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arXiv:1002.5041v2 [q-fin.PR] 30 Sep 2010

Arbitrage Opportunities in Misspecified

Stochastic Volatility Models

Rudra P. Jena∗

Peter Tankov∗†

Abstract

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

transaction costs.

Key words: stochastic volatility, model misspecification, volatility arbitrage,

butterfly, risk reversal, SABR model

2010 Mathematical Subject Classification: 91G20, 60J60

1Introduction

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.

E-mail: {jena,tankov}@cmap.polytechnique.fr

†Corresponding author

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Nov 1997May 2009

0

10

20

30

40

50

60

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

[10]) 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 [3] 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

model.

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 [4], market crash

fears [5] and liquidity effects such as transaction costs [22, 19, 8, 9], inability to trade in

continuous time [6] and market microstructure effects [24]. The literature is too vast to

cite even the principal contributions here

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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), [1] 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-

eters.

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

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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 [16]. 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

equation

dSt/St= µtdt + σt

?

1 − ρ2

tdW1

t+ σtρtdW2

t

(1)

where µ, σ and ρ ∈ [−1,1] are adapted processes such that

ˆt

0

(1 + S2

s)(1 + µ2

s+ σ2

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

zero P-a.s.

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

t,

(2)

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where atand bt> 0 are adapted processes such that

ˆt

0

(a2

s+ b2

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

t

(3)

t,

(4)

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].

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