arXiv:1002.5041v2 [q-fin.PR] 30 Sep 2010
Arbitrage Opportunities in Misspecified
Stochastic Volatility Models
Rudra P. Jena∗
There is vast empirical evidence that given a set of assumptions on
the real-world dynamics of an asset, the European options on this asset
are not efficiently priced in options markets, giving rise to arbitrage
opportunities. We study these opportunities in a generic stochastic
volatility model and exhibit the strategies which maximize the arbi-
trage profit. In the case when the misspecified dynamics is a classical
Black-Scholes one, we give a new interpretation of the butterfly and
risk reversal contracts in terms of their performance for volatility ar-
bitrage. Our results are illustrated by a numerical example including
Key words: stochastic volatility, model misspecification, volatility arbitrage,
butterfly, risk reversal, SABR model
2010 Mathematical Subject Classification: 91G20, 60J60
It has been observed by several authors [1, 3, 12] that given a set of assump-
tions on the real-world dynamics of the underlying, the European options
∗Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau France.
Nov 1997May 2009
Realized vol (2−month average)
Implied vol (VIX)
Figure 1: Historical evolution of the VIX index (implied volatility of options
on the S&P 500 index with 1 to 2 month to expiry, averaged over strikes, see
) compared to the historical volatility of the S&P 500 index. The fact
that the implied volatility is consistently above the historical one by several
percentage points suggests a possibility of mispricing.
on this underlying are not efficiently priced in options markets. Important
discrepancies between the implied volatility and historical volatility levels, as
illustrated in Figure 1, as well as substantial differences between historical
and option-based measures of skewness and kurtosis  have been docu-
mented. These discrepancies could be explained by systematic mispricings /
model misspecification in option markets, leading to potential arbitrage op-
portunities1. The aim of this paper is to quantify these opportunities within a
generic stochastic volatility framework, and to construct the strategies max-
imizing the gain. The arbitrage opportunities analyzed in this paper can be
called statistical arbitrage opportunities, because their presence / absence
depends on the statistical model for the dynamics of the underlying asset.
Contrary to model independent arbitrages, such as violation of the call-put
parity, a statistical arbitrage only exists in relation to the particular pricing
The issue of quantifying the gain/loss from trading with a misspecified
1There exist many alternative explanations for why the implied volatilities are con-
sistently higher than historical volatilities, such as, price discontinuity , market crash
fears  and liquidity effects such as transaction costs [22, 19, 8, 9], inability to trade in
continuous time  and market microstructure effects . The literature is too vast to
cite even the principal contributions here
model has so far mainly been studied in the case of the Black-Scholes model
with misspecified volatility [13, 25]. In this paper we go one step further, and
analyze the effects of misspecification of the volatility itself, the volatility of
volatility and of the correlation between the underlying asset and the volatil-
ity in a stochastic volatility model. Since these parameters may be observed
from a single trajectory of the underlying in an almost sure way, their mis-
specification leads, in principle, to an arbitrage opportunity. The questions
are whether this opportunity can be realized with a feasible strategy, and
how to construct a strategy maximizing the arbitrage gain under suitable
conditions guaranteeing the well-posedness of the optimization problem.
While the issue of consistency between real-world and risk-neutral prob-
ability measures has been given a rigorous treatment in several papers [1, 3,
12], the corresponding arbitrage trading strategies are usually constructed in
an ad-hoc manner [1, 17, 18]. For instance, when the risk-neutral skewness
is greater than the historical one (which roughly corresponds to correlation
misspecification in a stochastic volatility model),  suggest a strategy con-
sisting in buying all OTM puts and selling all OTM calls. Similarly, if the
risk-neutral kurtosis is greater than the historical one, the trading strategy
consists in selling far OTM and ATM options while simultaneously buying
near OTM options. In this paper we determine exactly which options must
be bought and sold to maximize arbitrage gains, depending on model param-
Our second objective is to analyze commonly used option trading strate-
gies, such as butterflies and risk reversals, and provide a new interpretation
of these structures in terms of their performance for volatility arbitrage. A
butterfly (BF) is a common strategy in FX option trading, which consists
in buying an out of the money call and an out of the money put with the
same delta value (in absolute terms) and selling a certain number of at the
money calls/puts. A risk reversal (RR) is a strategy consisting in buying an
out of the money call and selling an out of the money put with the same
delta value (in absolute terms). The financial engineering folklore goes that
“butterflies can be used to arbitrage misspecified volatility of volatility” and
“risk reversals can be used to arbitrage misspecified correlation”. In section 4,
we study these strategies and discuss their optimality for volatility trading.
During the last decade we have witnessed the appearence of a large spec-
trum of new products specifically designed for volatility and correlation trad-
ing, such as variance, volatility and correlation swaps. However, in most mar-
kets, European options continue to be much more liquid than exotic volatility
products and still constitute the most commonly used tool for volatility arbi-
trage. In this paper we therefore concentrate on arbitrage strategies involving
only the underlying asset and liquid European options.
The rest of the paper is structured as follows. In Section 2, we introduce
the generic misspecified stochastic volatility model. Section 3 defines the
admissible trading strategies and establishes the form of the optimal arbitrage
portfolio. Section 4 is dedicated to the special case when the misspecified
model is the constant volatility Black-Scholes model. This allows us to give
a new interpretation of butterfly spreads and risk reversals in terms of their
suitability for volatility arbitrage. Section 5 presents a simulation study of
the performance of the optimal arbitrage strategies in the framework of the
SABR stochastic volatility model . The purpose of this last section is not
to prove the efficiency of our strategies in real markets but simply to provide
an illustration using simulated data. A comprehensive empirical study using
real market prices is left for further research.
2A misspecified stochastic volatility framework
We start with a filtered probabiblity space (Ω,F,P,(Ft)t≥0) and consider a
financial market where there is a risky asset S, a risk-free asset and a certain
number of European options on S. We assume that the interest rate is zero,
and that the price of the risky asset S satisfies the stochastic differential
dSt/St= µtdt + σt
1 − ρ2
where µ, σ and ρ ∈ [−1,1] are adapted processes such that
(1 + S2
s)(1 + µ2
s)ds < ∞
a.s.for all t,
and (W1,W2) is a standard 2-dimensional Brownian motion. This integra-
bility condition implies in particular that the stock price process never hits
To account for a possible misspecification of the instantaneous volatility,
we introduce the process ˜ σt, which represents the instantaneous volatility
used by the option’s market for all pricing purposes. In particular, it is
the implied volatility of very short-term at the money options, and in the
sequel we call ˜ σ the instantaneous implied volatility process. We assume that
˜ σt= ˜ σ(Yt), where Y is a stochastic process with dynamics
dYt= atdt + btdW2
where atand bt> 0 are adapted processes such that
s)ds < ∞
a.s.for all t,
and ˜ σ : R → (0,∞) is a continuously differentiable Lipschitz function with
0 < σ ≤ ˜ σ(y) ≤ σ < ∞ and ˜ σ′(y) > 0 for all y ∈ R;
Further, to account for possible misspecification of the volatility of volatil-
ity b and of the correlation ρ, we assume that there exists another probability
measure Q, called market or pricing probability, not necessarily equivalent
to P, such that all options on S are priced in the market as if they were
martingales under Q. The measure Q corresponds to the pricing rule used
by the market participants, which may be inconsistent with the real-world
dynamics of the underlying asset (meaning that Q is not necessarily abso-
lutely continuous with respect to P). Under Q, the underlying asset and its
volatility form a 2-dimensional Markovian diffusion:
dSt/St= ˜ σ(Yt)
dYt= ˜ a(Yt,t)dt +˜b(Yt,t)d˜ W2
1 − ˜ ρ2(Yt,t)d˜ W1
t+ ˜ σ(Yt)˜ ρ(Yt,t)d˜ W2
where the coefficients ˜ a,˜b and ˜ ρ are deterministic functions and (˜W1,˜W2)
is a standard 2-dimensional Brownian motion under Q. Since ˜ σ is bounded,
the stock price process never hits zero Q-a.s.
The following assumptions on the coefficients of (3)–(4) will be used
throughout the paper:
i) There exists ε > 0 such that min(1− ˜ ρ(y,t)2,˜b(y,t)) ≥ ε for all (y,t) ∈
R × [0,T].
ii) The functions ˜ a(y,t),˜b(y,t), ˜ ρ(y,t) are twice differentiable with respect
to y; these coefficients as well as their first and second derivatives with
respect to y are bounded and Hölder continuous in y,t.
iii) The function ˜ σ is twice differentiable; this function as well as its first
and second derivative is bounded and Hölder continuous.
We suppose that a continuum of European options (indifferently calls or
puts) for all strikes and at least one maturity is quoted in the market. The
price of an option with maturity date T and pay-off H(ST) can be expressed
as a deterministic function of St, Ytand t:
P(St,Yt,t) = EQ[H(ST)|Ft].
Using standard methods (see e.g. ) one can show that under our as-
sumptions, for every such option, the pricing function P belongs to the class
C2,2,1((0,∞) × R × [0,T)) and satisfies the PDE
∂y+˜LP = 0,
where we define
∂y2+ S˜ σ(y)˜b˜ ρ∂2f
In addition (see ), the price of any such European option satisfies
∂y> 0,∀(S,y,t) ∈ (0,∞) × R × [0,T).
We shall use the following decay property of the derivatives of call and
put prices (see Appendix A for the proof).
Lemma 2.1. Let P be the price of a call or a put option with strike K and
maturity date T. Then
for all (y,t) ∈ R × [0,T). All the above derivatives are continuous in K and
the limits are uniform in S,y,t on any compact subset of (0,∞)×R×[0,T).
3The optimal arbitrage portfolio
We study the arbitrage from the perspective of the trader, who knows that
the market is using a misspecified stochastic volatility model to price the
options. We assume full observation: at every date t, the trader possesses
the information given by the σ-field Ftand knows the deterministic functions
˜ σ, ˜ ρ, ˜ a and˜b. In Section 5 we test the robustness of our results with respect
to this assumption.
Decay of the gamma
By direct differentiation of (27), we get
and it follows from (24) that
S2|T − t|
T − t
This proves the required decay properties, and the continuity of
follows from (24).
Decay of the vega
We denote U(S,y,t) :=
and u(x,y,t) :=
solutions of parabolic PDE [20, Corollary 2.4.1], we conclude that the deriva-
ferentiated term by term with respect to y, producing
. Using the regularity of coefficients and local regularity results for
∂y3exist, and therefore the operator (26) may be dif-
∂t+ A1u = −σσ′
A1=A + (ρbσ)′∂
All the primes denote the derivative w.r.t. y and the terminal condition is
u(S,y,T) ≡ 0, since the original terminal condition is independent of y.
The right-hand side of (30) satisfies
so from (24) we get
(T − t)
T − t
Let Γ1denote the fundamental solution of the parabolic equation with
the operator appearing in the left-hand side of (30). Using the estimates
of the fundamental solution in [21, section 4.13] (in particular, the Hölder
continuity) and the bound (32), we can show that the solution to (30) is
Using the boundedness of σ and σ′, the bound on the fundamental solution
and (32), and integrating out the variable v, we get
(T − r)
2(r − t)
T − r−c(x − z)2
r − t
Explicit evaluation of this integral then yields the bound
|u(x,y,t)| ≤ CK√T − te−cx2
|U(S,y,t)| ≤ CK√T − te−
c log2 S
from which the desired decay properties follow directly.
differentiate equation (30) with respect to x:
We denote w(x,y,t) =
∂xand W(S,y,t) =
∂t+ A1w = −σσ′
From (31) and (24),
T − texp
T − t
Similarly to the previous step we now get:
(T − r)(r − t)
T − r−c(x − z)2
r − t
and explicit evaluation of this integral yields the bounds
|w(x,y,t)| ≤ CKe−cx2
T−t,|W(S,y,t)| ≤ CK
c log2 S
equation (30) with respect to y:
We denote v(x,y,t) =∂u
∂y, V (S,y,t) =∂U
∂t+ A2v = −σσ′∂
− a′′u − (ρbσ)
∂x+ a′+ (bb′)′.
Once again, from (31) and (24),
T − texp
T − t
Using this bound together with (36) and (33) and proceding as in previous
steps, we complete the proof.
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