance swaps and the index price
itself is free of arbitrage.
Apart from the additional presence of the stock price, this resembles closely
the situation in Heath-Jarrow-Morton (HJM) interest rate theory where the aim
is to construct arbitrage-free price processes of zero bonds (see [HJM92]). We
carry this similarity further and introduce the forward variance curve (v(T ))T ≥0
of the log-returns of S, defined as
vt (T ) := ∂T Vt (T )
T, t ≥ 0
on some stochastic base W := (Ω, F∞ , P, F) which supports an extremal Brownian motion W .1
We then have a HJM-type result, namely that (under the assumptions of
the next section) v(T ) must be a martingale for each T and therefore has no
drift. This will be carried out in section 2, where we will also introduce the
“Musiela-parametrization” v̂ of v in terms of a fixed time-to-maturity x,
v̂t (x) := vt (x + t)
x, t ≥ 0 .
It is then also shown in theorem 2.2 that for all Brownian motions B on W
the market of all variance swaps (V (T ))T ∈[0,∞) and the B-“associated price
process” S, defined by
)
St := Et (X)
p
dXt := v̂t (0) dBt
is free of arbitrage. In such a case we call the curve v a variance curve model,
and B has the intuitive meaning of a “correlation structure”. We want to emphasize that these no-arbitrage-conditions are very straight forward to enforce,
in remarkable contrast to the severe difficulties in this respect with the aforementioned “stochastic implied volatility models”.
Remark 1.1 We want to stress that we do not attempt to develop a model to
price variance swaps – on the contrary, we assume that their market prices are
given. Instead, we want to make us of this information to construct a market
model of variance; see also section 2.1.3.
1 For
details and a precise setup, please refer to section 2.2.
3
Finite-Dimensional Realizations
In practise we are interested in forward variance curves which are given as a
functional of a finite-dimensional Markov-process: We aim to represent v̂ as
v̂t (x) = G(Zt ; x)
(1)
where G : Z × R0+ → R+
0 is a suitable positive function and where Z is an
Rm -valued Markov process which is a strong solution to an SDE
dZt = µ(t; Zt )dt +
n
X
σ j (t; Zt )dWtj
(2)
j=1
defined in terms of the n-dimensional standard Brownian motion W . A pair
(G, Z) is called consistent iff (1) defines a variance curve model. This leads to
the natural question:
Problem (P2)
When are a parameter process Z and a functional G consistent?
This will be addressed in section 2.3, and we will show in theorem 2.3 that
consistency is essentially implies
1
∂x G(Zt ; x) = µ(t; Zt ) ∂z G(Zt ; x) + σ 2 (t; Zt ) ∂zz G(Zt ) ,
2
and that a martingale-solution Z to this equation is a variance curve model.
Theorem 2.4 presents conditions on when Z will stay in the domain Z of G:
To this end we will assume that Z is a sub-manifold with boundary. Denoting
by Tz Z the tangent space of Z in z, we will show that Z stays in Z at least
locally (i.e. up to a strictly positive stopping time) if
)
µ(t; z) − 12 σz0 σ(t; z) ∈ Tz Z
(3)
σ(t; z) ∈ Tz Z
holds for all z ∈ intZ and t ≥ 0. Moreover, if Z is closed in the relative topology and a similar condition as (3) holds on its boundary, then the solution stays
globally in Z.2
These results are closely related to the concept of “finite-dimensional realizations” (FDR), as introduced by Björk/Christensen [BC99] and Björk/Svensson
[BS01] for HJM interest rate models: We say a variance curve model v̂ admits
an FDR, if for every z ∈ Z there exists a consistent pair (G, Z) such that
v̂t (·) = G(Zt ; ·) up to a strictly positive stopping time. Note that we now understand v̂t (·) and G(Zt ; ·) as functions, and therefore omit the argument x.
2 Refer
to section 2.3 for notation and precise definitions.
4
Problem (P3)
Given a family v̂ and a smooth functional G, when will v̂ admit an FDR in
terms of G?
Under the condition that G := { G(z) z ∈ Z } is a smooth manifold in a suitable
Hilbert space H, we will solve this question in section 3 by following closely
ideas by Björk/Christensen [BC99]: writing v̂ as a solution to an H-valued SDE
dv̂t = ∂x v̂t dt +
n
X
β̂tj (v̂t ) dWtj
(4)
j=1
we show in theorem 3.1 that it admits an FDR in terms of G if and only if
)
¡
¢
∂x G(z) − 12 β̂v0 β̂ G(z) ∈ Im [∂z G(z)]
¡
¢
β̂ G(z) ∈ Im [∂z G(z)]
In that case, we can also explicitly determine the diffusion-coefficients of Z in
terms of β̂ and G.
1.1
Applications
In the second part of the article, we will discuss examples (section 4): Following our conjecture above, we will particularly focus on (affine) exponentialpolynomial functionals G, a very well-known example of which is the forward
variance curve
G(z; x) = z2 + (z1 − z2 )e−z3 x
3
(z1 , z2 , z3 ) ∈ R+ .
Similar to Filipovic [F01] for interest rates, we find that in general the coefficients in the exponentials (ie, z3 above) must be constant if the curve is to
produce arbitrage-free variance swap prices.
In a brief discussion in section 4.3, we then explain how variance curve
models can be used to obtain hedges of payoffs in terms of stock price and
traded variance swaps.
Finally, we use our previous results and apply them to the standard market
practise of recalibration of various models. Taking Heston’s popular model [H93]
as an example, we show that mean-reversion should be kept constant during the
life of an exotic to ensure that its price process is a local martingale in the real
world of the institution 3 where it is the result of frequent recalibration.
1.2
Structure of the article
To gain some intuition, we will first review the reverse case if the stock price
process is given as a continuous martingale.
3 See
section 4.4 for details and terminology.
5
This will give us some heuristics upon which we build our model framework
in section 2.2: The introduction of variance swap term-structure, the definition
of the forward variance curve and the derivation of the corresponding HJMtype conditions, both for the general case and (in section 2.3) for the case of a
consistent pair (G, Z), thereby solving problems (P1) and (P2) above.
In section 3 we then proceed to solve (P3) by following [BC99] to derive the
conditions when a family v can be parameterized via a functional G.
The examples mentioned above will be presented in section 4 and we also
discuss how exotic options can be hedged in terms of stock price and variance
swaps in section 4.3. In the last section 4.4, we show that mean-reversion in
Heston-type models must be kept constant during recalibration.
We conclude in section five.
2
Variance Curve Models
We will assume a given probability space W = (Ω, F∞ , P, F) with an n-dimensional
Brownian motion W = (W j )j=1,...,n on which we want to model a market of
variance swaps and an index stock price process, which pays no dividends. We
also assume that the prevailing interest rates are zero and set the initial stock
price level to S0 = 1.
Our notations will mostly follow Revuz/Yor [RY99] and da Prato/Zabzyck
[PZ92]: We denote by HT2 the space of all square-integrable martingales up to
T , and by L2T the space of all predicable processes ϕ which are integrable by
RT
a Brownian motion, i.e. E[ 0 ||ϕs ||2 ds] < ∞ (the dimension of the Brownian
motion will be clear from the context).
The symbols HTloc and Lloc
T denote the respective local spaces, and we may
drop the T in case T = ∞.
We denote by B(U ) the Borel-σ-algebra of a topological space U and by P
the predictable σ-algebra on Ω × R+
0 . Given a process Z = (Zt )t≥0 , we define
its image measure by PZ [A] := P[Z ∈ A] where A ∈ B[C[0, ∞)]. We also denote
by E(X) the Dolean-Dade exponential of a process X.
We will use the symbols x∨y := max(x, y), x∧y := min(x, y) and x+ := x∨0.
The set of positive numbers x > 0 is denoted by R+ , while R+
0 is the set of nonnegative numbers.
Assumption 1 For ease of exposure we will assume that W is extremal on W,
i.e. that we can write each local martingale M as a stochastic integral over W .
2.1
Review of the Standard Case
For illustration, we assume in this subsection that we are given a continuous
stock price process as a positive local martingale S on W, such that its variance
swap prices are finite. This is the reverse case to the rest of the article, where
the variance curves will be given and S will be constructed accordingly.
6
This subsection is intended to build some understanding of the required
properties of a variance swap model, but from a logical point of view it can be
omitted and the reader may immediately proceed to section 2.2.
Proposition 2.1 If S is a positive local martingale, we can write it as
St = Et (X)
where
Xt =
(5)
Z tp
ζs dBs
(6)
0
√
for some ζs ∈ Lloc and a Brownian motion B.
We also have that X ∈ HT2 for all T < ∞.
Proof – Since S is a positive local martingale, we can write St = Et (X) for
some local martingale X (cf. [RY99] pg. 328, prop. 1.6). Following the arguments
Rt
of [RY99] pg. 203, let hXit = 0 ζs ds, which is a valid integrand for X since
R t −1
Rt
E[ 0ζs 1 ζs >0 dhXis ] = E[ 01 ζs >0 ds] ≤ t < ∞. Therefore, we can define
Z t
Z t
1
√ 1 ζs >0 dXt + 1 ζs =0 dWs1 .
Bt :=
ζs
0
0
¢
R t ¡ −2 2
We have hBit = 0 ζs ζs 1 ζs >0 + 1 ζs =0 ds = t, and since B is clearly adapted
and continuous, it is a Brownian motion with the required property (6).4
By assumption that the variance swap prices are finite, we also have that
RT
E[ 0 ζs ds] < ∞, ie X ∈ HT2 for all finite T .
¤
Hence, any positive continuous stock price process can be written as a
“stochastic volatility model”
p
dSt
= ζt dBt .
St
Note, however, that the joint process (S, ζ) will generally not be Markov.
2.1.1
Variance Swaps
Since variance swaps are assumed to be tradable at any time t, the time-t price
Vt (T ) is given as the expectation of the quadratic variation of log S under a
pricing measure P:
"Z
#
T
¯ ¤
¯
£
Vt (T ) = EP hlog SiT ¯ Ft = EP
ζs ds ¯ Ft .
(7)
0
It is clear from standard arbitrage-theory that the measure P is in general not
unique, i.e. the price processes V (T ) of the variance swaps with maturities T ≥ 0
are in general not determined by specifying the stock price process (S, ζ) alone.
4 Note
that we did not need the assumption that W is extremal.
7
It is therefore necessary to fix a pricing measure, which we will assume for
the remainder of this subsection is P. We want to stress that by constructing
directly the variance swap prices (which is the subject of this article), the prices
of variance swaps are given by the market and any pricing measure must have
property (7).
Given that V (T ) is a martingale and due to the extremality of W , we find
some b(T ) ∈ Lloc such that
m Z t
X
Vt (T ) = V0 (T ) +
bjs (T ) dWsj .
0
j=1
2.1.2
Forward Variance
We also see that at any time t, the curve Vt (·) is absolutely continuous with
respect to the Lebesgue measure λ, hence we can define the derivative along T ,
£ ¯ ¤
vt (T ) := ∂T Vt (T ) = E ζT ¯ Ft
T, t ≥ 0 .
(8)
which is called the fixed maturity T -forward variance seen at time t (vt (T ) is
well defined for T > t). Note the conceptual similarity with the forward rate in
interest rate modelling.
Proposition 2.2 (HJM-Condition for Forward Variance) For all T ≥ 0, the
process v(T ) = (vt (T ))t≥0 defined by (8) is martingale and can be written as
Z t
vt (T ) = v0 (T ) + βs (T ) dWs
(9)
0
with β(T ) := ∂T b(T ) ∈ L
2.1.3
loc 5
.
Pricing Variance Swaps using European Options
Neuberger [N92] has shown that the price of a variance swap in the present
framework can be computed as
V0 (T ) =
=
=
=
E [ hlog ST i ]
−2E [ log ST ]
·Z 1
¸
Z ∞
1
1
+
+
−2E
(K
−
S
)
dK
+
(S
−
K)
dK
T
T
2
2
0 K
1 K
Z ∞
Z 1
1
1
Put(K,T) dK − 2
Call(K,T) dK
−2
2
2
K
K
1
0
where Put(K,T) and Call(K,T) denote quoted market option prices, which need
to be available for this approach. As discussed in Buehler [B04], we usually
5 The fact that β(T ) = ∂ b(T ) follows from the uniqueness of the representation of a local
T
martingale on F with respect to W .
8
only have a discrete number of traded options on the market, but this approach
still allows to approximately price and hedge variance swaps using standard
instruments.6
We therefore do not attempt to develop a pricing scheme for variance swaps
(or forward variance swaps, for that matter) here. Rather, we can use the
market prices (quoted or computed by the above formula) as input to develop a
variance swap model which in turn can be used to price and hedge more exotic
options such as, for example (but of course not exclusively), options on realized
variance: E.g. a call on realized variance has payoff
ÃZ
!+
T
ζs ds − K
(10)
0
and a volatility swap pays out
s
Z
T
ζs ds .
0
It is natural to hedge such payoffs with the “forward” on the underlying quantity
hlog SiT , i.e. with variance swaps. Note that this is even more convincing if we
want to price options which pay out variance swaps, such as a call on a (forward
started) variance swap.
2.2
Definitions and Basic Properties
We now introduce our variance curve models: We want to specify the forward
variance price processes v(T ), which we imagine as the expected future instantaneous variance as in the previous section. This process should allows us to
construct a stock process such that the joint market of stock and variance swaps
is arbitrage-free.
Definition 2.1 (Variance Curve Model) We call a family v = (v(T ))T ≥0 a
Variance Curve Model on W if:
(a) For all T < ∞, v(T ) is a non-negative continuous martingale with representation
n
X
dvt (T ) =
βtj (T ) dWtj
(11)
j=1
for some β(T ) ∈ L
loc
(this is the “HJM-condition” for variance curves).
(b) For all T < ∞, the initial variance swap prices are finite,
Z
V0 (T ) :=
v0 (x) dx < ∞ .
0
6 For
T
details see Demeterfi et al. [DDKZ99].
9
(12)
(c) The process v· (·) is predictable (for example, if vt (·) is left-continuous).
By proposition 2.2 it is clear that forward variance must be a martingale, and
finiteness of the variance swap prices is a very natural assumption, if we want
to use them as liquid instruments. The last condition is technical and used in
definitions 2.2 below.
Proposition 2.3 The variance swap price processes V = (V (T ))T ≥0
Z
T
vt (s) ds
Vt (T ) :=
(13)
0
are martingales with dynamics dVt (T ) =
Pn
j
j
j=1 bt (T ) dWt
where bjt (T ) =
RT j
β (s) ds.
0 t
Proof – It is clear that
¯ ¤ by (13) is adapted and therefore is a martin£ V (T ) defined
gale with Vt (T ) = E VT (T ) ¯ Ft and V0 (T ) given by (12). The representation
of V via b follows easily from (11) and the uniqueness of the representation of
V (T ) with respect to W .
¤
Given a variance curve model, we call the positive process
ζt := vt (t)
(14)
the short variance of v. It is well defined by the requirements of definiton 2.1.
Since also
"Z
# Z
T
E
T
ζs ds
0
it follows that process
definition:
=
E [ vs (s) ] ds = V0 (T ) < ∞ ,
0
√
ζ is in L2T for all T < ∞. This justifies the following
Definition 2.2 (Associated Stock Price Process) For any variance curve model
v and an arbitrary real-valued Brownian motion B on W, the B-associated stock
price process is defined as the local martingale
Z tp
St := Et (X) with Xt :=
ζs dBs .
0
The process X is in L2T for all T < ∞ and a variance swap on S has the price
¯ ¤
¯ ¤
£
£
E hlog SiT ¯ Ft = E hXiT ¯ Ft = Vt (T ) ,
where V was defined in (13).
It follows then directly by construction
Theorem 2.1 (Variance Swap Market Model) Let v be a variance curve model,
B a Brownian motion and S its associated stock price proces.
Then, the joint market (S, V ) is free of arbitrage.
10
We see a very convenient property of the current model approach: Once the
variance curve model is fully specified by v0 and β, an associated stock price
process can easily be constructed to yield a full variance swap market model
which is free of arbitrage.
Remark 2.1 (Interpretation of B) Note that each B defined on the stochastic
base W can be written as
N
X
dBt =
ρjt dWtj
j=1
in terms of some stochastic “correlation vector” ρ ∈ L2 .
Since the Brownian motion B defines the “correlation structure” of S with
its variance process, it has the intuitive meaning of a “skew parameter”.
Note, however, while B can be chosen arbitrarily to yield a local martingale S,
more care must be taken if S is required to be a true martingale. For example, we can try to satisfy Kazamaki’s criterion, cf. Revuz/Yor [RY99] pg. 331, or
Novikov’s criterion, pg. 332. If the latter is satisfied then S is a true martingale
for all Brownian motions B.7
2.2.1
Fixed time-to-maturity
In the sprit of Musiela’s parametrization [M93] of forward rates, we now introduce the respective process for variance curve models:
Definition 2.3 We call
v̂t (x) := vt (t + x)
the fixed time-to-maturity forward variance, and V̂t (x) :=
time-to-maturity variance swap.
Rx
v̂ (s) ds the fixed
0 t
Note that above definition is valid for each fixed t and almost all ω. To define
a proper process v̂, we have to impose some additional regularity on v.
Proposition 2.4 Let v be a variance curve model. Assume that v0 is differentiable in T , that β in (11) is B[Rn ] × P-measurable and almost surely differentiable in T with
sZ
∞
∂T βtj (x)2 dx ∈ Lloc j = 1, . . . , n .
(15)
0
Pn R t
Then, ∂T vt (T ) is well defined as ∂T vt (T ) = ∂T v0 (T ) + j=1 0∂T βsj (T ) dWsj
and the fixed time-to-maturity forward variance v̂(x) is of the form
Z t
n Z t
X
v̂t (x) = v̂0 (x) + ∂x v̂s (x) ds +
β̂sj (x) dWsj
(16)
0
j=1
0
7 Also note recent results from Andersen et al. [AP04], who show that some commonly
mentioned stochastic volatility models do not necessarily yield true martingale stock price
processes.
11
where β̂tj (x) := βtj (t + x).
Proof – With the assumptions above, we have
v̂t (x)
=
(9)
=
=
(15)
=
vt (t + x)
Z t
v0 (t + x) + ∂T βs (u + x) dWu
0
Z t
v0 (x) + ∂T v0 (s + x) ds
0
¾
Z T½
Z t
+
βu (u + x) + ∂T βu (s + x) ds dWu
0
u
¾
Z t½
Z s
v0 (x) +
∂T v0 (s + x) + ∂T βu (s + x) dWu ds
0
Z
+
βu (u + x) dWu
0
=
0
T
Z
Z
t
v0 (x) +
0
Z
=
T
∂T vs (s + x) ds +
v̂0 (x) +
t
T
∂T v̂s (x) ds +
0
βu (u + x) dWu
0
Z
β̂u (x) dWu ,
0
as claimed. Note that (15) basically ensures that
tingale (see, for example, Protter [P04] p. 208).
Rs
∂ β (T ) dWu is a local mar0 T u
¤
The reverse of the previous proposition constitutes the HJM-condition for
the fixed time-to-maturity case: Assume we start with a family v̂, when does
vt (T ) := v̂t (T − t) define a variance curve model?
Theorem 2.2 (HJM-condition for Variance Curve Models) Let v̂ = (v̂(x))x≥0
be a family of non-negative adapted processes such that:
(a) The process v̂(x) has a representation
dv̂t (x) = ∂x v̂t (x) dt +
n
X
β̂tj (x) dWtj
(17)
j=1
Pn
β̂tj (x) dWtj is a martingale.
Rx
(b) The prices of variance swaps V̂0 (x) := 0 v̂0 (s) ds are finite for all x < ∞.
such that ẑt (x) :=
j=1
(c) The volatility coefficient β̂ in (17) satisfies
qR
∞
0
Then, the family v = (v(T ))T ∈[0,∞) given by
½
v̂t (T − t) t ≤ T
vt (T ) :=
v̂T (0)
t>T
12
∂x β̂tj (x)2 dx ∈ Lloc .
(18)
defines a variance curve model.
Proof – We have to satisfy the conditions of definition 2.1. The finiteness of
variance swap prices is satisfied by (a). Now assume v is defined by (18). As
before,
n
X
(19)
dvt (T ) = dv̂t (T − t) =
β̂tj (T − t) dWtj ,
j=1
i.e. v(T ) is a local martingale. Now note that (c) above on β̂ ensures that ∂x ẑ
is well-defined. Hence, we can compute
vT (T ) − vt (T )
(19)
=
n Z
X
j=1
=
t
n Z
X
j=1
=
T
β̂tj (T − t) dWtj
(
T
t
Z
β̂tj (T )
−
ẑT (T ) − ẑt (T ) −
∂x β̂tj (y) dy
T −t
n Z
X
j=1
=
)
T
T
dWtj
n
o
∂x ẑT (y) − ∂x ẑt (y) dy
T −t
ẑT (T − t) − ẑt (T − t) ,
so v(T ) is a true martingale since we assumed that ẑ is a true martingale. Finally, ζt := v̂t (0) is by construction well defined.
¤
This theorem allows us to specify v̂ instead of v. We will therefore also refer to
v̂ as a “variance curve model” if it satisfies the conditions of theorem 2.2.
Conclusion 2.1 Theorems 2.1 and 2.2 answer (P1) from the introduction.
Remark 2.2 Despite the introduction of forward rates in terms of fixed-timeto-maturity by Musiela, it is more common in interest-rate theory to deal with
fixed maturity objects because the maturities of underlying market instruments
are typically fixed points in time (such as LIBOR rates and Swaps).8
A variance curve, in contrast, is more naturally seen as a fixed time-tomaturity object, in particular given that the short end of the curve is the instantaneous variance of the log-price of the stock as seen in definition 2.2.
It should also be noted that an option on realized variance (such as a call
(10) for example) is not an option on a variance swap.
2.2.2
Fitting the initial term structure
Let fˆt (x) be the interest rate forward rate with time-to-maturity x observed at
some time t. An advantage of the HJM-approach for interest-rates is that it the
8 In
a typical LIBOR rate model, the short rate is not modelled.
13
current forward rate is given as
Z t³
n Z t
´
X
β̂sj (x) dWsj
fˆt (x) = fˆ0 (x) +
∂x fˆs (x) − αs (x) ds +
0
j=1
0
Rx
Pn
with HJM-drift αt (x) := j=1 β̂tj (x) 0 β̂tj (y) dy so that the initial curve fˆ0 can
be estimated from market quotes without imposing additional constraints on
the volatility structure β̂.
In contrast, our specification of v̂ must remain non-negative, which renders
specification of the volatility structure dependent on v̂0 .
In the main part of this article we will deal with finite-dimensional realizations of v̂, where this is not a concern (because we will write v̂ in terms of a
non-negative functional). However, if we were to work directly with v̂, we might
consider to parameterize it as
v̂t (x) = v̂0 (x)eŵt (x)
(20)
Proposition 2.5 Equation (20) defines a variance curve model v̂ iff it has a
representation


n
n
X
X
1
dŵt (x) = ∂x ŵt (x) −
γ̂tj (x)2  dt +
γ̂tj (x) dWtj
(21)
2 j=1
j=1
for γ ∈ Lloc such that eŵt (x) is a true martingale on [0, T ] for all finite T .
The condition that eŵt (x) is a true martingale is not trivial (see [RY99] chapter
VIII). However, if we want to allow arbitrary initial curves and be able to
chose the volatility structure independently from the chosen initial curve, this
approach can be employed. Unfortunately, it does not allow v̂t (0) to be zero,
hence models such as Heston’s are not covered in this setting.
Remark 2.3 In [D04], Dupire discusses a similar approach as in proposition 2.5
for the case of a one-dimensional Brownian motion driving the forward variance
and a second standard Brownian motion to used to obtain non-trivial correlation
with the stock price.
The article also contains details on hedging in such a framework.
Proof – [of the proposition] Let us assume that
dŵt (x) = α̂t (x) dt +
n
X
γtj (x) dWtj
j=1
Using Ito and assuming v̂0 ≡ 1, we have




n
n
X
X
1
γtj (x)2  dt
dv̂t (x) = v̂t (x) α̂t (x) dt +
γtj (x) dWtj  + v̂t (x) 
2
j=1
j=1
14
Set therefore β j := v̂γ j . Given theorem 2.2 we therefore must have
n
1 X j 2 ! ∂x vt (x)
α̂t (x) +
γ (x) =
= ∂x wt (x) ,
2 j=1 t
v̂t (x)
which, together with the martingale assumption, proves the result.
¤
Remark 2.4 Of course, we could also write
Rx
ŵ (y) dy
0 t
V̂t (x) = V̂0 (x) e
in which case we end up exactly in the interest-rate case. However, such an
expression is far less intuitive given the inherent property
Z x2
¯ ¤
£
V̂t (x2 ) − V̂t (x1 ) =
E ζt+x ¯ Ft dx
x1
for 0 ≤ x1 < x2 .
2.3
Consistent Variance Curve Functionals
In the previous section, we have discussed variance curve models which were
given in terms of general integrable processes. In practise, however, we usually want to concentrate on models which are given as functionals of a finitedimensional parameter process.
Consider the situation in the reality of a trading floor: We do not actually
see an infinite number of variance swap prices (V0 (T ))T ∈[0,∞) in the market.
Rather, a discrete set of swap prices will be interpolated by some functional
which is parameterized by a finite-dimensional parameter vector.9 We call such
functionals variance curve functionals:
Definition 2.4 (Variance Curve Functional) A Variance Curve Functional is
RT
+
a non-negative C 2,2 -function G : (z; x) ∈ Z × R+
G(z; x) dx < ∞ for
0 → R0
0
all (z, T ). The open subset Z ⊂ Rm is called the parameter space of G.
Given a functional G, we now have to find a parameter process Z = (Zt )t∈[0,∞)
such that
v̂t (x) := G(Zt ; x) , x ≥ 0 ,
forms a variance curve model. To avoid arbitrage, we need to meet the conditions of theorem 2.2:
9 In an abstract sense, an implied volatility parametrization which is then used to price
variance swaps by Neuberger’s formula is also such a functional.
15
Definition 2.5 (Consistent Parameter Process) A Consistent Parameter Process for (G, Z) is a diffusion process Z, such that for all Z0 ∈ Z we have Zt ∈ Z
and the family
v̂t (x) := G(Zt ; x) , x ≥ 0 ,
is a variance curve model.
The pair (G, Z) is the called consistent.
Naturally, we now have to ask whether a given curve functional G is consistent at all and if so, whether we can specify a particular consistent process. We
want to focus on diffusions of the type

Pn
dZt = µ(t; Zt ) dt + j=1 σ j (t; Zt ) dWtj 

µ : R+
0 × Z 7→ R
σ j : R+
0 × Z 7→ R


(22)
which admit a unique strong solution.10 The set of these processes will be
denoted by Ξ. Since any such Z is uniquely defined by the coefficients (µ, σ) we
may identify Z with (µ, σ).
We can now prove the following theorem, which is closely related to proposition 3.1.1 in [F01]. It is paves the way to answer problem (P2):
Theorem 2.3 (HJM-condition for Variance Curve Functionals) If a process Z ∈
Ξ is consistent with (G, Z), then necessarily
)
∂x G(Zt ; x) = µ(t; Zt ) ∂z G(Zt ; x) + 12 σ 2 (t; Zt ) ∂zz G(Zt ; x)
(23)
Zt ∈ Z
P ⊗ λ|[0,∞)2 -as for all Z0 ∈ Z.
Conversely, if Z ∈ Ξ solves (23) such that G(Z0 ; T ) = E [ G(ZT ; 0) ], then Z
is consistent with (G, Z).
The above equation is short-cut notation for
∂x G(Zt ; x) =
n
X
µi (t, Zt ) ∂zi G(Zt ; x)
i=1
+


m
n
X
X
1
 (σ j )i (t, Zt )(σ j )kt (Zt ) ∂zi zk G(Zt ; x) .
2
j=1
i,k=1
We wrote (σ j )i to denote the ith component of the m-dimensional vector σ j .
10 For
conditions, see for example Protter [P04] chapter V.3
16
Proof – First assume that Z is consistent with G. Then v̂t (x) := G(Zt ; x) is a
variance curve model and
µ
¶
n
X
1 2
dv̂t = µ ∂z G + σ ∂zz G dt +
σ j ∂zj G dWtj
2
j=1
proves (23) by theorem 2.2.
Now assume on the other hand that (23) holds. Using Ito and (23) it is clear
that vt (T ) := G(Zt ; T − t) is a local martingale for all T with
dvt (T ) =
n
X
βtj (T ) dWtj ,
βtj (T ) := σ j (t; Zt ) ∂zj G(Zt ; T − t) ∈ Lloc
T .
(24)
j=1
Finally, Z is a true martingale since G(Z0 ; T ) = E [ G(ZT ; 0) ].
¤
Theorem 2.3 gives us the required condition to problem (P2) when a pair (G, Z)
is consistent. However, it leaves the question open whether a process Z given
by a pair (µ, σ) leaves Z at some stage or not. This will be treated in theorem
2.4 below for the case where Z is a manifold with boundary.
But first we want to add two corollaries which allows us in many circumstances to focus on a simpler version of theorem 2.3, and that we can sometimes
focus on time-homogeneous systems only.
Definition 2.6 We call a process Z ∈ Ξ essentially diffuse if there exists a
state space D ⊂ Rm such that for almost all s ≥ 0, zs ∈ D and all t > s:
(a) The support of Zt started in Zs = zs is D.
(b) We have λm ¿ P[Zt ∈ ·|Zs = zs ] on D.
We denote by Ξ∗ the set of such processes.
Corollary 2.1 (HJM-condition as PDE) An essentially diffuse process Z ∈
Ξ∗ is consistent with G if and only if D ⊆ Z and it solves
1
∂x G(z; x) = µ(t; z) ∂z G(z; x) + σ 2 (t; z) ∂zz G(z; x)
2
¯
£
¤
+
¯
on D × R+
0 × R0 such that G(z, T ) = E G(ZT ; 0) Z0 = z .
(25)
Proof – Fix s ≥ 0 and zs ∈ Z. Since Z is essentially diffuse, let P[Zt ∈ ·|Zs =
zs ] = pt + ot where pt ≈ λm |D and ot ⊥ λm . Let ψt be the strictly positive
density of pt wrt to λm |D . Then, for any appropriate F ,
Z Z
Z Z
F (t, z)P[Zt ∈ dz|Zs = zs ] dt =
F (t, z)1 z∈D ψt (z) dz dt
R+ D
R+ Rm
Hence, if (23) holds P ⊗ λ-as then (25) holds Lebesgue-a.s., and by continuity
of the coefficients of (23) this extends to the entire of D × R+ . The reverse
17
assertion follows since Zt takes only values in D.
¤
If the coefficients of Z are regular enough, it follows that a consistent process
for G can be chosen as time-homogeneous:
Corollary 2.2 (Reduction to the time-homogeneous case) Let Z ∈ Ξ∗ with
Lipschitz coefficients (µ, σ) and assume that Z is consistent with G.
Then, let u ≥ 0 such that (25) holds for all (z, u, x). Then, the solution
(u)
(u)
Z = (Zt )t≥0 of the time-homogeneous system
(u)
dZt
(u)
= µ(u; Zt ) dt +
n
X
(u)
σ j (u; Zt ) dWtj
(26)
j=1
exists and is also consistent with G.
Proof – If (µ, σ) are Lipschitz, then the homogeneous coefficients (µ(u; ·), σ(u; ·))
are also Lipschitz, so their SDE (26) admits a unique strong solution, cf. Protter
[P04] pg. 250ff.
¤
2.3.1
Extensions to manifolds Z: When does Z stay in Z ?
Definition 2.7 (Invariant manifold) Let (µ, σ) ∈ Ξ.
A regular sub-manifold with boundary Z ⊂ Rm is called locally invariant to
Z if for any starting point Z0 ∈ Z, there exists a strictly positive stopping time
τ such that Zt ∈ Z for t < τ .
We call Z globally invariant or just invariant if we can set τ = ∞.
Recall that if Z is a d-dimensional regular sub-manifold with boundary ∂Z, then
Tx Z denotes the tangent space of Z in an interior point x ∈ Z. By definition,
the boundary ∂Z of Z is either empty or a (d − 1)-dimensional manifold, and
for a point x ∈ ∂M , its (d − 1)-dimensional tangent space with respect to ∂Z is
denoted by Tx ∂M . For any regular sub-manifold, its closure is denoted by Z̄,
and contains its boundary (which might be empty).
For any point x ∈ ∂Z, we can find a smooth map ϕ : U → V from an open
set U into a relatively open set V ⊂ Rd−1 × R+
0 , which generates the “inward
pointing” tangent space (Tx Z)≥0 of M in x. We will now follow Björk et al
[BC99] and derive a condition under which Z will stay in a regular manifold
with boundary Z.
Using indices to denote derivatives, assume σ ∈ C 1 and define the vector
 1

σz σ(t; z)


..
σz0 σ(t; z) := 

.
σzm σ(t; z)
18
with σzi σ(t; z) given as the product
(σzj )i σ j (t; z) :=
Pm
m
X
j i j
j=1 (σz ) σ (t; z),
where we define
(∂z` σ j )i (t; z)(σ j )` (t; z) .
(27)
`=1
Theorem 2.4 Assume that Z ∈ Rm is an d-dimensional regular sub-manifold
with boundary and let Z ∈ Ξ with coefficients (µ, σ) and σ ∈ C 1 .
Then, Z is locally invariant for (µ, σ) iff
)
µ(t; z) − 12 σz0 σ(t; z) ∈ Tz Z
(28)
σ j (t; z) ∈ Tz Z
for all z ∈ intZ and all t ≥ 0.
Moreover, if Z is closed in the relative topology, then it is globally invariant
if additionally
)
µ(t; z) − 12 σz0 σ(t; z) ∈ (Tz Z)≥0
(29)
σ j (t; z) ∈ Tz ∂Z
for all z ∈ ∂Z and all t ≥ 0.11
Proof – The proof is an application of Stratonovic-calculus. See also [FT04]
theorem 1.2 or [T05] pg. 19 for a similar statement. For notational simplicity,
we only consider the time-homogeneous case:
Step 1: We first look at the general case of a sub-manifold, i.e. assume
Z0 ∈ intZ. Then, there exists an open set U0 (in the relative topology) such
that Z0 ∈ U0 ⊂ Z. The solution to a Stratonovic-SDE
dYt = η(Yt ) dt + ς(Yt ) ◦ dWt
starting at Y0 = Z0 will stay in Z until it leaves U0 , if and only if
)
η(z) ∈ Tz Z
ς(z) ∈ Tz Z
for all z ∈ U0 : This follows since Stratonovic calculus obeys the same rules as
the standard calculus.12 Also, the exit time τ from U0 is strictly positive, so Z
is locally invariant for Y .
From that, it is also clear why the process may only stay locally in Z: If the
sub-manifold is open in the relative topology, Y can approach the boundary in
a sequence of steps and so finally leave the manifold via its boundary (think of
the circle around the open unit ball). Indeed, it can be shown that it can only
leave the manifold at a boundary.
Step 2: Now consider the case where Z is closed in the relative topology,
i.e. that ∂ Z̄ ⊂ Z. By standard calculus, the condition η(z) ∈ (Tz Z)≥0 ensures
11 Note
12 For
∂Z might be empty.
an introduction into this topic, see for example Roger/Williams [RW00] chapter V.
19
that the pure drift term stays on the manifold. The second condition ς(z) ∈
Tz ∂Z on the other hand ensures that the diffusion term does not drive the
solution out of the (d − 1)-dimensional manifold ∂Z, just as above, and the
diffusion Y cannot leave Z.
Step 3: The previous remarks can now be translated to our case using the
transition between Stratonovic and Ito integral by way of
1
H j ◦ dW j = H j dW j − dhH j , W j i .
2
Let i = 1, . . . , m and j = 1, . . . , n. Then, (σ j )i is the volatility coefficient of Z i
with respect to the jth Brownian motion. We obtain
Ito
dh(σ j )i (Z), W j it
=
dh
m Z
X
`=1
=
=
m
X
`=1
m
X
·
0
(∂z` σ j )i (Zs ) dZs` + (· · ·)dhZit , W j it
Z ·
n
X
j i
dh (∂z` σ ) (Zs )
(σ k )` (Zs )dWsk + (· · ·)dt , W j it
0
k=1
¡
¢
(∂z` σ j )i (Zs )(σ j )` (Zs ) dt
`=1
(27)
=:
(σzj )i σ j dt
Defining σz0 σ as above and using (22), we get
dZt =
µ
¶
n
X
1
µ(Zt ) − σz0 σ(Zt ) dt +
σ j (Zt ) ◦ dWtj ,
2
j=1
to which our previous remarks apply.
¤
We are hence able to answer (P2) satisfactory in the case where Z is a submanifold:
Conclusion 2.2 (Solution to problem (P2)) To check whether Z and G are
consistent, we apply theorem 2.3 (or corollary 2.1) to find out whether the process
is consistent on Z, and theorem 2.4 to determine whether it also stays (at least
locally) in Z. We have also shown that we can often focus on time-homogeneous
processes only.
3
Variance Curve Models in Hilbert Spaces
We will now focus on problem (P3): Given v̂ as a stochastic process in terms
of an SDE, and a curve functional G, under which conditions on the coefficients
of v̂ can we find a consistent parameter process Z such that
v̂t (x) = G(Zt ; x) ?
20
To be able to approach this question, we have to impose some regularity on
the possible curves of v̂. Indeed, we will employ the theory of stochastic
differential equations in Hilbert spaces, the standard reference on which being da Prato/Zabcyk [PZ92], also see Teichmann [T05]. We will closely follow
[BC99] and [BS01]. In particular [BC99] is a very readable introduction into
the topic.
We structure this section as follows: First we recall some basic notation,
before we then present theorem 3.1 which will answer (P3) at least locally.
3.1
Basic Notations
We remain on the space W = (Ω, F∞ , P, F) which supports an extremal ndimensional Brownian motion W . Additionally we assume that we are also
given a Hilbert-space H, which will contain our forward variance curves.
In H, we want to give meaning to a stochastic differential equation of the
type
n
X
dv̂t = ∂x v̂t dt +
β̂tj (v̂t ) dWtj
(30)
j=1
with locally Lipschitz vector fields β 1 , . . . , β n : U ⊂ H → H from an open set
U ⊂ dom(∂x ) into H (the operator ∂x : dom(∂x ) ⊂ H → H is the generator of
the strongly continuous semigroup (Tt )t≥0 of shift operators (Tt v̂)(x) := v̂(x +
t)).13
We only consider the time-homogeneous case, but the time-inhomogeneous
setting works similarly. We call a solution v̂ to (30) H-strong in v̂0 ∈ U if there
exists a strictly positive stopping time τ up to which
Z
v̂t = v̂0 +
t
∂x v̂s ds +
0
n Z
X
j=1
0
t
β̂sj (v̂t ) dWsj
(31)
´
i
hR ³
Pn
T
holds with P 0 k|v̂s k|H + j=1 ||β̂s (v̂s )||2H ds < ∞ = 1.
For such a process, the Stratonovic-form is given by
µ
¶
n
X
1 0
dv̂t = ∂x v̂t − β̂v̂ β̂(v̂t ) dt +
β̂tj (v̂t ) ◦ dWtj
2
j=1
(32)
where we understand as before
β̂v̂0 β̂(v̂t )
n
X
:=
(∂v̂ β j )(v̂) β j (v̂) .
j=1
Here, ∂v̂ β j denotes the Frechet-derivative of β j .
In the light of problem (P3), we will make the following assumptions for the
remainder of this section:
13 The
operator ∂x is formally given as ∂x v̂(x) := limu↓0
21
1
u
((Tu v̂)(x) − v̂(x)).
Assumptions 2 (Björk/Christensen [BC99])
(a) Z is an open connected subset of U .
(b) For each v̂0 = G(Z0 ) with Z0 ∈ Z, equation (30) has a unique H-strong
solution.
(c) The map G : Z → H is injective and C 2,2 .
(d) The derivative ∂z G : Z → H is injective.
(e) The derivative ∂x G : Z → H is a continuous map.
Fundamental Remark 3.1 Assumption 3. above is non-trivial.
One way around this is to use a Hilbert space, in which the operator ∂x
is bounded. If additionally β̂ 1 , . . . , β̂ m are smooth, then (32) has a H-strong
solution. This is discussed by Björk/Svensson in [BS01] remark 4.2 on pg. 21
who also provide a respective Hilbert space H in their definition 4.1 pg. 17.
Our assumptions allow us now to define the variance curve manifold
G := G(Z) ⊂ H
and we assume throughout that v̂0 ∈ intG.
Definition 3.1 (Locally Consistent and FDR) We say v̂ is locally consistent
with G if for all Z0 ∈ Z there exists an m-dimensional Markov process Z and
a strictly positive stopping time τ such that on t < τ , Z stays in Z, solves the
SDE
n
X
dZt = µ(Zt ) dt +
σ j (Zt ) dWtj
(33)
j=1
and satisfies
v̂t (x) = G(Zt ; x) .
(34)
We call the pair (G, Z) a finite dimensional representation or FDR of the variance curve v̂.
3.2
Invariant Manifolds
Theorem 3.1 (Björk/Christenssen-type Invariance) The process v̂ = (v̂t )t≥0
admits a finite dimensional representation if and only if
)
∂x G(z) − 21 β̂v0 β̂(G(z)) ∈ Img ∂z G(z)
(35)
β̂(G(z)) ∈ Img ∂z G(z)
holds for all z ∈ Z, and in this case Z is given as the solution to (33) with
)
µ(z) = (∂z G)(z)−1 ∂x G(z)
(36)
σ j (z) = (∂z G)(z)−1 β̂ j (G(z))
where µ and σ are locally Lipschitz.
22
Proof – Using Teichmann’s [T05] convenient notation, we write β̂ 0 (v̂) := ∂x v̂ −
1 0
1 0
0
2 β̂v β̂(v̂) and σ (z) := µ(z) − 2 σz σ(z) for the Stratonovic drifts of v̂ and Z,
respectively.
We follow the proof of theorem 1 in [BC99].
Step 1: We begin with the “only if” part: Hence we assume (34). Using
the Stratonovic-form of Z we get
dv̂t = ∂z G(Zt ) σ 0 (Zt ) dt +
n
X
∂z G(Zt )σ j (Zt ) ◦ dWtj .
(37)
j=1
Comparing this equation with (32) we find for z = G−1 (v̂) that
β̂ j (G(z)) = ∂z G(z)σ j (z)
(j = 0, 1, . . . , m)
hence β̂ j ∈ span (∂z1 G(z), . . . , ∂zm G(z)) for j = 0, . . . , m, in other words, (35).
Step 2: Now the “if” part: Assume that v̂ is the solution to (30) or, equivalently, (32), and that (35) holds.
Since ∂z G is assumed to be injective we find unique b(z) = (b0 (z), . . . , bn (z))
for any point z = G−1 (v̂) such that
β̂ j (v̂) = ∂z G(z) bj (z)
(j = 0, 1, . . . , m) .
(38)
In the light of (37), we would like to set σ j (z) := bj (z), but we have to prove
that the functions a and b can be chosen to allow a solution for (33).
To this end, we note that ∂z G is a bounded, injective operator with closed
range (because z ∈ Rm ) and therefore has a smooth left-inverse (∂z G)−1 (see
[BC99] lemma 1). Hence,
¡
¢
σ j (z) := (∂z G)−1 (z) β̂ j G(z)
is a locally smooth mapping σ j : z 7→ Rm , so µ and σ in (36) are locally Lipschitz – hence, (33) has a solution up to some explosion time τ > 0.14
¤
The proof of the previous theorem used similar techniques as theorem 2.4. However, a global existence result is more difficult to obtain because of the additional
need to prove existence of a solution to the SDE (33) with coefficients (36).
Conclusion 3.1 (Local Solution to problem (P3)) At least locally, theorem 3.1
solves problem (P3): A variance curve model admits an FDR (G, Z) if (35) is
satisfied, and in this case the parameter process Z is given as a solution to the
SDE of (36).
4
Applications
After we have solved the abstract problems (P1) - (P3), let us finally present
some examples. We mainly focus on exponential-polynomial variance curves,
but also discuss a few other approaches.
14 See,
for example, [P04] pg. 303.
23
4.1
Exponential-Polynomial Variance Curve Models
Definition 4.1 The family of Exponential-Polynomial Curve Funtional is pam
rameterized by z = (z1 , . . . , zm ; zm+1 , . . . , zd ) ∈ R+ × Rd−m and given as
m
X
G(z; x) =
pi (z; x)e−zi x
(39)
i=1
PN
where pi are polynomials of the form pi (x) = j=0 aij (z)xj with coefficents a
such that pi ≥ 0 λd -as. W.l.g. we can assume that deg(pi ) > deg(pi+1 ).15
(Also compare Bjoerk et al [BS01] and Filipovic [F01].)
We assume that any parameter process has Zti 6= Ztj for i 6= j, since otherwise
RT
we can just rewrite (39) accordingly. Also note that 0 G(z; x) dx < ∞ for all
T < ∞.
Lemma 4.1 Let Z be a parameter process consistent with an exponential-polynomial
variance curve functional.
Then, the first m coordinates Z 1 , . . . , Z m are constant.
Proof – We have
∂x G =
−zi
m
X
i=1
∂zj G =
pi (z; x)e−zi x +
m
X
∂x pi (z; x)e−zi x
i=1
m
X
−pi (z; x)xe−zi x 1 j≤m +
∂zj pi (z; x)e−zi x
(40)
i=1
∂z2j zj G =
m
X
¡
¢
pi (z; x)x2 e−zi x − 2∂zj pi (z; x)xe−zi x 1 j≤m +
∂z2j zj pi (z; x)e−zi x
i=1
We ignore the mixed terms ∂z2j zk G, since we can already see that ∂z2j zj G with
j ≤ m are the only terms in (25) which involve polynomials of degree degpi + 2
as factors in front of the exponentials e−zi x . Since we choose the zi distinct,
and because neither µ nor σ depends on x, this implies that σi2 = 0 for i ≤ m.
In other words, the states zi for i ≤ m cannot be random.
Next, we use (40) and find with the same reasoning (now applied to the
polynomials of degree degpi + 1) that µi = 0 for i ≤ m, so Z i must be a constant.
¤
We will now present two particular exponential-polynomial curve functionals.
In the light of lemma 4.1, we will keep the exponentials constant but investigate
the possible dynamics of the remaining parameters.
15 We
denote by deg(p) the degree of a polynomial p.
24
Example 1 (Linearly Mean-Reverting Variance Curve Models) The Functional
G(z; x) := z2 + (z1 − z2 )e−κx .
+
with z ∈ Z := R+
0 × R0 is consistent with Z if µ1 (t; z) = κ(z2 − z1 ) and
µ2 (t; z) = 0 (that is, Z 2 must be a martingale). The volatility parameters can
be freely specified, as long as Z 1 and Z 2 remain non-negative.
We call such a model a linearly mean-reverting variance curve model.
Proof – Corollary 2.1 implies that we have to match
!
−κ(z1 − z2 )e−κx = µ1 (t; z)e−κx + µ2 (t; z)(1 − e−κx )
Since the left hand side has no term constant in x, we must have µ2 (t; z) = 0
(i.e. Z 2 is martingale), and then µ1 (t; z) = κ(z2 − z1 ).
¤
√
A popular parametrization is σ2 = 0 and σ1 (t; z1 ) = ν z1 for some ν > 0,
which has been introduced by Heston [H93] (Z 1 is then the square of the shortvolatility of the associated stock price process). Another possible choice is
¶
µ
¶
µ
0
κ(z2 − z1 )
νz1α
µ(z) =
σ(z) =
ηρz2 η ρ̂z2
0
p
with constants α ∈ [ 12 , 2], ν, η ∈ R+ , ρ ∈ (−1, 0] and ρ̂ = 1 − ρ2 .
The next model is a generalization of the above. We will omit the proof
which works similar as above.
Example 2 (Double Mean-Reverting Variance Curve Models) Let c, κ > 0 con3
stant and let z = (z1 , z2 , z3 ) ∈ R+ . The Curve Functional
(
−cx
−e−κx
z3 + (z1 − z3 )e−κx + (z2 − z3 )κ e κ−c
(κ 6= c)
G(z; x) :=
(41)
z3 + (z1 − z3 − κx(z2 − z3 )) e−κx
(κ = c)
is consistent with any parameter process Z such that
dZt1
dZt2
dZt3
=
=
=
κ(Zt2 − Zt1 ) dt + σ1 (Zt ) dWt
c(Zt3 − Zt2 ) dt + σ2 (Zt ) dWt
σ3 (Zt ) dWt
and is called a double mean-reverting variance curve model.
It turns out to be a flexible and applicable model: At the time of writing,
the variance functional (41) fits the variance swap market of major indices well,
so this kind of double mean-reverting models is a good candidate for a variance
curve model (a similar model has been proposed by Duffie et al. [DPS00] who use
√
√
σ1 (z) = z1 and σ2 (z) = z2 with a particular sparse correlation structure).
Such a model has recently been successfully implemented to price and hedge
options on variance and related products. Higher order models with similar
structure can also be considered.
25
4.1.1
Exponential Curves
As in (39), let (pi )i=1,...,m be polynomials and let
g(z; x) =
m
X
pi (z; x)e−zi x
i=1
with z = (z1 , . . . , zm ; zm+1 , . . . , zd ) ∈ R+
m
× Rd−m . Set
G(z; x) := exp(g(z; x)) .
(42)
Using theorem 2.3, a necessary condition for a consistent pair is
©
ª
1
∂x g(z; x) = µ(z) ∂z g(z; x) + σ 2 (z) (∂z g(z; x))2 + ∂zz g(z; x) .
2
We hence find similar to the above results (also compare theorems 3.6.1 and
3.6.2 from Filipovic [F01] p.52ff):
Lemma 4.2 For each consistent parameter process Z for G, the coordinates
Z 1 , . . . , Z m are constant. Moreover, there must be at least one pair i 6= j such
that Z i = 2Z j , otherwise Z is entirely constant.
Example 3 (Exponential Mean-Reverting Models) Let
g(z; x) = z2 + (z1 − z2 )e−κx +
z3
(1 − e−2κx )
4κ
with z1 , z2 , z3 ∈ R+ .
Then, σ3 ≡ 0.
Under the assumption that µ1 (t; z) = κ(z2 −z1 ) is a mean-reverting
term, we
p
further find that 0 ≤ µ2 (t; z) ≤ 12 z3 for all z, so that σ1 (t; z) = z3 − µ2 (t; z),
p
σ2 (t; z) = σ1 (t; z) − z3 − 2µ2 (t; z) and µ3 (t; z) = −µ2 (t; z).
√
In the case µ2 = 0, this yields σ2 = 0 and then σ1 (t; z) = z3 , which is the
exponential Ornstein-Uhlenbeck stochastic volatility model discussed in depth
by Fouque et al in [FPS00].
4.2
Variance Swap Curves given in terms of Volatility
In this section, we briefly want to discuss a few variance curve term-structure
interpolation schemes in terms of volatility of the variance curve, i.e. schemes
where the variance swap price is given as a functional
V̂t (x) = Σ2 (Zt ; x)x
where
Σ(z; x) = z1 + z2 w(x)
(43)
for some “term-structure” function w.
Such curves arise if standard forms of implied volatility term-structure functionals are employed to model the variance swap curve. This approach might be
26
appealing if the implied volatility term-structure is well captured by a particular
choice of w.16 A few choices which have been considered for implied volatility
interpolation are
w0 (x) := 0
w1 (x) := ln(1 + x)
√
w2 (x) :=
²+x
√
w3 (x) := 1/ ² + x
(44)
(45)
(46)
(47)
The first case w0 is called the “Black&Scholes” case. See also Haffner p. 87 in
[H04] for a discussion of implied volatility term-structure interpolation.
In our previous notation,
¡
¢
G(z; x) := ∂x Σ(z; x)2 x = Σ(z; x)2 + 2Σ(z; x)Σ0 (z; x)x
i.e.
G(z; x)
= (z12 + 2z1 z2 w(x) + z22 w2 (x)) + 2(z1 + z2 w(x))z2 w0 (x)x
= z12 + 2z1 z2 (w(x) + w0 (x)x) + z22 (w2 (x) + 2w(x)w0 (x)x) (48)
We have to show:
!
∂x G = µ(t; z) ∂z G +
1 2
σ (t; z) ∂zz G
2
(49)
Corollary 4.1
(a) Case (44) is consistent iff µ1 (z1 ) = 0, i.e. Z 1 is a non-negative martingale.
(b) Case (44) is the only case of (44)-(47) which is consistent.
Proof – The first statement is obvious. We prove the second statement for w1 ,
since it works similarly for w2 and w3 . Hence let w(x) = w1 (x) = ln(1 + x).
Focusing on (49), we obtain
2z1 z2 {2w0 (x) + w00 (x)x}
©
ª
+2z22 2w0 (x)w(x) + w0 (x)2 x + w00 (x)w(x)x
∂z1 G = 2z1 + 2z2 {w(x) + w0 (x)x}
©
ª
∂z2 G = 2z1 {w(x) + w0 (x)x} + 2z2 w2 (x) + 2w(x)w0 (x)x
∂z1 z1 G = 2
©
ª
∂z2 z2 G = 2 w2 (x) + 2w(x)w0 (x)x
∂z1 z2 G = 2 {w(x) + w0 (x)x} .
∂x G =
We now show for w1 that z2 = 0 and µ1 = 0, i.e. that the functional degenerates
to the Black & Scholes case. First, we compute the derivatives
w1 (x) = ln(1 + x) , w10 (x) =
1
1+x
and
w100 (x) = −
1
.
(1 + x)2
16 Gatheral [G04] notes that in some stock price models, the shape of the term-structure of
implied volatility is similar to the shape of the variance swap curve
27
We therefore have
½
¾
½
¾
2
x
ln(1 + x) x − x ln(1 + x)
2
∂x G = 2z1 z2
−
+ 2z2 2
+
.
1 + x (1 + x)2
1+x
(1 + x)2
Now note that none of the terms ∂zi G or ∂z2i zj contains terms in
to satisfy (49), for all x ≥ 0, we must have
!
0 = −2z1 z2 x + 2z22 (x − x ln(1 + x)) .
1
(1+x)2 .
Hence,
(50)
Assume z2 6= 0. Then, (50) implies z1 = z2 (1 − ln(1 + x)), which is not possible. Hence z2 = 0 and the curve G reduces to the “Black & Scholes case” w0 . ¤
4.3
Hedging with Variance Curve Models
In this subsection, we want to mention that a variance curve model usually
allows to compute hedges of prices of contingent claims in terms of both the
stock price and the variance swaps.
The treatment in the following paragraph is of practical nature; a more
sorrow investigation of existence of hedges in terms of traded variance swaps in
arbitrary variance curve models is subject to further research.
Given a consistent pair (G, Z) we make the following simplifying assumptions:
Assumptions 3
(a) The correlation structure ρ in definition 2.2 is deterministic or functional
dependent on (S, Z). In this case, (S, Z) is still Markov.
(b) In each time interval [t0 , t0 + ε), it is possible to find fixed maturities
Tm > · · · > T1 > t0 + ε, possibly depending on Zt0 and t0 , such that the
function
³
´
G̃(Zt ; t) = G(Zt ; T1 − t), . . . , G(Zt ; Tm − t)
can be inverted for t ∈ [t0 , t0 +ε), that is we can recover Zt = G̃−1 (Vt (T1 )−
Vt (t), . . . , Vt (Tm )−Vt (t)) from the observed prices of variance swaps V̄ti :=
Vt (Ti ), i = 1, . . . , m. In short
Zt = G̃−1 (V̄t )
t ∈ [t0 , t0 + ε) .
(51)
We also assume that G̃−1 is C 1 in v.
Using the first assumption, the price of an European contingent claim H under
the martingale measure P is
£ ¯
¤
ht (St ; Zt ) := E H ¯ St ; Zt .
28
By standard arguments we get
dht (St ; Zt ) = ∂S ht (St ; Zt ) dSt +
m
X
∂zi ht (St ; Zt ) dZ̃ti
i=1
with martingales dZ̃ti = dZti −µi (Zt ) dt. Under the invertibility assumption (51)
we can then write dht (St ; Zt ) ≡ dht (St ; G−1 (Vt )) in each interval t ∈ [t0 , t0 + ε),
so we obtain
dht (St ; Zt ) ≡
m
X
¡
¢
¡
¢
∂S ht (St ; G−1 (Vt )) dSt +
∂vk ht St ; G−1 (Vt ) dV̄tk ,
|
{z
}
|
{z
}
k=1
Delta
“VarSwapDelta”
which is just the required hedge of H in terms of stock and variance swaps.
An extension of this approach to path-dependent options or options on realized
variance is straight forward.
4.4
Recalibration of Stochastic Volatility Models
Another important consequence of the observations in section 4.1 is that daily
recalibration of pricing models is subject to the same arbitrage conditions as
developed above:
We will assume that an institution has a pricing model M in place, which
is parameterized by some parameter z ∈ Z and which then allows to compute
prices of arbitrary claims on the stock price S, which in turn is given in the
“real” world as a continuous process on a stochastic base W = (Ω, F∞ , P, F).
As a guiding example for a pricing model, we will use Heston’s model: Let
W̃ = (Ω̃, F̃∞ , P̃, F̃) be the standard Wiener space with two-dimensional Brownian motion W̃ = (W̃ 1 , W̃ 2 ). Then define
p
dṽτ := κ(m − ṽτ ) dt + ν ṽτ dW̃τ1 ṽ0 = v0
(52)
p
p
1
2
2
dX̃τ :=
ṽτ d(ρW̃τ + 1 − ρ W̃τ ) X̃0 = 0
(53)
4
with z = (S0 , v0 ; κ, m, ν, ρ) ∈ Z = R+ × (−1, 0] such that S̃ = S0 E(X̃) is a
martingale.
We now consider this as a model with parameters z ∈ Z. It is clear that
inside the model in model time τ only the states S̃τ and vτ change, while the
remaining four parameter are assumed to be constant. (We have used the timeindex τ to underline the fact that τ denotes model time.)
On the space W̃, we denote by θ the shift-operator as in Revuz/Yor [RY99]
p. 35. Let H be a payoff dependent on the path of S̃, i.e. a measurable and
integrable map
H : C[0, ∞) → R .
Given a continuous non-random function (yu )u∈[0,t] with yt = S̃0 we can nonetheless define
H ◦ θtY (S̃) := H(y1 [0,t) + S̃ ◦ θ−t 1 [t,∞) ) .
29
This formalizes the idea of “gluing” y in front of S̃ to account for the unavailable
information before τ = 0.
Now assume that we are trading in the market W with the continuous stock
price process S and a constant cash bond of 1. Besides the stock S, we assume
that at any time t, prices of various reference instruments such as European
option prices are publicly quoted. We use their prices Pt = (Pti )i=1,...,Nt to
calibrate our pricing model, i.e. we run an algorithm
Ψ : Pt 7→ Zt ∈ Z
which uniquely determines some model parameters Zt . We use these parameters
to value our exotic option positions (the term “exotic” here means that no
market quotes are publicly available for the product). We denote by X the set
of payoffs of those exotic structures. At any time t, the stock price in W has
the known path s := (Su )0≤u≤t until t. Given that path, the price of an exotic
option H ∈ X is given by
i
h
πt [H] := ẼZt H ◦ θts (S̃)
(by using θs we have provided past information on S to the pricing model). It is
apparent that this procedure yields a new meta-model M by means of successive
recalibration: It is given by the stock price, the calibration instruments and,
clearly, the price processes of the exotics in our book:
M = (S; P ; π(H)H∈X ) .
Clearly, there are two categories of arbitrage in such a meta-model: The first,
which we want to call static arbitrage, is arbitrage within the underlying pricing
model M itself (such as violations of European call price relations etc). The
second, we call it dynamic arbitrage, relates to the meta-model:
Under which conditions do the price processes in M generate arbitrage?
In the light of our variance curve models, it is clear that the implied variance
curve functional of the meta-model must be consistent. This is a necessary
condition for any meta-model and can therefore be applied under the assumption
of continuity of the price process S and the parameter process Z. It follows:
Proposition 4.1 The Heston meta-model is not free of dynamic arbitrage if
the speed κ of mean-reversion in Zt is not kept constant.
A similar statement obviously holds for the “Double Mean-Reverting” and the
”Exponential Mean-Reverting” models (examples 2 and 3, respectively).
Remark 4.1 Of course, proposition 4.1 is a theoretical result. In a real world
application, “small” arbitrage-violations are common and cannot be avoided, if
only for the mere fact that no model reflects the true market dynamics correctly.
However, the above proposition shows that the speed of mean-reversion should
30
not vary too widely and it seems natural either to fix it (and readjust it when
market conditions substantially change) or to impose a penalty in the calibration
which ensures that it does not fluctuate too widely.
5
Conclusions
We have developed a framework for variance curve market models, both in infinite dimensions and in finite-dimensional representations. We have derived
necessary and sufficient conditions to ensure that a variance curve functional is
consistent with some parameter process and applied the results to various examples which are used in practise. We have then shown that at least theoretically,
mean-reversion in the analyzed models such as Heston should be kept constant. A specific variance curve functional, namely the “Double mean-reverting
model”, has been proposed as an application.
Further directions of research are the introduction of jumps, questions of
completeness, from a practical point of view, the extension of this approach to
incorporate more information on the correlation structure ρ appears interesting,
as is also a more detailled discussion of hedging with variance curve models than
in section 4.3.
Acknowledgements
I want to thank Prof.Dr. Alexander Schied (TU Berlin) for his patient support
and discussions. I also want to thank Prof. Dr. Josef Teichmann (TU Vienna) for
his helpful advice and Dr. Dr. Marcus Overhaus and the Deutsche Bank Global
Equity Derivatives Quantitative Research team for help and support.
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