Arbitrage Bounds For Prices of Weighted Variance Swaps

a
r
X
i
v
:
1
0
0
1
.
2
6
7
8
v
3

[
q
-
f
i
n
.
P
R
]

1
8

S
e
p

2
0
1
2
Arbitrage bounds for prices of weighted variance swaps

Mark Davis
Jan Ob l oj
Vimal Raval
Abstract
We develop a theory of robust pricing and hedging of a weighted variance swap given market
prices for a nite number of comaturing put options. We assume the put option prices do not
admit arbitrage and deduce no-arbitrage bounds on the weighted variance swap along with super-
and sub- replicating strategies that enforce them. We nd that market quotes for variance swaps are
surprisingly close to the model-free lower bounds we determine. We solve the problem by transforming
it into an analogous question for a European option with a convex payo. The lower bound becomes
a problem in semi-innite linear programming which we solve in detail. The upper bound is explicit.
We work in a model-independent and probability-free setup. In particular we use and extend
Follmers pathwise stochastic calculus. Appropriate notions of arbitrage and admissibility are intro-
duced. This allows us to establish the usual hedging relation between the variance swap and the log
contract and similar connections for weighted variance swaps. Our results take form of a FTAP: we
show that the absence of (weak) arbitrage is equivalent to the existence of a classical model which
reproduces the observed prices via riskneutral expectations of discounted payos.
Key Words: Weighted variance swap, weak arbitrage, arbitrage conditions, model-independent
bounds, pathwise Ito calculus, semi-innite linear programming, fundamental theorem of asset pric-
ing, model error.
1 Introduction
In the practice of quantitative nance, the risks of model error are by now universally appreciated.
Dierent models, each perfectly calibrated to market prices of a set of liquidly traded instruments, may
give widely dierent prices for contracts outside the calibration set. The implied hedging strategies,
even for contracts within the calibration set, can vary from accurate to useless, depending on how well
the model captures the sample path behaviour of the hedging instruments. This fundamental prob-
lem has motivated a large body of research ranging from asset pricing through portfolio optimisation,
risk management to macroeconomics (see for example Cont (2006), Follmer et al. (2009), Acciaio et al.
(2011), Hansen and Sargent (2010) and the references therein). Another stream of literature, to which
this paper is a contribution, develops a robust approach to mathematical nance, see Hobson (1998),
Davis and Hobson (2007), Cox and Ob loj (2011a). In contrast to the classical approach, no probabilistic
setup is assumed. Instead we suppose we are given current market quotes for the underlying assets and
some liquidly traded options. We are interested in no-arbitrage bounds on a price of an option implied
by this market information. Further, we want to understand if, and how, such bounds may be enforced,
in a model-independent way, through hedging.
In the present paper we consider robust pricing and hedging of a weighted variance swap when prices
of a nite number of comaturing put options are given. If (S
t
, t [0, T]) denotes the price of a nancial
asset, the weighted realized variance is dened as
(1.1) RV
T
=
n
i=1
h(S
ti
)
_
log
S
ti
S
ti1
_
2
,
where t
i
is a pre-specied sequence of times 0 = t
0
< t
1
< < t
n
= T, in practice often daily sampling,
and h is a given weight function. A weighted variance swap is a forward contract in which cash amounts
This paper was previously circulated under the title Arbitrage Bounds for Prices of Options on Realized Variance.
Department of Mathematics, Imperial College London, London SW72AZ, UK (mark.davis@imperial.ac.uk).
Mathematical Institute, Oxford-Man Institute of Quantitative Finance and St Johns College, University of Oxford,
Oxford OX1 3LB, UK (obloj@maths.ox.ac.uk)
Imperial College London. Work supported by EPSRC under a Doctoral Training Award.
1
Arbitrage bounds for prices of options on realised variance 2
equal to A RV
T
and A P
RV
T
are exchanged at time T, where A is the dollar value of one variance
point and P
RV
T
is the variance swap price, agreed at time 0. Three representative cases considered in
this paper are the plain vanilla variance swap h(s) 1, the corridor variance swap h(s) = 1
I
(s) where I
is a possibly semi-innite interval in R
+
, and the gamma swap in which h(s) = s. The reader can consult
Gatheral (2006) for information about variance swaps and more extended discussion of the basic facts
presented below. In this section we restrict the discussion to the vanilla variance swap; we return to the
other contracts in Section 4.
In the classical approach, if we model S
t
under a risk-neutral measure Q as
1
S
t
= F
t
exp(X
t
1
2
X
t
),
where F
t
is the forward price (assumed to be continuous and of bounded variation) and X
t
is a continuous
martingale with quadratic variation process X
t
, then S
t
is a continuous semimartingale and
log
S
ti
S
ti1
= (g(t
i
) g(t
i1
)) + (X
ti
X
ti1
),
with g(t) = log(F
ti
)
1
2
X
ti
. For m = 1, 2 . . . let s
m
i
, i = 0, . . . , k
m
be the ordered set of stopping times
in [0, T] containing the times t
i
together with times
m
0
= 0,
m
k
= inft >
m
k1
: [X
t
X
m
k1
[ > 2
m
wherever these are smaller than T. If RV

m
T
denotes the realized variance computed as in (1.1) but using
the times s
m
i
, then RV
m
T
X
T
= log S
T
almost surely, see Rogers and Williams (2000), Theorem
IV.30.1. For this reason, most of the pricing literature on realized variance studies the continuous-time
limit X
T
rather than the nite sum (1.1) and we continue the tradition in this paper.
The key insight into the analysis of variance derivatives in the continuous limit was provided by
Neuberger (1994). We outline the arguments here, see Section 4 for all the details. With S
t
as above and
f a C
2
function, it follows from the general It o formula that Y
t
= f(S
t
) is a continuous semimartingale
with dY
t
= (f
(S
t
))
2
dS
t
. In particular, with the above notation
d(log S
t
) =
1
S
t
dS
t

1
2S
2
t
dS
t
=
1
S
t
dS
t

1
2
dlog S
t
,
so that
(1.2) log S
T
= 2
_
T
0
1
S
t
dS
t
2 log(S
T
/S
0
)
This shows that the realized variation is replicated by a portfolio consisting of a self-nancing trading
strategy that invests a constant $2D
T
in the underlying asset S together with a European option whose
exercise value at time T is (S
T
) = 2 log(S
T
/F
T
). Here D
T
is the time-T discount factor. Assuming
that the stochastic integral is a martingale, (1.2) shows that the risk-neutral value of the variance swap
rate P
RV
T
is
(1.3) P
RV
T
= E
_
log S
T
_
= 2E[log(S
T
/F
T
)] = 2
1
D
T
P
log
,
that is, the variance swap rate is equal to the forward value P
log
/D
T
of two log contractsEuropean
options with exercise value log(S
T
/F
T
), a convex function of S
T
. (1.3) gives us a way to evaluate P
RV
T
in any given model. The next step is to rephrase P
log
in terms of call and put prices only.
Recall that for a convex function f : R
+
R the recipe f
(a, b] = f
+
(b) f
+
(a) denes a positive
measure f
(dx) on B(R
+
), equal to f
(x)dx if f is C
2
. We then have the Taylor formula
(1.4) f(x) = f(x
0
) +f
+
(x
0
)(x x
0
) +
_
x0
0
(y x)
+
(dy) +
_

x0
(x y)
+
f
(dy).
Applying this formula with x = S
T
, x
0
= F
T
and f(s) = log(s/S
0
), and combining with (1.2) gives
(1.5)
1
2
log S
T
=
_
T
0
1
S
t
dS
t
log(F
T
/S
0
) + 1 S
T
/F
T
+
_
FT
0
(K S
T
)
+
K
2
dK +
_

FT
(S
T
K)
+
K
2
dK.
Assuming that puts and calls are available for all strikes K R
+
at traded prices P
K
, C
K
, (1.5) provides
a perfect hedge for the realized variance in terms of self-nancing dynamic trading in the underlying (the
1
For example, St = (e
(rq)t
S
0
)(e
Wt
1
2
2
t
) in the Black-Scholes model, with the conventional notation.
rst four terms) and a static portfolio of puts and calls and hence uniquely species the variance swap
rate P
RV
T
as
(1.6) P
RV
T
=
2
D
T
_

0
1
K
2
(P
K
1
(KFT )
+C
K
1
(K>FT )
)dK.
Our objective in this paper is to investigate the situation where, as in reality, puts and calls are
available at only a nite number of strikes. In this case we cannot expect to get a unique value for P
RV
T
as in (1.6). However, from (1.2) we expect to obtain arbitrage bounds on P
RV
T
from bounds on the value
of the log contracta European option whose exercise value is the convex function log(S
T
/F
T
). It
will turn out that the weighted variance swaps we consider are associated in a similar way with other
convex functions. In Davis and Hobson (2007) conditions were stated under which a given set of prices
(P
1
, . . . , P
n
) for put options with strikes (K
1
, . . . , K
n
) all maturing at the same time T is consistent
with absence of arbitrage. Thus the rst essential question we have to answer is: given a set of put
prices consistent with absence of arbitrage and a convex function
T
, what is the range of prices P
for
a European option with exercise value
T
(S
T
) such that this option does not introduce arbitrage when
traded in addition to the existing puts? We answer this question in Section 3 following a statement of
the standing assumptions and a summary of the Davis and Hobson (2007) results in Section 2.
The basic technique in Section 3 is to apply the duality theory of semi-innite linear programming
to obtain the values of the most expensive sub-replicating portfolio and the cheapest super-replicating
portfolio, where the portfolios in question contain static positions in the put options and dynamic trading
in the underlying asset. The results for the lower and upper bounds are given in Propositions 3.1 and
3.5 respectively. (Often, in particular in the case of plain vanilla variance swaps, the upper bound
is innite.) For completeness, we state and prove the fundamental Karlin-Isii duality theorem in an
appendix, Appendix A. With these results in hand, the arbitrage conditions are stated and proved in
Theorem 3.6.
In this rst part of the paperSections 2 and 3we are concerned exclusively with European options
whose exercise value depends only on the asset price S
T
at the common exercise time T. In this case
the sample path properties of S
t
, t [0, T] play no role and the only relevant feature of a model
is the marginal distribution of S
T
. For this reason we make no assumptions about the sample paths.
When we come to the second part of the paper, Sections 4 and 5, analysing weighted variance swaps,
then of course the sample path properties are of fundamental importance. Our minimal assumption is
that S
t
, t [0, T] is continuous in t. However, this is not enough to make sense out of the problem.
The connection between the variance swap and the log contract, given at (1.2), is based on stochastic
calculus, assuming that S
t
is a continuous semimartingale. In our case the starting point is simply a set
of prices. No model is provided, but we have to dene what we mean by the continuous-time limit of the
realized variance and by continuous time trading. An appropriate framework is the pathwise stochastic
calculus of Follmer (1981) where the quadratic variation and an Ito-like integral are dened for a certain
subset Q of the space of continuous functions C[0, T]. Then (1.2) holds for S() Q and we have the
connection we sought between variance swap and log contract. For weighted variance swaps the same
connection exists, replacing log by some other convex function
T
, as long as the latter is C
2
. In the
case of the corridor variance swap, however,
T
is discontinuous and we need to restrict ourselves to a
smaller class of paths / for which the It o formula is valid for functions in the Sobolev space J
2
. All of this
is consistent with models in which S
t
is a continuous semimartingale, since then P : S(, ) / = 1.
We discuss these matters in a separate appendix, Appendix B.
With these preliminaries in hand we formulate and prove, in Section 4 the main result of the paper,
Theorem 4.3, giving conditions under which the quoted price of a weighted variance swap is consistent
with the prices of existing calls and puts in the market, assuming the latter are arbitrage-free. When the
conditions fail there is a weak arbitrage, and we identify strategies that realize it.
In mathematical nance it is generally the case that option bounds based on super- and sub-replication
are of limited value because the gap is too wide. However, here we know that as the number of put options
increases then in the limit there is only one price, namely, for a vanilla variance swap, the number P
RV
T
of (1.6), so we may expect that in a market containing a reasonable number of liquidly-traded puts the
bounds will be tight. In Section 5 we present data from the S&P500 index options market which shows
that our computed lower bounds are surprisingly close to the quoted variance swap prices.
This paper leaves many related questions unanswered. In particular we would like to know what
happens when there is more than one exercise time T, and how the results are aected if we allow jumps
in the price path. The latter question is considered in the small time-to-expiry limit in the recent paper
Keller-Ressel and Muhle-Karbe (2011). A paper more in the spirit of ours, but taking a complementary
approach, is Hobson and Klimmek (2011)[HK] . In our paper the variance swap payo is dened by the
continuous-time limit, and we assume that prices of a nite number of traded put and/or call prices are
known. By contrast, HK deal with the real variance swap contract (i.e., discrete sampling) but assume
that put and call prices are known for all strikes K R
+
, so in practice the results will depend on how
the traded prices are interpolated and extrapolated to create the whole volatility surface. Importantly,
HK allow for jumps in the price process; this is indeed the main focus of their work. The price bounds
they obtain are sharp in the continuous-sampling limit. Combining our methods with those of HK would
be an interesting, and possibly quite challenging, direction for future research, see Remark 4.6 below.
Exact pricing formulas for a wide variety of contracts on discretely-sampled realized variance, in a
general Levy process setting, are provided in Crosby and Davis (2011).
2 Problem formulation
Let (S
t
, t [0, T]) be the price of a traded nancial asset, which is assumed non-negative: S
t
R
+
=
[0, ). In addition to this underlying asset, various derivative securities as detailed below are also traded.
All of these derivatives are European contracts maturing at the same time T. The present time is 0. We
make the following standing assumptions as in Davis and Hobson (2007):
(i) The market is frictionless: assets can be traded in arbitrary amounts, short or long, without transaction
costs, and the interest rates for borrowing and lending are the same.
(ii) There is no interest rate volatility. We denote by D
t
the market discount factor for time t, i.e. the
price at time 0 of a zero-coupon bond maturing at t.
(iii) There is a uniquely-dened forward price F
t
for delivery of one unit of the asset at time t. This will
be the case if the asset pays no dividends or if, for example, it has a deterministic dividend yield. We let
t
denote the number of shares that will be owned at time t if dividend income is re-invested in shares;
then F
0
= S
0
and F
t
= S
0
/(D
t
t
) for t > 0.
We suppose that n put options with strikes 0 < K
1
< . . . < K
n
and maturity T are traded, at
time-0 prices P
1
, . . . , P
n
. In (Davis and Hobson, 2007), the arbitrage relations among these prices were
investigated. The facts are as follows. A static portfolio X is simply a linear combination of the traded
assets, with weights
1
, . . . ,
n
, , on the options, on the underlying asset and on cash respectively, it
being assumed that dividend income is re-invested in shares. The value of the portfolio at maturity is
(2.1) X
T
=
n
i=1
i
[K
i
S
T
]
+
+
T
S
T
+D
1
T
and the set-up cost at time 0 is
(2.2) X
0
=
n
i=1
i
P
i
+S
0
+.
Note that it does not make any sense a priori to speak about X
t
the value of the portfolio at any
intermediate time as this is not determined by the market input and we do not assume the options
are quoted in the market at intermediate dates t (0, T). To make sense to statements like X
T
is
non-negative we need to specify the universe of scenarios we consider. Later, in Section 4, this will mean
the space of of possible paths (S
t
: t T). However here, since we only allow static trading as in (2.1),
we essentially look at a oneperiod model and we just need to specify the possibly range of values for S
T
and we suppose that S
T
may take any value in [0, ).
A model / is a ltered probability space (, F = (T
t
)
t[0,T]
, P) together with a positive F-adapted
process (S
t
)
t[0,T]
such that S
0
coincides with the given time-0 asset price. Given market prices of options,
we say that / is a market model for these options if M
t
= S
t
/F
t
is an F-martingale (in particular, S
t
is
integrable) and the market prices equal to the Pexpectations of discounted payos. In particular, / is a
market model for put options if P
i
= D
T
E[K
i
S
T
]
+
for i = 1, . . . , n. We simply say that / is a market
model if the set of market options with given prices is clear from the context. It follows that in a market
model we have joint dynamics on [0, T] of all assets prices such that the initial prices agree with the
market quotes and the discounted prices are martingales. By the (easy part of the) First Fundamental
Theorem of Asset Pricing (Delbaen and Schachermayer, 1994) the dynamics do not admit arbitrage.
Our main interest in the existence of a market model and we want to characterise it in terms of the
given market quoted prices. This requires notions of arbitrage in absence of a model. We say that there
is model-independent arbitrage if one can form a portfolio with a negative setup cost and a non-negative
r
2
r
1
k
1
k
2
1 k
3
r
3
Figure 1: Normalized put prices (k
i
, p
i
) consistent with absence of arbitrage. An additional put with
price r = 0 and strike k [0, ] does not introduce arbitrage.
payo. There is weak arbitrage if for any model there is a static portfolio (
1
, . . . ,
n
, , ) such that
X
0
0 but P(X
T
0) = 1 and P[X
T
> 0] > 0. In particular, a model independent arbitrage is a
special case of weak arbitrage, as in Cox and Ob loj (2011b). Throughout the paper, we say that prices
are consistent with absence of arbitrage when they do not admit a weak arbitrage.
It is convenient at this point to move to normalized units. If for i = 1, . . . , n we dene
(2.3) p
i
=
P
i
D
T
F
T
, k
i
=
K
i
F
T
,
then, in a market model, we have P
i
= D
T
F
T
E[K
i
/F
T
M
T
]
+
and hence
p
i
= E[k
i
M
T
]
+
.
Also, we can harmlessly introduce another put option with strike k
0
= 0 and price p
0
= 0. Let
n = maxi : p
i
= 0, n = infi : p
i
= k
i
1(= + if = ) and K = [k
n
, k
n
], where it is hereafter
understood as K = [k
n
, ) if n = .
The main result of Davis and Hobson (2007, Thm. 3.1), which we now recall, says that there is no
weak arbitrage if and only if there exists a market model. The conditions are illustrated in Figure 1.
Proposition 2.1 Let r : [0, k
n
] R
+
be the linear interpolant of the points (k
i
, p
i
), i = 0, . . . , n. Then
there exists a market model if and only if r() is a non-negative, convex, increasing function such that
r(0) = 0, r(k) [k 1]
+
and r
(k
nn
) < 1, where r
denotes the left-hand derivative. If these con-

ditions hold except that n = and r
(k
n
) = 1 then there is weak arbitrage. Otherwise there is a
model-independent arbitrage. Furthermore, if a market model exists then one may choose it so that the
distribution of S
T
has nite support.
Remarks. (i) In Davis and Hobson (2007), the case of call options was studied. The result stated
here follows by put-call parity, valid in view of our frictionless markets assumption. The normalised call
price is c
i
= p
i
+ 1 k
i
.
(ii) Of course, no put option would be quoted at zero price, so in applications n = 0 always. As will
be seen below, it is useful for analysis to include the articial case n > 0.
Throughout the rest of the paper, we assume that the put option prices (P
1
, . . . , P
n
) do not admit a
weak arbitrage and hence there exists a market model consistent with the put prices. Let / be a market
model and let be the distribution of M
T
in this model. Then satises
_
R
+
1 (dx) = 1 (2.4a)
_
R
+
x(dx) = 1 (2.4b)
_
R
+
[k
i
x]
+
(dx) = p
i
, i = 1, . . . , n. (2.4c)
Conversely, given a probability measure on R
+
which satises the above we can construct a market
model / such that is the distribution of M
T
. For example, let (, F, P, (W
t
)
tR
+) be the Wiener space.
By the Skorokhod embedding theorem (cf. (Ob loj, 2004)), there is a stopping time such that W

and (W
t
) is a uniformly integrable martingale. It follows that we can put M
t
= 1 + W
(t/(Tt))
for
t [0, T). This argument shows that the search for a market model reduces to a search for a measure
satisfying (2.4). We will denote by M
P
the set of measures satisfying the conditions (2.4).
Lemma 2.2 For any M
P
, (R
+
K) = 0.
Proof. That [0, k
n
) = 0 when n > 0 follows from (2.4c) with i = n. When n n we have c
n
= 0, i.e.
there is a free call option with strike k
n
and we conclude that (k
n
, ) = 0.
The question we wish to address is whether, when prices of additional options are quoted, consistency
with absence of arbitrage is maintained. As discussed in Section 1, we start by considering the case where
one extra option is included, a European option maturing at T with convex payo.
3 Hedging convex payos
Suppose that, in addition to the n put options, a European option is oered at price P
at time 0, with
exercise value
T
(S
T
) at time T, where
T
is a convex function. We can obtain lower and upper bounds
on the price of
T
by constructing sub-replicating and super-replicating static portfolios in the other
traded assets. These bounds are given in Sections 3.1 and 3.2 respectively and are combined in Section
3.3 to obtain the arbitrage conditions on the price P
.
We work in normalised units throughout, that is, the static portfolios have time-T values that are
linear combinations of cash, underlying M
T
and option exercise values [k
i
M
T
]
+
. The prices of units
of these components at time 0 are D
T
, D
T
and D
T
p
i
respectively, where a unit of cash is $1. Indeed, to
price M
T
observe that $1 invested in the underlying at time 0 yields
T
S
T
/S
0
= S
T
/F
T
D
T
= M
T
/D
T
at time T. To achieve a consistent normalization for
T
we dene the convex function as
(3.1) (x) =
1
F
T
T
(F
T
x).
In a market model / we have P
= D
T
E[
T
(S
T
)] = D
T
F
T
E[(M
T
)], so the normalized price is
p
=
P
D
T
F
T
= E[(M
T
)]
and the cost for delivering a payo (M
T
) is D
T
p
.
3.1 Lower bound
A sub-replicating portfolio is a static portfolio formed at time 0 such that its value at time T is majorized
by (M
T
) for all values of M
T
. Obviously, a necessary condition for absence of model independent
arbitrage is that D
T
p
be not less than the set-up cost of any sub-replicating portfolio. It turns out
that the options k
i
with i n or i n are redundant, so the assets in the portfolio are indexed by
k = 1, . . . , m where
m = (n + 1) n n + 1
and the time-T values of these assets, as functions of x = M
T
are
a
1
(x) = 1 (Cash)
a
2
(x) = x (Underlying) (3.2)
a
i+2
(x) = [k
n+i
x]
+
, i = 1, . . . , m2 (Options).
We let a(x) be the m-vector with components a
k
(x). Note that a
m
(x) is equal to [k
n1
x]
+
if n n
and to [k
n
x]
+
otherwise. The set-up costs for the components in (3.2), as observed above, are D
T
, D
T
and D
T
p
n+i
respectively. The corresponding forward prices are as in (2.4):
b
1
= 1
b
2
= 1 (3.3)
b
i+2
= p
n+i
, i = 1, . . . , m2.
We let b denote the m-vector of the forward prices. A static portfolio is dened by a vector y whose kth
component is the number of units of the kth asset in the portfolio. The forward set-up cost is y
T
b and
the value at T is y
T
a(M
T
).
With this notation, the problem of determining the most expensive sub-replicating portfolio is equiv-
alent to solving the (primal) semi-innite linear program
P
LB
: sup
yR
m
y
T
b subject to y
T
a(x) (x) x K.
The constraints are enforced only for x K. If n > 0 [ n n] we have a free put with strike k
n
[call with
strike k
n
] and, since is convex, we can extend the sub-replicating portfolio to all of R
+
at no cost.
The key result here is the basic duality theorem of semi-innite linear programming, due to Isii (1960)
and Karlin, see Karlin and Studden (1966). This theorem, stated as Theorem A.1, and its proof are given
in Appendix A. The dual program corresponding to P
LB
is
D
LB
: inf
M
_
K
(x)(dx) subject to
_
K
a(x)(dx) = b
0
,
where M is the set of Borel measures such that each a
i
is integrable. The constraints in D
LB
can be
expressed as satisfying (2.4) for n < i < n. This is simply equivalent to M
P
since, as shown in
Lemma 2.2, any M
P
has support in K. Let V
L
P
and V
L
D
be the values of the primal and dual problems
respectively. It is a general and easily proved fact that V
L
P
V
L
D
. The duality gap is V
L
D
V
L
P
. The
Karlin-Isii theorem gives conditions under which there is no duality gap and we have existence in P
LB
.
Proposition 3.1 We suppose as above that (x) is a convex function on R
+
, nite for all x > 0, and
that (k
i
, p
i
) is a set of normalised put option strike and price pairs which do not admit a weak arbitrage.
If (x) is unbounded as x 0 and n = 0 then we further assume that p
1
/k
1
< p
2
/k
2
. Then V
L
D
= V
L
P
and
there exists a maximising vector y. The most expensive sub-replicating portfolio of a European option with
payo
T
(S
T
) at maturity T is the static portfolio X
as in (2.1) with weights
= F
T
D
T
y
1
,
= y
2
/
T
,
n+i
= y
2+i
for i = 1, . . . , m2 and
i
= 0 otherwise. For this portfolio, X
0
= D
T
F
T
V
L
D
.
If there is existence in the dual problem D
LB
then there is an optimal measure
which is a nite
linear combination of Dirac measures
m
j=1
w
j
xj
(dx) such that each interval [k
j
, k
j+1
) contains at
most one point x
j
. For this measure
(3.4)
(x) > 0 X
ST =FT x
=
n
i=1
i
[K
i
F
T
x]
+
+
T
F
T
x +
D
1
T
=
T
(F
T
x).
Proof. The rst part of the proposition is an application of Theorem A.1. The primal problem P
LB
is feasible because any support line corresponds to a portfolio (containing no options). The functions
a
1
, . . . , a
m
are linearly independent. Recall from Proposition 2.1 that if (k
i
, p
i
) do not admit weak
arbitrage then there is a measure satisfying the conditions (2.4) and such that is a nite weighted sum
of Dirac measures. It follows that
_
R
+
[(x)[(dx) < unless one of the Dirac measures is placed at x = 0
and is unbounded at zero. If n > 0 then there is no mass on the interval [0, n), hence none at zero. When
n = 0 we always have p
1
/k
1
p
2
/k
2
. If p
1
/k
1
= p
2
/k
2
then the payo [k
2
M
T
]
+
k
2
[k
1
M
T
]
+
/k
1
has
null cost and is strictly positive on (0, k
2
). Since p
1
> 0 there must be some mass to the left of k
1
, and this
mass must be placed at 0, else there is an arbitrage opportunity. But then
_
d = + and V
L
D
= +.
The condition in the proposition excludes this case. In every other case there is a realizing measure such
that (0) = 0. Indeed, if p
1
/k
1
< p
2
/k
2
then the extended set of put prices (k, 0), (k
1
, p
1
), . . . , (k
n
, p
n
)
is consistent with absence of arbitrage if k [0, ], where = (k
1
p
2
k
2
p
1
)/(p
2
p
1
) (see Figure 1). Any
model realizing these prices puts weight 0 on the interval [0, k). Thus V
L
D
is nite under the conditions we
have stated. It remains to verify that the vector b belongs to the interior of the moment cone M
m
dened
at (A.1). For this, it suces to note that for all i such that k
i
(k
n
, k
n
) it holds that [k
i
1]
+
< p
i
< k
i
,
and so the condition is satised. We now conclude from Theorem A.1 that V
L
P
= V
L
D
and that we have
existence in the primal problem. The expressions for
etc. follow from the relationships (2.3) between

normalized and un-normalized prices.
Assume now that the dual problem has a solution. Any optimal measure
M
P
satises
_
K
(x)
(dx) = inf
MP
__
K
(x)(dx)
_
.
Recall K = [k
n
, k
n
] and partition K into intervals I
n+1
, . . . , I
nn+1
dened by
I
i
= [k
i1
, k
i
) for i = n + 1, . . . n n and I
nn+1
= [k
nn
, k
n
),
so I
nn+1
= if n n. Lemma 3.2 below asserts that we may take
atomic with at most one atom in

each of the intervals I
i
. By denition the optimal subhedging portfolio X
satises X
T

T
(S
T
) while
our duality result shows that the
expectations of these random variables coincide. Hence X
T
=
T
(S
T
)
a.s. for M
T
= S
T
/F
T
distributed according to
, and (3.4) follows.

Lemma 3.2 Let M
P
and suppose
_
K
[(x)[(dx) < . Dene
(3.5) J
= i n n + 1[(I
i
) > 0.
Now let
be the measure
iI
(I
i
)
xi
,
in which
x
denotes the Dirac measure at x, and for an index i J, x
i
=
I
i
xd(x)
(Ii)
. Then
M
P
and
(3.6)
_
K
(x)
(dx)
_
K
(x)(dx).
Proof of lemma: The inequality (3.6) follows from the conditional Jensen inequality. A direct com-
putation shows that
satises (2.4), so that
M
P
.
It remains now to understand when there is existence in the dual problem. We exclude the case when
T
is ane on some [z, ) which is tedious. We characterise the existence of a dual minimiser in terms
of properties of the solution to the primal problem and also present a set of sucient conditions. Of the
conditions given, (i) would never be encountered in practice (it implies the existence of free call options)
and (iii),(iv) depend only on the function
T
and not on the put prices P
i
. The examples presented in
Section 3.1.1 show that if these conditions fail there may still be existence, but this will now depend on
the P
i
. Condition (ii) is closer to being necessary and sucient, but is not stated in terms of the basic
data of the problem.
Proposition 3.3 Assume
T
is not ane on some half-line [z, ). Then, in the setup of Proposition
3.1, the existence of a minimiser in the dual problem D
LB
fails if and only if
(3.7) n = and X
ST =s
=
T
s +
D
1
T
<
T
(s), for all s K
n
.
In particular, each of the following is a sucient condition for existence of a minimiser in D
LB
:
(i) n < ;
(ii) we have
(3.8) X
ST =Kn
=
T
K
n
+
D
1
T
<
T
(K
n
) and lim
s
T
(s)
T
s = ;
(iii) for any y <
T
(K
n
) there is some x > K
n
such that the point (K
n
, y) lies on a support line to
T
at x;
(iv)
T
satises
(3.9)
_

0
x
T
(dx) = + .
Proof: We consider two cases.
Case 1: n n, i.e. condition (i) holds. In this case the support of any measure M
P
is contained
in the nite union I
n+1
I
n
of bounded intervals, and
n
i=n+1
(I
i
) = 1. Further, p
n1
> k
n
1
and (2.4b) together imply that (I
n
) > 0. Let
j
, j = 1, 2, . . . be a sequence of measures such that
_
K
(x)
j
(dx) inf
MP
__
K
(x)(dx)
_
as j .
By Lemma 3.2, we may and do assume that each
j
is atomic with at most one atom per interval. We
denote w
i
j
=
j
(I
i
) and let x
i
j
denote the location of the atom in I
i
. For deniteness, let x
i
j
= , where
is some isolated point, if
j
has no atom in I
i
, i.e., if w
i
j
= 0. Let A be the set of indices i such that x
i
j
converges to as j , i.e. x
i
j
,= for only nitely many j, and let B be the complementary set of
indices. Then there exists j
such that
iB
w
i
j
= 1 for j > j
. Because the w
i
j
and x
i
j
are contained in
compact intervals there exists a subsequence j
k
, k = 1, 2 . . . and points w
i
, x
i
such that w
i
j
k
w
i
and
x
i
j
k
x
i
as k . It is clear that
iB
w
i
= 1 and that
V
L
D
= lim
k
_
K
(x)
j
k
(dx) =
_
K
(x)
(dx)
where
(dx) =
iB
w
i
x
i
(dx). Similarly, the integrals of 1, x and [k

i
x]
+
converge, so
M
P
.
Finally, since the intervals I
i
are open on the right, it is possible that x
i
I
i+1
. If also x
i+1
I
i+1
we
can invoke Lemma 3.2 to conclude that this 2-point distribution in I
i+1
can be replaced by a 1-point
distribution without increasing the integral. Thus
retains the property of being an atomic measure

with at most one atom per interval. We have existence in the dual problem and, by the arguments above,
X
T
=
T
(S
T
) d
-a.s.
Case 2: n = . Let X
T
(s) denote the payo of the portfolio X
at time T when S
T
= s. Suppose that
a minimiser
exists in D
LB
, which we may take as atomic. Since n = there exists x (k
n
, ) such
that
(x) > 0 and hence, by (3.4), X
T
(F
T
x) = (F
T
x). This shows that (3.7) fails since F
T
x > K
n
.
It remains to show the converse: that if n = but (3.7) fails then a minimiser
exists. Given
our assumption on
T
, if (3.7) fails then K
:= sups 0 : X
T
(s) =
T
(s) [K
n
, ). We continue
with analysis similar to Case 1 above. Here we have intervals I
n
, . . . , I
n
covering K
n
= [k
n
, k
n
), but also
a further interval I
n+1
= [k
n
, ) which does not necessarily have zero mass. Since I
n+1
is unbounded,
the argument above needs some modication. We take a minimizing sequence of point mass measures
j
as in Case 1. If x
n+1
j
converge (on a subsequence) to a nite x
n+1
then we can restrict our attention

to a compact [0, x
n+1
+ 1] and everything works as in Case 1 above. Suppose to the contrary that

liminf x
n+1
j
= , as j . We rst apply the arguments in Case 1 to the sequence
j
of restrictions
of
j
to K
n
. Everything is the same as above except that now
i
w
i
j
1. A subsequence converges to a
sub-probability measure
on K
n
, equal to a weighted sum of Dirac measures as above. Dene
if
1

(K
n
) = 1 and otherwise
+ (1
1
)
x
, where x I
n+1
is to be determined. Whatever
the value of x,
satises conditions (2.4a) and (2.4c) (the put values depend only on
).
Since
j
M
P
,
n+1
i=1
w
i
j
x
i
j
= 1 and in particular
1
n
i=1
w
i
j
x
i
j
w
n+1
j
k
n
.
Taking the limit along the subsequence we conclude that
(3.10) 1
n
i=1
w
i
x
i
k
n
_
1
n
i=1
w
i
_
= k
n
(1
1
).
By the convergence argument of Case 1,
2

_
kn
0
x
(dx) 1. If
1
< 1 we have only to choose
x = (1
2
)/(1
1
) to ensure that the forward condition (2.4b) is also satised. The inequality (3.10)
guarantees that x k
n
. Thus
M
P
and, since n = , x > k
n
and hence K
> K
n
.
We now show that the complementary case
1
= 1 contradicts K
[K
n
, ). Indeed, if
1
= 1 we
have, since n = ,
k
n
1 < p
n
=
_

0
[k
n
x]
+
(dx) =
_
kn
0
(k
n
x)
(dx) = k
n

n
i=1
w
i
x
i
.
Observe that then
(3.11) w
n+1
j
x
n+1
j
= 1
n
i=1
w
i
j
x
i
j

j
1
n
i=1
w
i
x
i
= p
n
k
n
+ 1 > 0.
Since K
[K
n
, ), taking =
T
(K
+ 1) X
(K
+ 1) =
T
(K
+ 1)
T
> 0, we have
(3.12)
T
(s) X
T
(s) (s K
+ 1), s K
+ 1.
Dene now a new function

T
by
T
(s) =
T
(s)1
sK
+ (
T
s +
D
1
T
)1
s>K
so that we have X
T
(S
T
)
T
(S
T
) and, by denition, for S
T
> K
we have X
T
(S
T
) <
T
(S
T
).
It follows that X
is also the most expensive subreplicating portfolio for

T
(s) and hence the primal and
dual problems for and for

(x) =
1
FT
T
(F
T
x) have all the same value. Writing this explicitly and using
(3.12) and(3.11) gives:
0 = lim
j
_

0
((x)

(x))
j
(dx) = lim
j
((x
n+1
j
)

(x
n+1
j
))w
n+1
j
lim
j
(x
n+1
j
K
1)w
n+1
j
= (p
n
k
n
+ 1) > 0,
a contradiction.
We turn to showing that each of (i)(iv) is sucient for existence of a dual minimiser .
2
Obviously
n < is sucient as observed above. We now show that either of (ii) or (iii) implies that K
[K
n
, )
and that (3.12) holds. In the light of the arguments above this will be sucient. X
T
(s) is linear on
[K
n
, ) and
T
is convex hence the dierence of the two either converges to a constant or diverges
to innity. In the latter case the dierence grows quicker than a linear function, more precisely (3.12)
holds. The former case is explicitly excluded in (ii) and is contradictory with (iii) as the line (s) :=
T
s +
D
1
T
+ lim
u
(
T
(u) X
T
(u)) is asymptotically tangential to
T
(s) as s and hence
the condition in (iii) is violated for any y < (K
n
). In particular, it follows that K
< K
n
then we can
add to X
a positive number of call options with strike K

n
and obtain a subreplicating portfolio with
an initial cost strictly larger (recall that n = ) than X
which contradicts the optimality of X
. This
completes the proof that either (ii) or (iii) is sucient for existence.
Finally, we argue that (iii) and (iv) are equivalent. In (iii) the point x satises
T
(x) +(K
n
x) = y, for some [
T
(x),
T
(x+)].
This equation has a solution x for all y <
T
(K
n
) if and only if lim
x
T
(x) x
T
(x) = . The
equivalence with (iv) follows integrating by parts
_

Kn
x
T
(dx) =
_
x
T
(x)
T
(x)
_
Kn
.
3.1.1 Examples with one put option

We consider now examples in which just one put option price is specied. We illustrate dierent cases
when existence in the dual problem holds or fails. For simplicity assume all prices are normalised, i.e.
D
T
= F
T
= 1, and we are given only a single put option with strike k = 1.2. The convex function takes
the form (x) = 1/x +ax
b
. Computation of the most expensive sub-hedging portfolio can be done by a
simple search procedure.
Consider rst the case a = 0, so that (x) = 1/x. The results are shown in Figure 2 with data shown
in Table 1: p is the put price, x
0
, x
1
the points of tangency, w
0
, w
1
the implied probability weights on
x
0
, x
1
and , the units of, respectively, cash and forward in the portfolio.
p = 0.4 is a regular case: we have tangent lines at x
0
, x
1
and the solution to the dual problem puts
weights 8/9, 1/9 respectively on these points. As p increases it is advantageous to include more puts in
the portfolio, so x
1
increases. At p = 0.6 we reach a boundary case where = = 0, and the put is
correctly priced by the Dirac measure with weight 1 at x
0
= 0.6 = k/2. Obviously, this measure does
not correctly price the forward, but it does correctly price the portfolio since = 0. When p > 0.6, the
only way to increase the put component further is to take < 0 (and then clearly the optimal value of
is 0.) When p = 0.7 the optimal value is = 0.8 and we nd that in this and every other such case
the implied weight is w
0
= 1, as the general theory predicts.
Next, take a = 0.25, b = 1, so that (x) = 1/x+0.25x, has its minimum at x = 2 and is asymptotically
linear. We take p = 0.7. Taking the expectation with respect to the limiting measure
gives the value

of the lower sub-hedge in Figure 3, but this is not optimal: we can add a maximum number 0.25 of call
p x
0
x
1
value w
0
w
1
0.4 0.75 3 1.2222 0.6667 -0.1111 0.8889 0.1111
0.6 0.6 - 1.6667 0 0 1.00 -
0.7 0.5 - 2.00 -0.8 0 1.00 -
Table 1: Data for Figure 2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
1
0
1
2
3
4
5

p=0.6
p=0.7
p=0.4
Figure 2: Most expensive subhedging portfolios for (x) = 1/x given a single put option with strike 1.2
and prices p = 0.4, 0.6, 0.7
options, which have positive value, giving the upper sub-hedge in Figure 3. This is optimal, but does not
correspond to any dual measure.
Finally, let a = 0.0625, b = 2, giving (x) = 1/x + 0.0625x
2
. The minimum is still at x = 2 but
is asymptotically quadratic. This function satises condition (iii) of Proposition 3.3 and there is dual
existence for every arbitrage-free value of p. The optimal portfolio for p = 0.7 is shown in Figure 4.
3.2 Upper bound
To compute the cheapest super-replicating portfolio we have to solve the linear program
P
UB
: inf
yR
m
y
T
b subject to y
T
a(x) (x) x K.
The corresponding dual program is
D
UB
: sup
M
_
K
(x)(dx) subject to
_
K
a(x)(dx) = b,
where, by Lemma 2.2, we may replace K by R
+
. By (1.4) we have for x R
+
(x) = (1) +
(1+)(x 1) +
_
(0,1]
[k x]
+
(dk) +
_
(1,)
[x k]
+
(dk).
Consider M
P
and let p
(k) =
_
[k x]
+
(dx), c
(k) =
_
[x k]
+
(dk) be the (normalised) prices of
puts and calls. Integrating the above against gives
(3.13)
_
(x)(dx) = (1) +
_
(0,1]
p
(k)
(dk) +
_
(1,)
c
(k)
(dk).
Recall that c
(k) = p
(k)+1k and hence maximising in c
(k) or in p
(k) is the same. p
(k) is a convex
function dominated by the linear interpolation of points (k
i
, p
i
), i = 0, 1, . . . , n, extended to the right of
(k
n
, p
n
) with slope 1. Since we assume the given put prices do not admit weak arbitrage, it follows from
2
In fact for this part of the Proposition we do not need to impose any additional conditions on
T
apart from convexity.
0 2 4 6 8 10
1
0
1
2
3
4
5
6
Figure 3: The most expensive subhedging portfolio (upper dashed line) for (x) = 1/x + 0.25x given
a single put option with strike 1.2 and price p = 0.7. The lower dashed line is the portfolio priced at
_
(x)
(dx) which is suboptimal.

0 2 4 6 8 10
2
1
0
1
2
3
4
5
6
7
Figure 4: Most expensive subhedging portfolios for (x) = 1/x+0.0625x
2
given a single put option with
strike 1.2 and price p = 0.7.
(Davis and Hobson, 2007) (see also Proposition 2.1 above) that this upper bound is attained either exactly
or in the limit. More precisely, if n = one can take
z
M
P
supported on k
n
, , . . . , , k
n
, z, for z large
enough, which attain the upper bound on [0, k
n
] and asymptotically induce the upper bound on (k
n
, )
as z . It follows from (3.13) that the value of the dual problem is V
U
D
= lim
z
_
K
(x)
z
(dx).
If n n one can take z = k
n
and
k n
attains the upper bound. It follows from (3.13) that then
V
U
D
=
_
K
(x)
k n
(dx).
From this observations, one expects that y the solution to the primal problem will correspond to
a (normalised) portfolio y
T
a(x) which linearly interpolates (k
i
, (k
i
)), i = k
n
, . . . , n n and (if n = )
extends linearly to the right as to dominate (k). The function x y
T
a(x) is piecewise linear with a
nite number of pieces and no such function can majorize the convex function over R
+
unless (0) <
and
() = < . In general we impose:

(a) n > 0 or
_
n = 0 and (0) <
_
and
(b) n n or
_
n = and
() <
_
.
(3.14)
We have the following result.
Proposition 3.4 If condition (3.14) holds then there exists a solution y to the linear program P
UB
. The
function y
T
a(x) is the linear interpolation of the points (k
n
, (k
n
)), , . . . , (k
n n
, (k
n n
)) together with,
if n = , the line l(x) = y
T
a(k
n
) + (x k
n
)
() for x k
n
. Primal and dual problem have the same
value V
P
U
= V
D
U
and the existence of a maximiser in the dual problem fails if and only if n = and
is not ane on [k
n
, ).
If the condition (3.14) is not satised, there is no feasible solution and V
D
U
= .
Proof. As argued above, (3.14) is a necessary condition for existence of a feasible solution. Suppose,
for example, that n = =
(). Then (z k
n
)
z
(z) = c
n
and hence z
z
(z) c
n
as z .
Together with
() = this implies that V

D
U
= . Other cases are similar.
Suppose (3.14) holds and rst consider the case when n = . is bounded on [k
n
, k
n
] and the linear
interpolation is well dened as is the extension beyond k
n
. Further, there exists some constant such
that y
T
a(x) (x) for all x R
+
. The weight y
2+j
on the jth put option is the change in slope
at k
n+j
, the underlying weight y
2
is equal to , and at x = k
n
we have
n
= y
1
+ y
2
k
n
, so the cash
weight is y
1
=
n
k
n
. The value of the objective function is, by denition,
y
T
b =
_
R
+
y
T
a(x)(dx),
for any M
P
. In particular, since y
T
a(k
i
) = (k
i
), taking z large enough, we have
y
T
b
0
=
_
R
+
y
T
a(x)
z
(dx) =
n
i=n
(k
i
)
z
(k
i
) + y
T
a(z)
z
(z)
=
_
R
+
(x)
z
(dx) + ( y
T
a(z) (z))
z
(z)
_
R
+
(x)
z
(dx) +(z).
(3.15)
Recall that
_
x
z
(dx) = 1 and in particular (z) 0 as z . Taking the limit as z in the
above, we conclude that y
T
b
0
V
U
D
. The basic inequality V
U
D
V
U
P
between the primal and dual values
then implies that y is optimal for P
UB
and V
U
D
= V
U
P
. The existence of the solution to D
UB
fails unless
there exists z k
n
with y
T
a(z) = (z), which happens if and only if (z) is ane on [k
n
, ).
When n n the arguments are analogous, except that now we need to ensure y
T
a(x) (x) only for
x K = [k
n
, k
n
]. The dual problem has a maximiser as observed in the remarks above the Proposition.
The primal problem also has a solution y but it is not unique. Indeed, let y
0
= (k
n
, 1, 0, . . . , 0, 1) and
observe that y
0
T
b = 0 and y
0
T
a(x) 0 for x K. In consequence, we can add to y multiples of y
0
without aecting its performance for P
UB
.
We can now summarize the results for the cheapest super-replicating portfolio as in (2.1),(2.2). The
dierence with the above is that we need to ensure super-replication for all possible values of S
T
and not
only for S
T
[K
n
, K
n
]. Since the payo X
T
, as a function of S
T
, is piecewise linear with a nite number
of pieces it is necessary that
T
satises
(3.16)
T
(0) < and
T
() = < .
Under this condition, the above results show that the cheapest super-replicating portfolio has a payo
which linearly interpolates (K
i
,
T
(K
i
)), i = 0, . . . , n and extends to the right of K
n
with slope
T
().
Proposition 3.5 If (3.16) holds then there is a cheapest super-replicating portfolio (
i
) of the
European option with payo
T
(S
T
) whose initial price is
X
0
= sup
z
D
T
_
R
+
T
(F
T
x)
z
(dx).
The underlying component is
=
T
, the cash component is
= D
T
(
T
(K
n
) K
n
) and the option
components are
i
=

T
(K
j+1
)
T
(K
j
)
K
j+1
K
j

T
(K
j
)
T
(K
j1
)
K
j
K
j1
.
If condition (3.16) is not satised, there is no super-replicating portfolio.
3.3 Arbitrage conditions
With the above results in hand we can state the arbitrage relationships when a European option whose
exercise value at T is a convex function
T
(S
T
) can be traded at time 0 at price P
in a market where
there already exist traded put options, whose prices P
i
are in themselves consistent with absence of
arbitrage. Recalling the notation of Propositions 3.1 and 3.5, X
0
and X
0
are respectively the setup costs
of the most expensive sub-replicating and cheapest super-replicating portfolios, with X
0
= + when no
super-replicating portfolio exists.
Theorem 3.6 Assume the put prices do not admit a weak arbitrage. Consider a convex function
T
and
suppose that if
T
is ane on some halfline [z, ) then it is strictly convex on [0, z). The following are
equivalent:
1. The prices P
, P
1
, . . . , P
n
do not admit a weak arbitrage.
2. There exists a market model for put options in which P
= D
T
E[
T
(S
T
)],
3. The following condition (3.17) holds and either P
(X
0
, X
0
), or P
= X
0
and existence holds in
D
LB
, or P
= X
0
< and existence holds in D
UB
.
(3.17) P
2
>
K
2
K
1
P
1
if n = 0 and is unbounded at the origin.
If (3.17) holds and P
/ [X
0
, X
0
] then there is a model-independent arbitrage. If (3.17) holds and
P
= X
0
or X
0
and existence fails in D
LB
or in D
UB
respectively, or if (3.17) fails, then there is a weak
arbitrage.
Remark 3.7 We note that the robust pricing and hedging problem solved above is essentially invariant
if
T
is modied by an ane factor. More precisely, if we consider a European option with payo
1
T
(S
T
) =
T
(S
T
) + S
T
+ and let P
1 denote its price then the prices P

1
, . . . , P
n
, P
are consistent
with absence of arbitrage if and only if P
1
, . . . , P
n
, P
1 = P
+S
0
/
T
+D
T
are.
Proof. Suppose rst that condition (3.17) holds. We saw in the proof of Proposition 3.1 that this condition
(under its equivalent form p
2
> (k
2
/k
1
)p
1
) guarantees the existence of a sub-replicating portfolio with
value X
0
. If P
(X
0
, X
0
) then there exists > 0 such that P
(X
0
+ , X
0
) and, since
there is no duality gap, there are measures
1
,
2
M
P
such that D
T
E
1
[
T
(S
T
)] < X
0
+ and
D
T
E
2
[
T
(S
T
)] > X
0
. A convex combination of
1
and
2
then satises D
T
E
[
T
(S
T
)] = P
and
one constructs a market model, for example by using Skorokhod embedding as explained in Section 2
above. If existence holds in D
LB
then it was shown in the proof of Proposition 3.1 that the minimizing
measure
satises
X
0
= D
T
_
K
T
(F
T
x)
(dx),
so that if P
= X
0
then
is a martingale measure that consistently prices the convex payo

T
and the
given set of put options. The same argument applies on the upper bound side.
Next, suppose that condition (3.17) holds and P
= X
0
but no minimizing measure exists in D
LB
.
Let / be a model and the distribution of S
T
under /. We can, at zero initial cost, buy
T
(S
T
) and
sell the portfolio X
T
and this strategy realizes an arbitrage under / if (
T
(S
T
) > X
T
) > 0. Suppose
now that (
T
(S
T
) > X
T
) = 0 and consider two cases. First, if
T
is not ane on some [z, ) then,
by Proposition 3.3, n = and
T
(S
T
) > X
T
for S
T
K
n
so that in particular ([K
n
, )) = 0. A
strategy of going short a call option with strike K
n
(which has strictly positive price since n = ) gives
an arbitrage since S
T
< K
n
a.s. in /. Second, suppose that
T
(x) 0 for x z but
T
(x) > 0 for x < z.
If ([K
n
, )) = 0 then we construct the arbitrage as previously so suppose this does not hold. Recall the
atomic measure
dened in Case 2 in the proof of Proposition 3.3 and that

1
= 1 as we do not have
existence of a minimiser for the dual problem. (x) in (3.1) is strictly convex on [0, z) and linear on [ z, )
with z = z/F
T
. It is not hard to see that
has to have an atom in some x
n
(k
n1
, k
n
) and that either
z = x
n
or else z k
n
. Otherwise we could modify
to obtain a minimiser for the dual problem. Strict

convexity of
T
on [0, z) implies that there exist at most n points s
1
, . . . , s
m
such that s
i
(K
i1
, K
i
)
and
T
(S
T
) strictly dominates X
T
for other values of S
T
z and hence ([0, z)) = (s
1
, . . . , s
n
). It
follows that support of
is a subset of s
1
/F
T
, . . . , s
n
/F
T
= x
n
. Let > 0 and consider a portfolio Y
with
Y
T
= Y
T
(S
T
) =
n
i=1
i
(K
i
S
T
)
+
+, such that Y
T
(s
j
) = 0, j = 1, . . . n.
This uniquely species
i
R. The payo of Y is simply a zigzag line with kinks in K
i
, zero in each s
i
and equal to for S
T
K
n
. It follows that Y
T
0 = 1 and Y
T
> 0 > 0 as ([K
n
, )) > 0.
However
prices all put options correctly so that the initial price of Y is

Y
0
=
n
i=1
i
P
i
+ D
T
= D
T
n
i=1
i
_
ki
0
(K
i
F
T
m)
(dm) +D
T
=D
T
_
kn
0
_
n
i=1
i
(K
i
F
T
m)
+
+
_
(dm) = D
T
_
kn
0
Y
T
(F
T
m)
(dm) = 0,
(3.18)
by construction of Y since
(s
1
/F
T
, . . . , s
n
/F
T
) = 1 as remarked above, and where we used
1
= 1.
It follows that Y is an arbitrage strategy in /.
Now suppose that condition (3.17) holds and P
= X
0
< and there is no maximising measure in the
dual problem D
UB
. Then, by Proposition 3.4, n = and
T
(S
T
) < X
T
for S
T
> K
n
. Straightforward
arguments as in the rst case above show that there is a weak arbitrage.
Finally, if P
< X
0
a model independent arbitrage is given by buying the European option with payo
T
(S
T
) and selling the subheding portfolio. This initial cost is negative while the payo, since X
subhedges
T
(S
T
), is non-negative. If P
> X
0
we go short in the option and long in the superhedge.
Now suppose (3.17) does not hold, so that P
2
/K
2
= P
1
/K
1
(this is the only case other than (3.17)
consistent with absence of arbitrage among the put options). Consider portfolios with exercise values
H
1
(S
T
) = [K
2
S
T
]
+
K
2
K
1
[K
1
S
T
]
+
H
2
(S
T
) =
T
(S
T
)
T
(K
2
)
0
(K
2
)(S
T
K
2
)
1
P
1
(P
(K
2
)
0
(F
T
K
2
)) [K
1
S
T
]
+
,
where
0
denotes the left derivative. The setup cost for each of these is zero, and H
1
(s) > 0 for s (0, K
2
)
while H
2
(s) as s 0. There is a number 0 such that H(s) = H
1
(s)+H
2
(s) > 0 for s (0, K
2
).
Weak arbitrage is realized by a portfolio whose exercise value depends on a given model /and is specied
via
X
T
(S
T
) =
_
[K
2
S
T
]
+
if P[S
T
[0, K
2
)] = 0
H(S
T
) if P[S
T
[0, K
2
)] > 0.
This completes the proof.
4 Weighted variance swaps
We come now to the second part of the paper where we consider weighted variance swaps. The main
idea, as indicated in the Introduction, is to show that a weighted variance swap contract is equivalent
to a European option with a convex payo and hence their prices have to be equal. The equivalence
here means that the dierence of the two derivatives may be replicated through trading in a model-
independent way. In order to formalise this we need to dene (continuous) trading in absence of a model,
i.e. in absence of a xed probability space. This poses technical diculties as we need to dene pathwise
stochastic integrals.
One possibility is to dene stochastic integrals as limits of discrete sums. The resulting object may de-
pend on the sequence of partitions used to dene the limit. This approach was used in Bick and Willinger
(1994) who interpreted the dierence resulting from dierent sequences of partitions as brokers method
of implementing continuous time trading order. They were then interested in what happens if they apply
the pricing-through-replication arguments on the set of paths with a xed (
2
) realised quadratic vari-
ation and wanted to recover Black-Scholes pricing and hedging. However for our purposes the ideas of
Bick and Willinger (1994) are not suitable. We are interested in a much wider set of paths and then the
replication of a weighted variance swap combining trading and a position in a European option would
depend on the broker (i.e. sequence of partitions used). Instead, as in Lyons (1995), we propose an ap-
proach inspired by the work of Follmer (1981). We restrict the attention to paths which admit quadratic
variation or pathwise local time. For such paths we can develop pathwise stochastic calculus including
It o and Tanaka formulae. As this subject is self-contained and of independent interest we isolate it in
Appendix B. Insightful discussions of this topic are found in Bick and Willinger (1994) and Lyons (1995).
To the standing assumptions (i)(iii) of Section 2 we add another one:
(iv) (S
t
: t T) /
+
the set of strictly positive, continuous functions on [0, T] which admit a nite,
non-zero, quadratic variation and a pathwise local time, as formally dened in Denitions B.1,B.3 and
Proposition B.4 of Appendix B.
Thus, our idea for the framework, as opposed to xing a specic model /, is to assume we are given
a set of possible paths for the price process: (S
t
: t T) T. This could be, for example, the space
of continuous non-negative functions, the space of functions with nite non-zero quadratic variation or
the space of continuous functions with a constant xed realised volatility. The choice of T is supposed
to reect our beliefs about characteristics of price dynamics as well as modelling assumptions we are
willing to take. Our choice above, T = /
+
, is primarily dictated by the necessity to develop a pathwise
stochastic calculus. It would be interesting to understand if an appropriate notion of no-arbitrage implies
(iv). A recent paper of Vovk (2011), based on a game-theoretic approach to probability, suggests one
may exclude paths with innite quadratic variation through a no-arbitrage-like restriction, an interesting
avenue for further investigation.
We introduce now a continuous time analogue of the weighted realised variance (1.1). Namely, we
consider a market in which, in addition to nite family of put options as above, a w-weighted variance
swap is traded. It is specied by its payo at maturity T:
(4.1) RV
w
T
P
RV(w)
T
:=
_
T
0
w(S
t
/F
t
)dlog S
t
P
RV(w)
T
,
where P
RV(w)
T
is the swap rate, and has null entry cost at time 0. The above simplies (1.1) in two ways.
First, similarly to the classical works on variance swaps going back to Neuberger (1994), we consider a
continuously and not discretely sampled variance swap which is easier to analyse with tools of stochastic
calculus. Secondly, the weighting in (1.1) is a function of the asset price h(S
ti
) and in (4.1) it is a
function of the ratio of the actual and the forward prices w(S
t
/F
t
). This departure from the market
contract denition is unfortunate but apparently necessary to apply our techniques. In practice, if w(S
t
)
is the function appearing in the contract denition we would apply our results with w(x) = w(S
0
x), so
that w(S
t
/F
t
) = w((S
0
/F
t
)S
t
). Since maturity times are short and, at present, interest rates are low, we
have S
0
/F
t
1. See below and Section 5 for further remarks.
Our assumption (iv) and Proposition B.6 imply that (log S
t
, t T) /. Theorem B.5 implies that
(4.1) is well dened as long as w L
2
loc
, we can integrate with respect to S
t
or M
t
and obtain an It o
formula. This leads to the following representation.
Lemma 4.1 Let w : R
+
[0, ) be a locally square integrable function and consider a convex C
1
function
w
with
w
(a) =
w(a)
a
2
. The extended Ito formula (B.1) then holds and reads
(4.2)
w
(M
T
) =
w
(1) +
_
T
0
w
(M
u
)dM
u
+
1
2
_
[0,T]
w(M
u
)dln M
u
.
The function
w
is specied up to an addition of an ane component which does not aect pricing
or hedging problems for a European option with payo
w
, see Remark 3.7 above. In what follows we
assume that w and
w
are xed. Three motivating choices of w, as discussed in the Introduction, and
the corresponding functions
w
, are:
1. Realised variance swap: w 1 and
w
(x) = ln(x). In this case there is of course no distinction
between w and the contract function w.
2. Corridor variance swap: w(x) = 1
(0,a)
(x) or w(x) = 1
(a,)
(x), where 0 < a < and
w
(x) =
_
ln
_
x
a
_
+
x
a
1
_
w(x).
Here we would take a = b/S
0
if the contract corridor is (0, b) or (b, )
3. Gamma swap: w(x) = S
0
x and
w
(x) = S
0
(xln(x) x).
Clearly, (4.2) suggests that we should consider portfolios which trade dynamically and this will allow
us to link w-weighted realised variance RV
w
T
with a European option with a convex payo
w
. Note
however that it is sucient to allow only for relatively simple dynamic trading where the holdings in the
asset only depend on assets current price. More precisely, we extend the denition of portfolio X from
static portfolios as in (2.1)-(2.2) to a class of dynamic portfolios. We still have a static position in traded
options. These are options with given market prices at time zero and include n put options but could
also include another European option, a weighted variance swap or other options. At time t we also hold
t
(M
t
) assets S
t
and
t
/D
t
in cash. The portfolio is self-nancing on (0, T] so that
(4.3)
t
:= (M
0
)S
0
+(0, S
0
) +
_
t
0
(M
u
)dM
u
t
(M
t
)S
t
D
t
, t (0, T],
and where is implicitly assumed continuous and with a locally square integrable weak derivative so
that the integral above is well dened, cf. Theorem B.5. We further assume that there exist: a linear
combination of options traded at time zero with total payo Z = Z(S
t
: t T), a convex function G and
constants

,

such that
(4.4)
t
(M
t
)S
t
+
t
/D
t
Z G(M
t
)/D
t
+

t
S
t
+

/D
t
, t T.
Such a portfolio X is called admissible. Observe that, in absence of a model, the usual integrability of
Z is replaced by having nite price at time zero. In the classical setting, the admissibility of a trading
strategy may depend on the model. Here admissibility of a strategy X may depend on which options are
assumed to trade in the market. The presence of the term G(M
t
) on the RHS will become clear from the
proof of Theorem 4.3 below. It allows us to enlarge the space of admissible portfolios for which Lemma
4.2 below holds.
The two notions of arbitrage introduced in Section 2 are consequently extended by allowing not only
static portfolios but possibly dynamic admissible portfolios as above. All the previous results remain
valid with the extended notions of arbitrage. Indeed, if given prices admit no dynamic weak arbitrage
then in particular they admit no static weak arbitrage. And for the reverse, we have the following general
result.
Lemma 4.2 Suppose that we are given prices for a nite family of co-maturing options
3
. If a market
model / exists for these options then any admissible strategy X satises E[D
T
X
T
] X
0
. In particular,
the prices do not admit a weak arbitrage.
Proof. Let / be a market model and X be an admissible strategy. We have X
T
= Z
1
+Y
T
, where Z
1
is a linear combination of payos of traded options and Y
t
=
t
(M
t
)S
t
+
t
/D
t
satises (4.4). Using
(4.3) it follows that
D
t
Y
t
= (1)S
0
+(0, S
0
) +
_
t
0
(M
u
)dM
u
D
t
Z G(M
t
) +

+
S
0
M
t
+

.
We may assume that G 0, it suces to replace G by G
+
. / is a market model and in particular Z
is an integrable random variable. Since the traditional stochastic integral and our pathwise stochastic
integral coincide a.s. in /, see Theorem B.7, we conclude that D
t
Y
t
is a local martingale and so is
N
t
:= D
t
Y
t

+
S
0
M
t
. We will argue that this implies EN
t
N
0
. Let
n
be the localising sequence
for N so that EN
tn
= N
0
. In what follows all the limits are taken as n . Fatous lemma shows
that EN
+
t
liminf EN
+
tn
and EN
t
liminf EN
tn
. By Jensens inequality the process G(M
t
) is a
submartingale, in particular the expectation is increasing and EG(M
t
) EG(M
T
) which is nite since
/ is a market model. Using the Fatou lemma we have
limEG(M
tn
) EG(M
t
) = Eliminf G(M
tn
) liminf EG(M
tn
),
showing that limEG(M
tn
) = EG(M
t
). Observe that N
t
is bounded below by

Z G(M
t
), where

Z is
an integrable random variable and G is convex. Using Fatou lemma again we can write
EG(M
t
)
+
+E
liminf EN
tn
= liminf E
_
G(M
tn
)
+
+

Z
tn
_
E
_
G(M
t
)
+
+

Z
t
_
which combined with the above gives EN
t
= liminf EN
tn
and in consequence EN
t
liminf EN
tn
=
N
0
, as required. This shows that E[D
T
Y
T
] Y
0
. Since in a market model expectations of discounted
payos of the traded options coincide with their initial prices it follows that E[D
T
X
T
] X
0
0. In
particular if X
T
0 and X
0
0 then X
T
= 0 a.s. and the prices do not admit a weak arbitrage.
Having extended the notions of (admissible) trading strategy and arbitrage, we can now state the
main theorem concerning robust pricing of weighted variance swaps. It is essentially a consequence of
the hedging relation (4.2) and the results of Section 3.
3
These could include European as well as exotic options.
Theorem 4.3 Suppose in the market which satises assumptions (i)(iv) the following are traded at time
zero: n put options with prices P
i
, a wweighted variance swap with payo (4.1) and a European option
with payo
T
(S
T
) = F
T
w
(M
T
) and price P
w
. Assuming the put prices do not admit a weak arbitrage,
the following are equivalent
1. The option prices (European options and weighted variance swap) do not admit a weak arbitrage.
2. P
1
, . . . , P
n
, P
w
do not admit a weak arbitrage and
(4.5) P
RV(w)
T
=
2P
w
D
T
F
T
2
w
(1).
3. A market model for all n + 2 options exists.
Remark 4.4 It is true that under (4.5) a market model for P
1
, . . . , P
n
, P
RV(w)
T
exists if and only if a
market model for P
1
, . . . , P
n
, P
w
exists. By Theorem 3.6, this is yet equivalent to P
1
, . . . , P
n
, P
w
being
consistent with absence of arbitrage. However it is not clear if this is equivalent to P
1
, . . . , P
n
, P
RV(w)
T
being consistent with absence of arbitrage. This is because the portfolio of the variance swap and dynamic
trading necessary to synthesise
T
(S
T
) payo may not be admissible when
T
(S
T
) is not a traded option.
Remark 4.5 The formulation of Theorem 4.3 involves no-arbitrage prices but these are enforced via
robust hedging strategies detailed in the proof. They involve the European option with payo
T
(S
T
)
which in practice may not be traded and should be super-/sub- replicated using Propositions 3.1 and 3.4.
Remark 4.6 . It would be interesting to combine our study with the results of Hobson and Klimmek
(2011) [HK] already alluded to in the Introduction. We note however that this may not be straightforward
since the European option constituting the static part of the hedge in HK need not be convex and the
variance kernels we consider are not necessarily monotone (in the terminology of HK), for example the
Gamma swap y(log(y/x))
2
. Finally we note that the bounds in HK are attained by models where quadratic
variation is generated entirely by a single large jump which is a radical departure from the assumption
of continuous paths. Whether it is possible to obtain sharper bounds which only work for reasonable
discontinuous paths is an interesting problem. We leave these challenges to future research.
Proof. We rst show that 2 = 3. Suppose that P
1
, . . . , P
n
, P
w
do not admit a weak arbitrage and
let / be a market model which prices correctly the n puts and the additional European option with
payo
T
(S
T
). Note that, from the proof of Theorem 3.6, / exists and may be taken to satisfy (i)(iv).
Consider the It o formula (4.2) evaluated at
n
T = inft 0 : M
t
/ (1/n, n) T instead of T. The
continuous function
w
is bounded on (1/n, n), the stochastic integral is a true martingale and taking
expectations we obtain
E[
w
(M
nT
)] =
w
(1) +
1
2
E
_
_
nT
0
w(M
u
)dM
u
_
.
Subject to adding an ane function to
w
we may assume that
w
0. Jensens inequality shows
that
w
(M
nT
), n 2, is a submartingale. Together with the Fatou lemma this shows that the LHS
converges to E[
w
(M
T
)] as n . Applying the monotone convergence theorem to the RHS we obtain
E[
w
(M
T
)] =
w
(1) +
1
2
E[RV
w
T
] ,
where either both quantities are nite or innite. Since / is a market model for puts and
T
(S
T
), we
have E[
w
(M
T
)] = P
w
/(D
T
F
T
). Combining (4.5) with the above it follows that E
_
RV
w
T
P
RV(w)
T
_
= 0
and hence / is a market model for puts,
T
(S
T
) and the wwighted variance swap.
Lemma (4.2) implies that 3 =1. We note also that, if we have a market model / for all the puts,
w
(S
T
) option and the wwighted variance swap then by the above (4.5) holds. Then Lemma (4.2) also
implies that 3 = 2.
It remains to argue that 1 = 2 i.e. that if P
1
, . . . , P
n
, P
w
are consistent with absence of arbitrage
but (4.5) fails then there is a weak arbitrage. Consider a portfolio X with no put options, (0, S
0
) =
2D
T
(
w
(1) S
0
w
(1)) +D
T
P
RV(w)
T
and (m) = 2D
T
w
(m). The setup cost of X is X
0
= D
T
(P
RV(w)
T
+
2
w
(1)). By assumption,
w
is C
1
with
w
(x) = w(x)/x
2
L
2
loc
so that can apply Theorem B.5. Then,
using (4.3) and (4.2) we obtain
X
t
=
t
(M
t
)S
t
+
t
/D
t
= (S
0
)S
0
/D
t
+(0, S
0
)/D
t
+
1
D
t
_
t
0
(M
u
)dM
u
=
D
T
D
t
_
2
w
(1) + 2
_
t
0
w
(M
u
)dM
u
+P
RV(w)
T
_
=
2D
T
D
t
w
(M
t
)
D
T
D
t
_
[0,t]
w(S
u
)dln M
u
+
D
T
D
t
P
RV(w)
T
.
Observing that 1 D
T
/D
t
D
T
and
_
[0,t]
w(S
u
)dln M
u
is increasing in t it follows that both X and
X are admissible. Suppose rst that
(4.6) P
RV(w)
T
<
2P
w
D
T
F
T
2
w
(1).
Consider the following portfolio Y : short 2/F
T
options with payo
T
(S
T
), long portfolio X and long a
wweighted variance swap. Y is admissible, the initial cost is
Y
0
= 2P
w
/F
T
+X
0
= 2P
w
/F
T
+D
T
(P
RV(w)
T
+ 2
w
(1)) < 0,
while Y
T
= 0 and hence we have a model independent arbitrage. If a reverse inequality holds in (4.6)
then the arbitrage is attained by Y .
5 Computation and comparison with market data

5.1 Solving the lower bound dual problem
When one of the sucient conditions given in Proposition 3.3 is satised, for existence in the lower bound
dual problem, then this problem can be solved by a dynamic programming algorithm. We briey outline
this here, referring the reader to Raval (2010) for complete details. For simplicity, we restrict attention
to the practically relevant case n = 0, n = . Of course, once the dual problem is solved, the maximal
sub-hedging portfolio is immediately determined.
The measure
to be determined satises
_
(x)
(dx) = inf
MP
__
R
+
(x)(dx)
_
.
and we recall from Lemma 3.2 that we can restrict our search to measures of the form (dx) =
n+1
i=1
w
i
i
(dx) where
i
[k
i1
, k
i
), w
i
0,
i
w
i
= 1. We denote
0
= 0 and, for i 1,
i
=
i
1
w
j
,
the cumulative weight on the interval [0, k
i
). For consistency of the put prices r
1
, . . . , r
n
with absence of
arbitrage, Proposition 2.1 dictates that
i
A
i
=
_
r
i
r
i1
k
i
k
i1
,
r
i+1
r
i
k
i+1
k
i
_
for 1 i < n
and
n
A
n
=
_
r
n
r
n1
k
n
k
n1
, 1
_
.
Given = (
1
, . . . ,
n
) (the nal weight is of course w
n+1
= 1
n
), the positions
i
are determined by
pricing the put options. We nd that when
i1
<
i
i
=
i
(
i1
,
i
) = k
i
+

i1
(k
i
k
i1
) (r
i
r
i1
)
i

i1
for i = 1, . . . , n,
n+1
=
n+1
(
n
) = k
n
+
1 +r
n
k
n
1
n
The measure corresponding to policy is thus
(5.1)
n
i=1
(
i

i1
)
i(i1,i)
+ (1
n
)
n+1(n)
.
It follows that the minimisation problem inf
MP
_
K
(x)(dx) has the same value as
(5.2) v
0
= inf
1A1
. . . inf
nAn
_
n
i=1
(
i

i1
)(
i
(
i1
,
i
)) + (1
n
)(
n+1
(
n
))
_
.
We can solve this by backwards recursion as follows. Dene
(5.3)
V
n
(
n
) = (1
n
)
n+1
(
n
)
V
j
(
j
) = inf
j+1Aj+1, j+1j
(
j+1

j
)(
i
(
j
,
j+1
)) +V
j+1
(
j+1
), j = n 1, . . . , 0.
Then V
0
(0) = v
0
. For a practical implementation one has only to discretize the sets A
j
, and then (5.3)
reduces to a discrete-time, discrete-state dynamic program in which the minimization at each step is just
a search over a nite number of points.
5.2 Market data
The vanilla variance swap is actively traded in the over-the-counter (OTC) markets. We have collected
variance swap and European option data on the S&P 500 index from the recent past. Using put option
prices, the lower arbitrage-bound for the variance swap rate is computed in each case, and summarised
in Table 2. One sees that the traded price of the variance swap frequently lies very close to the lower
bound. Nonetheless, under our standing assumptions of frictionless markets, all but one of the prices
were consistent with absence of arbitrage. The crash in October 2008 of the S&P 500, and indeed the
nancial markets in general, gave rise to signicant increase in expected variance, which can be seen in
the cross-section of data studied. One data point, the 3-month contract on 20/12/2008, lies below our
lower bound and, at rst sight, appears to represent an arbitrage opportunity. However, this data point
should probably be discarded. First, practitioners tell us that this was a day of extreme disruption in
the market, and indeed the nal column of Table 2, giving the number of traded put prices used in the
calculations, shows that December 19 and 20 were far from typical days. Second, the fact that the quotes
for 19 and 20 December are exactly the same makes it almost certain that the gure for 20 December is
a stale quote, not a genuine trade. It is a positive point of our method that we are able to pick up such
periods of market dislocation, just from the raw price data.
A further point relates to our discussion in Section 4 about the weight function w and its relation to
the contract weight w. This is a moot point here, since w = w = 1, but note from the nal column in
Table 2 that Libor rates are generally around 3% (albeit with some outliers), while the S&P500 dividend
yield increased from around 2.0% to around 3.1% over the course of 2008. Since the rate closely matches
the dividend yield we have F
t
= S
0
to a close approximation for t up to a few months.
A The Karlin & Isii Theorem
Let a
1
, . . . , a
m
, f be real-valued, continuous functions on K R
d
and let M denote the collection of all
nite Borel measures on K fullling the integrability conditions
_
K
[a
i
(x)[(dx) < for i = 1, . . . , m.
For a xed vector b = (b
1
, . . . , b
m
)
T
, and letting a(x) = (a
1
(x), . . . , a
m
(x))
T
for x K, consider the
optimisation problem
(P) : sup
yR
m
y
T
b s.t. y
T
a(x) f(x) x K.
Now, dene
(D) : inf
M
_
K
f(x)(dx) s.t.
_
K
a(x)(dx) = b,
where the constraint should be interpreted as
_
K
a
i
(x)(dx) = b
i
, i = 1, . . . , m. The values of the
problems (P) and (D) will respectively be denoted by V (P) and V (D). Finally, M
m
R
m
will denote
the moment cone dened by
(A.1) M
m
=
_
b = (
b
1
, . . . ,
b
m
)
T
[
b
i
=
_
K
a
i
(x)(dx), i = 1, . . . , m, M
_
.
Table 2: Historical variance swap (VS) quotes for the S&P 500 index and the lower bound (LB) for
it, implied by the bid prices of liquid European put options with the same maturity. The units for
the variance price and LB are volatility percentage points, 100
_
P
VS
T
, and M stands for months. The
European option price data is courtesy of UBS Investment Bank, and the variance-swap data was provided
by Peter Carr.
Term Quote date VS quote LB # puts Libor
2M 21/04/2008 21.24 20.10 50 2.79
2M 21/07/2008 22.98 22.51 50 2.79
2M 20/10/2008 48.78 46.58 93 4.06
2M 20/01/2009 52.88 47.68 82 1.21
3M 31/03/2008 25.87 23.59 42 2.78
3M 20/06/2008 22.99 21.21 46 2.76
3M 19/09/2008 26.78 25.68 67 3.12
3M 19/12/2008 45.93 45.38 112 1.82
3M 20/12/2008 45.93 65.81 137 1.82
6M 24/03/2008 25.81 25.34 33 2.68
6M 20/06/2008 23.38 23.20 38 3.10
Theorem A.1 Suppose
1. a
1
, . . . , a
m
are linearly independent over K,
2. b is an interior point of M
m
and
3. V (D) is nite.
Then V (P) = V (D) and (P) has a solution.
Remarks (i) The beauty in the Karlin & Isii theorem is that the proof draws upon no more than a
nite dimensional separating hyperplane theorem. Proofs can be found in Glasho (1979) and Karlin &
Studden Karlin and Studden (1966, Chapter XII, Section 2). We follow the latter below.
(ii) Condition 1 of the theorem ensures that the moment cone M
m
of (A.1) is m-dimensional, i.e. not
contained in some lower-dimensional hyperplane.
Proof of Theorem A.1. Dene the enlarged moment cone
M =
_
b = (
b
1
, . . . ,
b
m+1
)
T
:

b =
_
K
_
a(x)
f(x)
_
(dx), M
_
,
and let

M denote its closure. Then

M is a closed convex cone and, moreover, the vector (b
1
, . . . , b
m
, V (D))
lies on its boundary. There exists a supporting hyperplane to

M through this vector, specied by real
constants z
i
m+1
1
not all zero, such that
m
i=1
z
i
b
i
+z
m+1
V (D) = 0 (A.2)
and
m+1
i=1
z
i
b
i
0,
b

M. (A.3)
In particular, on considering Dirac measures, one has
(A.4)
m
i=1
z
i
a
i
(x) +z
m+1
f(x) 0, x K.
We now show that z
m+1
> 0. Indeed, for any > 0, the vector (b
1
, . . . , b
m
, V (D) ) lies in the
half-space complimentary to (A.3). Therefore
m
i=1
z
i
b
i
+z
m+1
(V (D) ) < 0, > 0.
This clearly implies z
m+1
0. However, z
m+1
= 0 is not possible, since this would contradict the
assumption that b lies in the interior M
m
, which is m dimensional. Thus, it must be that z
m+1
> 0.
Then from (A.4), it follows that
f(x)
m
i=1
y
i
a
i
(x),
where y
i
:= z
i
/z
m+1
for i = 1, . . . , m. Then (A.2) becomes
V (D) =
m
i=1
y
i
b
i
,
and this completes the proof.
B Pathwise stochastic calculus
This section describes a non-probabilistic approach to stochastic calculus, due to Follmer (1981), that
will enable us to dene the continuous-time limit of the nite sums (1.1) dening the realized variance,
without having to assume that the realized price function t S
t
is a sample function of a semimartingale
dened on some probability space.
For T > 0 let T = [0, T] with Borel sets B
T
. A partition is a nite, ordered sequence of times
= 0 = t
0
< t
1
< < t
k
= T with mesh size m() = max
1jk
(t
j
t
j1
). We x a nested sequence
of partitions
n
, n = 1, 2, . . . such that lim
n
m(
n
) = 0. All statements below relate to this specic
sequence. An obvious choice would be the set of dyadic partitions t
n
j
= jT/2
n
, j = 0, . . . , 2
n
, n = 1, 2, . . ..
Denition B.1 A continuous function X : T R has the quadratic variation property if the sequence
of measures on (T , B
T
)
n
=
tjn
(X(t
j+1
) X(t
j
))
2
tj
(where
t
denotes the Dirac measure at t) converges weakly to a measure , possibly along some sub-
sequence. The distribution function of is denoted X
t
, and we denote by Q the set of continuous
functions having the quadratic variation property.
Theorem B.2 ((Follmer, 1981)) For X Q and f C
2
, the limit
_
t
0
f
(X
s
)dX
s
:= lim
n
tjn
f
(X
tj
)(X
tj+1
X
tj
)
is well dened and satises the pathwise Ito formula:
(B.1) f(X
t
) f(X
0
) =
_
t
0
f
(X
s
)dX
s
+
1
2
_
t
0
f
(X
s
)dX
s
.
Proof. (Sketch). The proof proceeds by writing the expansion
(B.2) f(X
t
) f(X
0
) =
j
f
(X
tj
)(X
tj+1
X
tj
) +
1
2
j
f
(X
tj
)(X
tj+1
X
tj
)
2
+
j
R(X
tj
, X
tj+1
)
where R(a, b) ([b a[)(b a)
2
with an increasing function, (c) 0 as c 0. As m() 0,
the third term on the right of (B.2) converges to 0 and the second term converges to the second term in
(B.1). This shows that the rst term converges, so that essentially the pathwise stochastic integral is
dened by (B.1) for arbitrary C
1
functions f
.
An important remark is that X
t
, being continuous, achieves its minimum and maximum X
, X
in
[0, T], so only the values of f in the compact interval [X
, X
] (depending on the path X) are relevant

in (B.1).
For the applications in Section 4 we need an Ito formula valid when f
is merely locally integrable,

rather than continuous as it is in (B.1). In the theory of continuous semimartingales, such an extension
proceeds via local time and the Tanaka formulasee Theorem VI.1.2 of Rogers and Williams (2000).
Here we present a pathwise version, following the diploma thesis of Wuermli (1980). For a C
2
function
f we have f(b) = f(a) +
_
b
a
f
(y)dy, and applying the same formula to f
we obtain
f(b) f(a) =
_
b
a
_
f
(a) +
_
y
a
f
(u)du
_
dy = f
(a)(b a) +
_
b
a
(b u)f
(u)du
= f
(a)(b a) +
_

1
[ab,ab]
(u)[b u[f
(u)du.
Hence for any partition = t
j
of [0, t] we have the identity
(B.3) f(X
t
) f(X
0
) =
j
f
(X
tj
)(X
tj+1
X
tj
) +
_

j
_
1
[X
min
j
,X
max
j
]
(u)[X
tj+1
u[
_
f
(du),
where X
min
j
= minX
tj
, X
tj+1
and X
max
j
= maxX
tj
, X
tj+1
. We dene
(B.4) L
t
(u) = 2
j
1
[X
min
j
,X
max
j
]
(u)[X
tj+1
u[,
and note that L
t
(u) = 0 for u / (X
, X
).
Denition B.3 Let / be the set of continuous paths on [0, T] such that the discrete pathwise local time
L
n
t
(u) converges weakly in L
2
(du) to a limit L
t
(), for each t [0, T].
Proposition B.4 / Q. For X / and t [0, T] we have the occupation density formula
(B.5)
_
A
L
t
(u)du =
_
t
0
1
A
(X
s
)dX
s
, A B(R).
Proof. Suppose X /, and let
n
be a sequence of partitions of [0, T] with m(
n
) 0. From (B.3) we
have, for f C
2
and t [0, T],
(B.6) f(X
tn
) f(X
0
)
j
f
(X
tj
)(X
tj+1
X
tj
) =
1
2
_

L
n
t
(u)f
(u)du,
where

t
n
is the nearest partition point in
n
to t. As n , the rst term on the left of (B.6)
converges to f(X
t
) and, since f
L
2
([X
, X
], du), X / implies that the right-hand side converges

to (1/2)
_
L
t
(u)f
(u)du. Hence the integral term on the left of (B.6) also converges to, say, I
t
. If we
take f(x) = x
2
then the left-hand side of (B.6) is equal to
j:tj+1
tn
X
2
tj+1
X
2
tj
2X
tj
(X
tj+1
X
tj
) =
j:tj+1
tn
(X
tj+1
X
tj
)
2
,
showing that (a) X Q and (b) I
t
=
_
t
0
f
(X
s
)dX
s
, the Follmer integral of Theorem B.2. It now follows
from (B.6) that
(B.7)
_

L
t
(u)f
(u)du =
_
t
0
f
(X
s
)dX
s
,
i.e., for any continuous function g we have
_

L
t
(u)g(u)du =
_
t
0
g(X
s
)dX
s
.
Approximating the indicator function 1
A
by continuous functions and using the monotone convergence
theorem, we obtain (B.5).
For the next result, let J
2
be the set of functions f in C
1
(R) such that f
is weakly dierentiable
with derivative f
in L
2
loc
(R).
Theorem B.5 If X / the Follmer integral extends in such a way that the pathwise Ito formula (B.1)
is valid for f J
2
.
Proof. Let be a mollier function, a non-negative C
function on R such that (x) = 0 for [x[ > 1 and

_
(x)dx = 1, dene
n
(x) = n(nx) and let f
n
(x) =
_
f(xy)
n
(y)dy. Then f
n
, f
n
converge pointwise
to f, f
and f
n
f
weakly in L
2
. From Proposition B.4 we know that X Q and as f
n
C
2
we have,
using (B.1) and (B.7),
f
n
(X
t
) f
n
(X
0
) =
_
t
0
f
n
(X
s
)dX
s
+
1
2
_

L
t
(u)f
n
(u)du.
As n , the left-hand side converges to f(X
t
) f(X
0
) and the second term on the right converges to
1
2
_
L
t
(u)f
(u)du, and we can dene

_
t
0
f
n
(X
s
)dX
s
= lim
n
_
f
n
(X
t
) f
n
(X
0
)
1
2
_

L
t
(u)f
n
(u)du
_
.
It now follows from Proposition B.4 that the It o formula (B.1) holds for this extended integral.
We need one further result.
Proposition B.6 Let X / and let f : R R be a monotone C
2
function. Then Y = f(X) / and
the pathwise local times are related by
(B.8) L
Y
t
(u) = [f
(f
1
(u))[ L
X
t
(f
1
(u)).
Proof. We assume f is increasingthe argument is the same if it is decreasingand denote v = f
1
(u),
so u = f(v). For the partition
n
we have for xed t T , from (B.4),
L
Y,n
t
(u) =
j
1
{f(X
min
n,j
)f(v)f(X
max
n,j
)}
[f(X
t
n
j+1
) f(v)[
=
j
1
{X
min
n,j
)vX
max
n,j
)}
[f
(v)(X
t
n
j+1
v)) +
1
2
f
()(X
t
n
j+1
v)
2
[
= f
(v)
j
1
{X
min
n,j
)vX
max
n,j
)}
X
t
n
j+1
v +
f
2f
(X
t
n
j+1
v)
2
(B.9)
for between v and X
t
n
j+1
. Noting that f
()/2f
(v) is bounded for , v [X
, X
] and that
lim
n
max
j
[X(t
n
j+1
) X(t
n
j
)[ = 0, we easily conclude that if X / then the expression at (B.9)
converges in L
2
(dv) to f
(v)L
X
t
(v), so that Y / with local time given by (B.8).
We note that in the above result it suces to assume that f is dened on [X
, X
]. This allows us to
apply the Proposition for f = log and and stock price trajectories in Section 4. Indeed, from (B.8) we
obtain the elegant formula
L
log X
t
(u) = e
u
L
X
t
(e
u
).
Finally, we build the connection between the pathwise calculus and the classical It o (stochastic)
calculus.
Theorem B.7 Let (X
t
, t [0, T]) be a continuous semimartingale on some complete probability space
(, T, P). Then there is a set N T such that PN = 0 and for / N the path t X(t, ) belongs to
/ (and hence to Q), and L
t
(u) dened in Denition B.3 coincides with the semimartingale local time of
X
t
at u.
Proof. First, X
t
Q a.s. Indeed, Theorem IV.1.3 of Revuz and Yor (1994) asserts that discrete approx-
imations to the quadratic variation always converge in probability, so a sub-sequence converges almost
surely, showing that X
t
Q in accordance with Denition B.1. It is shown by Follmer (1981) that the
It o integral and the pathwise integral dened by (B.1) coincide almost surely. Every semimartingale S
t
has an associated local time which satises the occupation density formula (B.5). It remains to show that
with probability 1 the discrete approximations L
t
dened by (B.4) converge in L
2
(du). This is proved
in Wuermli (1980), by detailed estimates which we cannot include here.
References
Acciaio, B., H. Follmer and I. Penner (2011): Risk assessment for uncertain cash ows: Model ambiguity,
discounting ambiguity, and the role of bubbles, Finance and Stochastics 16(4): 669709.
Bick, A. and W. Willinger (1994): Dynamic spanning without probabilities, Stochastic Proc. Appl.
50(2), 349374.
Cont, R. (2006): Model uncertainty and its impact on the pricing of derivative instruments, Math.
Finance 16(3), 519 547.
Cox, A. M. G. and J. Ob loj (2011a): Robust hedging of double touch barrier options, SIAM Journal on
Financial Mathematics 2, 141182.
Cox, A. M. G. and J. Ob loj (2011b): Robust hedging of double no-touch barrier options, Finance and
Stochastics 15(3), 573605.
Crosby, J. and M. Davis (2012): Variance derivatives: pricing and convergence, Available at
http://ssrn.com/abstract=2049278.
Davis, M. and D. Hobson (2007): The range of traded option prices, Mathematical Finance 17, 114.
Delbaen, F. and W. Schachermayer (1994): A general version of the fundamental theorem of asset
pricing, Math. Ann. 300(3), 463520.
Follmer, H. (1981): Calcul dIt o sans probabilites, in: Seminaire de Probabilites XV (Univ. Stras-
bourg, 1979/1980), vol. 850 of Lecture Notes in Math., Springer, Berlin, 143150, (Available at
www.numdam.org. English translation in Sondermann (2006)).
Follmer, H., A. Schied, and S. Weber (2009): Robust preferences and robust portfolio choice, in: em-
phMathematical Modelling and Numerical Methods in Finance, eds. P. Ciarlet, A. Bensoussan and Q.
Zhang, vol. 15 of Handbook of Numerical Analysis. Elsevier, 2987.
Gatheral, J. (2006): The Volatility Surface: a Practitioners Guide. Wiley, New York.
Glasho, K. (1979): Duality theory of semi-innite programming, in: Semi-innite programming (Proc.
Workshop, Bad Honnef, 1978), vol. 15 of Lecture Notes in Control and Information Sci. Springer,
Berlin, 116.
Hansen, L. P. and T. J. Sargent (2010): Wanting robustness in macroeconomics, in: Handbook of Mon-
etary Economics, eds. B. M. Friedman and M. Woodford, vol. 3 of Handbook of Monetary Economics,
Elsevier, 1097 1157.
Hobson, D. and M. Klimmek (2012): Model independent hedging strategies for variance swaps, Finance
and Stochastics 16(4), 611649.
Hobson, D. G. (1998): Robust hedging of the lookback option, Finance and Stochastics 2(4), 329347.
Isii, K. (1960): The extrema of probability determined by generalized moments. I. Bounded random
variables, Ann. Inst. Statist. Math. 12, 119134; errata, 280.
Karlin, S. and W. Studden (1966): Tchebyche systems: With applications in analysis and statistics,
Pure and Applied Mathematics, vol. XV. Wiley Interscience.
Keller-Ressel, M. and J. Muhle-Karbe (2012): Asymptotic and exact pricing of options on variance, to
appear in Finance and Stochastics, available at arXiv:1003.5514v3.
Lyons, T. J. (1995): Uncertain volatility and the risk-free synthesis of derivatives, Applied Math Finance
2(2), 117133.
Neuberger, A. (1994): The log contract: A new instrument to hedge volatility, Journal of Portfolio
Management 20(2), 7480.
Ob loj, J. (2004): The Skorokhod embedding problem and its ospring, Probability Surveys 1, 321392.
Raval, V. (2010): Arbitrage bounds for prices of options on realized variance, Ph.D. thesis, University of
London (Imperial College).
Revuz, D. and M. Yor (1994): Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin.
Rogers, L. C. G. and D. Williams (2000): Diusions, Markov processes, and martingales, Vol. 2, Cam-
bridge University Press.
Sondermann, D. (2006): Introduction to stochastic calculus for nance: A new didactic approach, vol.
579 of Lecture Notes in Economics and Mathematical Systemtems. Springer-Verlag, Berlin.
Wuermli, M. (1980): Lokalzeiten f ur Martingale, unpublished diploma thesis supervised by Professor H.
Follmer, Universitat Bonn.
Vovk, V. (2012): Continuous-time trading and the emergence of probability, Finance and Stochastics
16(4): 561609.

Arbitrage Bounds For Prices of Weighted Variance Swaps

Uploaded by

Copyright:

Available Formats

Arbitrage Bounds For Prices of Weighted Variance Swaps

Uploaded by

Document Information

Original Description:

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Arbitrage Bounds For Prices of Weighted Variance Swaps

Uploaded by

Copyright:

Available Formats

a

Department of Mathematics, Imperial College London, London SW72AZ, UK (mark.davis@imperial.ac.uk).

wherever these are smaller than T. If RV

denotes the left-hand derivative. If these con-

as in (2.1) with weights

etc. follow from the relationships (2.3) between

atomic with at most one atom in

expectations of these random variables coincide. Hence X

, and (3.4) follows.

satises (2.4), so that

(dx). Similarly, the integrals of 1, x and [k

retains the property of being an atomic measure

(x) > 0 and hence, by (3.4), X

then we can restrict our attention

+ 1] and everything works as in Case 1 above. Suppose to the contrary that

is also the most expensive subreplicating portfolio for

a positive number of call options with strike K

which contradicts the optimality of X

3.1.1 Examples with one put option

gives the value

(k)+1k and hence maximising in c

(k) is the same. p

(dx) which is suboptimal.

() = < . In general we impose:

() = this implies that V

1 denote its price then the prices P

is a martingale measure that consistently prices the convex payo

dened in Case 2 in the proof of Proposition 3.3 and that

has to have an atom in some x

to obtain a minimiser for the dual problem. Strict

prices all put options correctly so that the initial price of Y is

5 Computation and comparison with market data

] (depending on the path X) are relevant

is merely locally integrable,

(y)dy, and applying the same formula to f

], du), X / implies that the right-hand side converges

function on R such that (x) = 0 for [x[ > 1 and

(u)du, and we can dene

(v) is bounded for , v [X

You might also like