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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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Using standard methods (see e.g. [15]) 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

˜ a∂P

∂y+˜LP = 0,

(5)

where we define

˜Lf =∂f

∂t+S2˜ σ(y)2

2

∂2f

∂S2+

˜b2

2

∂2f

∂y2+ S˜ σ(y)˜b˜ ρ∂2f

∂S∂y.

In addition (see [26]), the price of any such European option satisfies

∂P

∂y> 0,∀(S,y,t) ∈ (0,∞) × R × [0,T).

(6)

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

lim

K→+∞

∂P(S,y,t)

∂y

∂2P(S,y,t)

∂y2

∂2P(S,y,t)

∂S2

= lim

K→0

∂P(S,y,t)

∂y

∂2P(S,y,t)

∂y2

∂2P(S,y,t)

∂S2

= 0,

lim

K→+∞

= lim

K→0

= 0,

lim

K→+∞

= lim

K→0

= 0,

and

lim

K→+∞

∂2P(S,y,t)

∂S∂y

= lim

K→0

∂2P(S,y,t)

∂S∂y

= 0

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.

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To benefit from the market misspecification, our informed trader sets up

a dynamic self-financing delta and vega-neutral portfolio Xtwith zero initial

value, containing, at each date t, a stripe of European call or put options

with a common expiry date T. In addition, the portfolio contains a quantity

−δtof stock and some amount Btof cash.

To denote the quantity of options of each strike, we introduce a pre-

dictable process (ωt)t≥0taking values in the space M of signed measures on

[0,∞) equipped with the total variation norm ? · ?V. We refer to [7] for

technical details of this construction and rigorous definitions of the stochas-

tic integral and the self-financing portfolio in this setting. Loosely speaking,

ωt(dK) is the quantity of options with strikes between K and K+dK, where

a negative measure corresponds to a short position. We shall see later that

for the optimal strategies the measure ω is concentrated on a finite number

of strikes.

The quantity of options of each strike is continuously readjusted meaning

that old options are sold to buy new ones with different strikes. In practice,

this readjustment will of course happen at discrete dates due to transac-

tion costs and we analyze the performance of our strategies under discrete

rebalancing in section 5. The portfolio is held until date T∗< T.

Finally, unless explicitly mentioned otherwise, we assume that at all

times, the total quantity of options of all strikes in the portfolio is equal

to 1:

?ωt?V ≡

ˆ

|ωt(dK)| ≡ 1

(7)

for all t. This position constraint ensures that the profit maximization prob-

lem is well posed in the presence of arbitrage opportunities (otherwise the

profit could be increased indefinitely by increasing the size of positions). If

the portfolio is rebalanced completely at discrete equally spaced dates, and

the transaction cost per one unit of option is constant, this assumption im-

plies that independently of the composition of the portfolio, the transaction

cost paid at each readjustment date is constant, and therefore does not in-

fluence the optimization problem. This constraint is also natural from the

point of view of an option exchange or a market maker who wants to struc-

ture standardized option spreads to satisfy a large number of retail clients,

because the number of options in such a spread must be fixed.

Remark 3.1. For an individual trader, who is trying to optimize her option

portfolio, it may also be natural to impose a margin constraint, that is, a

constraint on the capital which the trader uses to meet the margin require-

ments of the exchange. We discuss option trading under margin constraint

in Section 3.1.

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The value of the resulting portfolio is,

Xt=

ˆ

PK(St,Yt,t)ωt(dK) − δtSt+ Bt,

where the subscript K denotes the strike of the option. Together with Lemma

2.1, our assumptions ensure (see [7, section 4]) that the dynamics of this

portfolio are,

dXt=

?ˆ

?ˆ

ωt(dK)LPK

ωt(dK)∂PK

?

?

dt +

?ˆ

ωt(dK)∂PK

∂S

?

dSt

+

∂y

dYt− δtdSt

where

Lf =∂f

∂t+S2

tσ2

2

t

∂2f

∂S2+b2

t

2

∂2f

∂y2+ Stσtbtρt

∂2f

∂S∂y

To make the portfolio instantaneosly risk-free, we choose,

ˆ

ωt(dK)∂PK

∂y

= 0,

ˆ

ωt(dK)∂PK

∂S

= δt

to eliminate the dYtand dStterms. We emphasize that these hedge ratios are

always computed using the market parameters i.e. ˜ a,˜b and ˜ ρ. The portfolio

dynamics then become

dXt=

´ωt(dK)LPKdt,

(8)

and substituting the equation (5) into this formula and using the vega-

neutrality, we obtain the risk-free profit from model misspecification:

dXt=

ˆ

ωt(dK)(L −˜L)PKdt.

(9)

or, at the liquidation date T∗,

XT∗ =

ˆT∗

0

ˆ

ωt(dK)(L −˜L)PKdt,

(10)

where,

(L −˜L)PK=S2

t(σ2

t− ˜ σ2(Yt))

2

∂2PK

∂S2+(b2

t−˜b2

2

t)∂2PK

∂y2

+ St(σtbtρt− ˜ σ(Yt)˜bt˜ ρt)∂2PK

∂S∂y

(11)

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To maximize the arbitrage profit at each date, we therefore need to solve the

following optimization problem:

Maximize

Pt=

ˆ

ˆ

ωt(dK)(L −˜L)PK

(12)

subject to

|ωt(dK)| = 1

and

ˆ

ωt(dK)∂PK

∂y

= 0.

(13)

Remark 3.2. It should be pointed out that in this paper the arbitrage portfo-

lio is required to be instantaneously risk-free and the performance is measured

in terms of instantaneous arbitrage profit. This corresponds to the standard

practice of option trading, where the trader is usually only allowed to hold

delta and vega netral portfolios and the performance is evaluated over short

time scales. This rules out strategies which are locally risky but have a.s.

positive terminal pay-off, such as the strategies based on strict local martin-

gales, which may be admissible under the standard definition of arbitrage.

Optimality of arbitrage strategies for an equity market under the standard

definition of arbitrage has recently been studied in [14].

The following result shows that a spread of only two options (with strikes

and weights continuously readjusted) is sufficient to solve the problem (12).

Proposition 3.3. The instantaneous arbitrage profit (12) is maximized by

ωt(dK) = w1

tδK1

t(dK) − w2

tδK2

t(dK),

(14)

where δK(dK) denotes the unit point mass at K, (w1

optimal weights given by

t,w2

t) are time-dependent

w1

t=

∂PK2

∂y

+∂PK2

∂PK1

∂y∂y

,w2

t=

∂PK1

∂y

+∂PK2

∂PK1

∂y∂y

,

and (K1

t,K2

t) are time-dependent optimal strikes given by

(K1

t,K2

t) = arg max

K1,K2

∂PK2

∂y(L −˜L)PK1−∂PK1

∂y(L −˜L)PK2

+∂PK2

∂y

∂PK1

∂y

.

(15)

Proof. Step 1. We first show that the optimization problem (15) is well-

posed, that is, the maximum is attained for two distinct strike values. Let

F(K1,K2) denote the function to be optimized in (15). From the property

(6) and Lemma 2.1 it follows that F is continuous in K1and K2. Let us show

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that for every ε > 0 we can find an interval [a,b] such that |F(K1,K2)| ≤ ε

for all (K1,K2) / ∈ [a,b]2. We introduce the shorthand notation

g(K) =∂PK

∂y

F(K1,K2) =g(K2)f(K1) − g(K1)f(K2)

g(K1) + g(K2)

,f(K) = (L −˜L)PK,

.

Fix ε > 0. By Lemma 2.1, there exists N > 0 with |f(K)| ≤ ε for all

K : K / ∈ [e−N,eN]. Let δ = inf|logK|≤Ng(K). Property (6) entails that

δ > 0. Then we can find M > 0 such that g(K) ≤

K : K / ∈ [e−M,eM] (if the denominator is zero, any M > 0 can be used). It

follows that for all (K1,K2) / ∈ [e−N,eN] × [e−M,eM],

the same manner, we can find a rectangle for the second term in F, and,

taking the square containing both rectangles, we get the desired result.

εδ

sup|logK|≤N|f(K)|for all

???

g(K2)f(K1)

g(K1)+g(K2)

??? ≤ ε. In

Suppose now that for some K1and K2, F(K1,K2) > 0. Then, by taking ε

sufficiently small, the above argument allows to find a compact set containing

the maximum of F, and hence the maximum is attained for two strikes which

cannot be equal since F(K,K) ≡ 0. If, on the other hand, F(K1,K2) ≤ 0

for all K1,K2, then this means that F(K1,K2) ≡ 0 and any two strikes can

be chosen as maximizers of (15).

Step 2. We now show that the two-point solution suggested by this propo-

sition is indeed the optimal one. Let ωtbe any measure on R+satisfying the

constraints (13). Let ωt= ω+

define ν+

ω+

ω−

to conditions (6) and (13). From the same conditions,

t− ω−

t :=

t be the Jordan decomposition of ωtand

ω−

t

t(R+). These measures are well-defined due

t :=

ω+

t

t(R+)and ν−

ω−

t(R+) =

´

∂PK

∂ydν+

t

∂PK

∂ydν−

´

∂PK

∂ydν+

t+´

t

,ω+

t(R+) =

´

∂PK

∂ydν−

t

∂PK

∂ydν−

´

∂PK

∂ydν+

t+´

t

.

The instantaneous arbitrage profit (12) can then be written as

Pt=

ˆ

´

(L −˜L)PKdω+

∂PK

∂ydν−

t

t−

ˆ

(L −˜L)PKdω−

t

=

´(L −˜L)PKdν+

´

t−´

t+´

∂PK

∂ydν+

∂PK

∂ydν−

t

´(L −˜L)PKdν−

t

∂PK

∂ydν+

t

.

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Then,

Pt=

´gdν−

t

´fdν+

´gdν+

t−´gdν+

t+´gdν−

t

´fdν−

t

t

,

≤

´ν−

t(dK)?g(K) +´gdν+?supK1

g(K1)´fdν+

g(K1) +´gdν+

≤ sup

K1

g(K1)f(K2) − f(K1)g(K2)

g(K1) + g(K2)

g(K1)´fdν+

t−f(K1)´gdν+

g(K1)+´

t

gdν+

t

´gdν+

t+´gdν−

t

= sup

K1

t− f(K1)´gdν+

t

t

´ν+

t(dK){g(K1) + g(K)}supK2

g(K1)f(K2)−f(K1)g(K2)

g(K1)+g(K2)

g(K1) +´gdν+

t

= sup

K1,K2

Since for the solution (14) the above sup is attained, this solution is indeed

optimal.

The optimal strategy of Proposition 3.3 is adapted, but not necessarily

predictable. In particular, if the maximizer in (15) is not unique, the optimal

strikes may jump wildly between different maximizers leading to strategies

which are impossible to implement. However, in practice the maximizer is

usually unique (see in particular the next section), and the following result

shows that this guarantees the weak continuity, and hence predictability, of

the optimal strategy.

Proposition 3.4. Let t∗< T, and assume that the coefficients σt, btand ρt

are a.s. continuous at t∗and that the maximizer in the optimization problem

(15) at t∗is a.s. unique. Then the optimal strikes (K1

uous at t∗, and the optimal strategy (ωt) is a.s. weakly continuous at t∗.

t,K2

t) are a.s. contin-

Proof. To emphasize the dependence on t, we denote by Ft(K1,K2) the func-

tion being maximized in (15). By the continuity of σt, bt ρt, St and Yt,

there exists a neighborhood (t1,t2) of t∗(which may depend on ω ∈ Ω), in

which these processes are bounded. Since the convergence in Lemma 2.1

is uniform on compacts, similarly to the proof of Proposition 3.3, for every

ε > 0, we can find an interval [a,b] (which may depend on ω ∈ Ω) such that

|Ft(K1,K2)| ≤ ε ∀(K1,K2) / ∈ [a,b]2and ∀t ∈ (t1,t2). This proves that the

optimal strikes (K1

t,K2

t) are a.s. bounded in the neighborhood of t∗.

Let (tn)n≥1 be a sequence of times converging to t∗and (K1

the corresponding sequence of optimal strikes. By the above argument, this

sequence is bounded and a converging subsequence (K1

tn,K2

tn)n≥1

tnk,K2

tnk)k≥1can be

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extracted. Let (˜K1,˜K2) be the limit of this converging subsequence. Since

(K1

strikes (k1,k2),

tnk,K2

tnk) is a maximizer of Ftnkand by continuity of F, for arbitrary

Ft∗(˜K1,˜K2) = lim

k→∞Ftnk(K1

tnk,K2

tnk) ≥ lim

k→∞Ftnk(k1,k2) = Ft∗(k1,k2).

This means that (˜K1,˜K2) is a maximizer of Ft∗(·,·), and, by the uniqueness

assumption,˜K1= K1

deduce that K1

of ωtfollows directly.

t∗ and˜K2= K2

t→ K1

t∗. By the subsequence criterion we then

t→ K2

t∗ and K2

t∗ a.s. as t → t∗. The weak convergence

3.1Trading under margin constraint

Solving the optimization problem (12) under the constraint (7) amounts to

finding the optimal (vega-weighted) spread, that is a vega-neutral combina-

tion of two options, in which the total quantity of options equals one. This

approach guarantees the well-posedness of the optimization problem and pro-

vides a new interpretation for commonly traded option contracts such as

butterflies and risk reversals (see next section).

From the point of view of an individual option trader, an interesting alter-

native to (7) is the constraint given by the margin requirements of the stock

exchange. In this section, we shall consider in detail the CBOE minimum

margins for customer accounts, that is, the margin requirements for retail in-

vestors imposed by the Chicago Board of Options Exchange [11, 27]. Under

this set of rules, margin requirements are, by default, calculated individually

for each option position:

• Long positions in call or put options must be paid for in full.

• For a naked short position in a call option, the margin requirement at

time t is calculated using the following formula [27].

Mt= PK

t + max(αSt− (K − St)1K>St,βSt) := PK

where PKis the option price, α = 0.15 and β = 0.1.

t + λK

t,

• The margin requirement for a “long call plus short underlying posi-

tion” is equal to the call price plus short sale proceeds plus 50% of the

underlying value.

• The margin requirement for a "short call plus long underlying" is 50%

of the underlying value.

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In addition, some risk offsets are allowed for certain spread options, but we

shall not take them into account to simplify the treatment. For the same

reason, we shall assume that all options held by the trader are call options.

Let ωt= ω+

hedged option portfolio can be written as

t− ω−

t be the Jordan decomposition of ωt. The value of a delta-

Xt= Bt+

ˆ ?

ˆ ?

ˆ ?

PK− St∂PK

?

−PK

∂S

?

ωt(dK)

= Bt+

PK

1 −∂PK

?

∂S

?

+∂PK

∂S(PK− St)

?

?

ω+

t(dK)

+

1 −∂PK

∂S

+∂PK

∂S(St− PK)

?

ω−

t(dK)

The margin requirement for this position is

Mt=

ˆ ?

ˆ ?

:=

PK+∂PK

∂S(1 + γ)St

?

?

ω+

t(dK)

+

(λK+ PK)1 −∂PK

ˆ

∂S

?

+ γSt∂PK

∂S

?

ω−

t(dK)

ˆ

β+

t(K)ω+

t(dK) +β−

t(K)ω−

t(dK)

with γ = 0.5. Supposing that the trader disposes of a fixed margin account

(normalized to 1), the optimization problem (12)–(13) becomes

Maximize

Pt=

ˆ

ˆ

ωt(dK)(L −˜L)PK

subject to

β+

t(K)ω+

t(dK) +

ˆ

β−

t(K)ω−

t(dK) = 1

and

ˆ

ωt(dK)∂PK

∂y

= 0.

This maximization problem can be treated similarly to (12)–(13). Assume

that at time t, the problem

arg sup

K1,K2

∂PK2

∂y(L −˜L)PK1−∂PK1

β−(K2)∂PK1

∂y(L −˜L)PK2

+ β+(K1)∂PK2

∂y∂y

.

(16)

admits a finite maximizer (K1

is maximized by the two-strike portfolio (14) with

t,K2

t). Then the instantaneous arbitrage profit

w1

t=

β+(K1

t)∂PK2

+ β+(K1

∂y

β+(K2

t)∂PK1

∂y

t)∂PK2

∂y

,w2

t=

β+(K2

t)∂PK1

+ β+(K1

∂y

β+(K2

t)∂PK1

∂y

t)∂PK2

∂y

.

13

Page 14

However, since β+(K) → 0 as K → +∞, the problem (16) may not admit a

finite maximizer for some models. For instance, in the Black-Scholes model,

a simple asymptotic analysis shows that β+(K) converges to zero faster than

∂PK

∂y

as K → +∞, while

This means that by keeping K2fixed and choosing K1large enough, the

instantaneous arbitrage profit (16) can be made arbitrarily large. In financial

terms this means that since the margin constraint does not limit the size of

the long positions, it may be optimal to buy a large number of far out of the

money options. So the margin constraint alone is not sufficient to guarantee

the well-posedness of the problem and a position constraint similar to (7) is

necessary as well (in particular, the position constraint limits the transaction

costs). If both the margin constraint and the position constraint are imposed,

the optimal portfolio will in general be a combination of three options with

different strikes.

∂PK

∂y

converges to zero faster than∂2PK

∂S∂yand

∂2PK

∂y2 .

While our analysis focused on CBOE margining rules for retail investors,

a number of other exchanges, for example, CME for large institutional ac-

counts, use an alternative margining system called Standard Portfolio Analy-

sis of Risk (SPAN). Under this system, the margin requirement is determined

as the worst-case loss of the portfolio over a set of 14 market scenarios. In

this case, position constraints are also necessary to ensure the well-posedness

of the optimization problem. See [2] for a numerical study of optimal option

portfolios under SPAN constraints in the binomial model.

4 The Black-Scholes case

In this section, we consider the case when the misspecified model is the Black-

Scholes model, that is, ˜ a ≡˜b ≡ 0. This means that at each date t the market

participants price options using the Black-Scholes model with the volatility

value given by the current instantaneous implied volatility ˜ σ(Yt). The process

Ytstill has a stochastic dynamics under the real-world probability P. This

is different from the setting of [13], where one also has a = b = 0, that is,

the instantaneous implied volatility is deterministic. In this section, since we

do not need to solve partial differential equations, to simplify notation, we

set ˜ σt≡ Yt, that is, the mapping ˜ σ(·) is the identity mapping. Recall the

formulas for the derivatives of the call/put option price in the Black-Scholes

14

Page 15

model (r = 0):

∂P

∂σ= Sn(d1)√τ = Kn(d2)√τ,

∂2P

∂σ∂S= −n(d1)d2

˜ σ

∂2P

∂S2=n(d1)

∂2P

S˜ σ√τ,

,

∂σ2=Sn(d1)d1d2√τ

˜ σ

,

where d1,2=

normal density.

m

˜ σ√τ±

˜ σ√τ

2, τ = T − t, m = log(S/K) and n is the standard

We first specialize the general optimal solution (14) to the Black-Scholes

case. All the quantities σt, ˜ σt,bt,ρt,Stare, of course, time-dependent, but

since the optimization is done at a given date, to simplify notation, we omit

the subscript t.

Proposition 4.1. The optimal option portfolio maximizing the instanta-

neous arbitrage profit (12) is described as follows:

• The portfolio consists of a long position in an option with log-moneyness

m1= z1˜ σ√τ−˜ σ2τ

m2= z2˜ σ√τ −˜ σ2τ

f(z1,z2) =(z1− z2)(z1+ z2− w)

2and a short position in an option with log-moneyness

2, where (z1,z2) is a maximizer of the function

ez2

1/2+ ez2

2/2

with w =

˜ σbτ+2σρ

b√τ

.

• The weights of the two options are chosen to make the portfolio vega-

neutral.

We define by Poptthe instantaneous arbitrage profit realized by the optimal

portfolio.

Proof. Substituting the Black-Scholes values for the derivatives of option

prices, and making the change of variable z =

problem to solve becomes:

?b√τ

from which the proposition follows directly.

m

˜ σ√τ+˜ σ√τ

2, the maximization

max

z1,z2bS

n(z1)n(z2)

n(z1) + n(z2) 2˜ σ(z2

1− z2

2) −bτ

2(z1− z2) − ρσ

˜ σ(z1− z2)

?

, (17)

Remark 4.2. A numerical study of the function f in the above proposition

shows that it admits a unique maximizer in all cases except when w = 0.

15

Page 16

Remark 4.3. From the form of the functional (17) it is clear that one should

choose options with the largest time to expiry τ for which liquid options are

available.

Remark 4.4. The gamma term (second derivative with respect to the stock

price) does not appear in the expression (17), because the optimization is

done under the constraint of vega-neutrality, and in the Black-Scholes model

a portfolio of options with the same maturity is vega-neutral if and only if

it is gamma neutral (see the formulas for the greeks earlier in this section).

Therefore in this setting, to arbitrage the misspecification of the volatility

itself, as opposed to skew or convexity, one needs a portfolio of options with

more than one maturity.

The variable z =

to the Delta of the option: for a call option ∆ = N(z). This convenient

parameterization corresponds to the current practice in Foreign Exchange

markets of expressing the strike of an option via its delta. Given a weight-

ing measure ωt(dK) as introduced in section 2, we denote by ¯ ωt(dz) the

corresponding measure in the z-space. We say that a portfolio of options

is ∆-symmetric (resp., ∆-antisymmetric) if ¯ ωtis symmetric (resp., antisym-

metric).

log(S/K)

σ√τ

+σ√τ

2

introduced in the proof is directly related

The following result clarifies the role of butterflies and risk reversals for

volatility arbitrage.

Proposition 4.5. Let Poptbe the instantaneous arbitrage profit (12) realized

by the optimal strategy of Proposition 4.1

1. Consider a portfolio (RR) described as follows:

• If bτ/2 + ρσ/˜ σ ≥ 0

– buy1

or, equivalently, delta value N(−1) ≈ 0.16

– sell

or, equivalently, delta value N(1) ≈ 0.84.

• if bτ/2 + ρσ/˜ σ < 0 buy the portfolio with weights of the opposite

sign.

2units of options with log-moneyness m1= −˜ σ√τ −˜ σ2τ

2units of options with log-moneyness m2= ˜ σ√τ −˜ σ2τ

2,

1

2,

Then the portfolio (RR) is the solution of the maximization problem

(12) under the additional constraint that it is ∆-antisymmetric.

2. Consider a portfolio (BF) described as follows

16

Page 17

• buy x units of options with log-moneyness m1 = z0˜ σ√τ − ˜ σ2τ,

or, equivalently, delta value N(z0) ≈ 0.055, where z0≈ 1.6 is a

universal constant, solution to

z2

2e

0

z2

0

2 = e

z2

0

2 + 1

• buy x0units of options with log-moneyness m2= −z0˜ σ√τ − ˜ σ2τ,

or, equivalently, delta value N(−z0) ≈ 0.945

• sell 1 − 2x0 units of options with log-moneyness m3= −˜ σ2τ

equivalently, delta value N(0) =1

to make the portfolio vega-neutral, that is, x0=

2

or,

2, where the quantity x0is chosen

1

2(1+e−z2

0/2)≈ 0.39.

Then, the portfolio (BF) is the solution of the maximization problem

(12) under the additional constraint that it is ∆-symmetric.

3. Define by PRRthe instantaneous arbitrage profit realized by the portfolio

of part 1 and by PBFthat of part 2. Let

|˜ σbτ + 2σρ|

|˜ σbτ + 2σρ| + 2bK0√τ

where K0is a universal constant, defined below in the proof, and ap-

proximately equal to 0.459. Then

α =

PRR≥ αPopt

and

PBF≥ (1 − α)Popt.

Remark 4.6. In case of a risk reversal, the long/short decision depends on

the sign of ρ +˜ σbτ

means that when the implied volatility surface is flat and ρ < 0, if b is big

enough, contrary to conventional wisdom, the strategy of selling downside

strikes and buying upside strikes will yield a positive profit, whereas the

opposite strategy will incur a loss. This is because the risk reversal (RR) has

a positive vomma (second derivative with respect to Black-Scholes volatility).

Let Pxbe the price of the risk reversal contract where one buys a call with

delta value x <1

2σrather than on the sign of the correlation itself. This

2and sells a call with delta value 1 − x. Then

∂2Px

∂σ2= Sn(d)˜ στd,d = N(1 − x) > 0.

Therefore, from equation (11) it is clear that if b is big enough, the effect

of the misspecification of b will outweigh that of the correlation leading to a

positive instantaneous profit.

17

Page 18

Proof. Substituting the Black-Scholes values for the derivatives of option

prices, and making the usual change of variable, the maximization problem

(12) becomes:

maxSb2√τ

2˜ σ

ˆ

ˆ

z2n(z)¯ ωt(dz) − Sb(bτ/2 + ρσ/˜ σ)

ˆ

zn(z)¯ ωt(dz)

(18)

subject to

n(z)¯ ωt(dz) = 0,

ˆ

|¯ ωt(dz)| = 1.

1. In the antisymmetric case it is sufficient to look for a measure ¯ ω+

(0,∞) solution to

t on

max−Sb(bτ/2 + ρσ/˜ σ)

ˆ

(0,∞)

zn(z)¯ ω+

t(dz)

subject to

ˆ

(0,∞)

|¯ ωt(dz)| =1

2.

The solution to this problem is given by a point mass at the point

z = argmaxzn(z) = 1 with weight −1

bτ/2 + ρσ/˜ σ < 0. Adding the antisymmetric component of ¯ ω+

proof of part 1 is complete.

2if bτ/2 + ρσ/˜ σ ≥ 0 and

1

2if

t, the

2. In the symmetric case it is sufficient to look for a measure ¯ ω+

solution to

maxSb2√τ

2˜ σ

[0,∞)

subject to

[0,∞)

By the same argument as in the proof of Proposition 3.3, we get that

the solution to this problem is given by two point masses at the points

z1and z2given by

ton [0,∞)

ˆ

z2n(z)¯ ω+

t(dz)

ˆ

n(z)¯ ω+

t(dz) = 0,

ˆ

[0,∞)

|¯ ω+

t(dz)| =1

2.

(z1,z2) = argmaxn(z1)z2

2n(z2) − n(z2)z2

n(z1) + n(z2)

1n(z1)

= argmax

z2

2− z2

z2

1

2 + e

1

z2

e

2

2

,

from which we immediately see that z1= 0 and z2coincides with the

constant z0introduced in the statement of the proposition. Adding the

symmetric part of the measure ¯ ω+

t, the proof of part 2 is complete.

3. To show the last part, observe that the contract (BF) maximizes the

first term in (18) while the contract (RR) maximizes the second term.

The values of (18) for the contract (BF) and (RR) are given by

PBF=Sb2√τ

˜ σ√2πe−z2

0/2,PRR=Sb|bτ/2 + ρσ/˜ σ|

√2π

e−1

2.

18

Page 19

and therefore

PRR

PBF+ PRR=

Since the maximum of a sum is always no greater than the sum of

maxima, Popt≤ PBF+ PRRand the proof is complete.

|˜ σbτ + 2σρ|

|˜ σbτ + 2σρ| + 2bK0√τ

with

K0= e

1

2−z2

0

2.

Remark 4.7. Let us sum up our findings about the role of butterflies and risk

reversals for arbitrage:

• Risk reversals are not optimal and butterflies are not optimal unless

ρ = −b˜ στ

• Nevertheless there exists a universal risk reversal (16-delta risk reversal

in the language of foreign exchange markets, denoted by RR) and a

universal butterfly (5.5-delta vega weighted buttefly, denoted by BF)

such that one can secure at least half of the optimal profit by investing

into one of these two contracts. Moreover, for each of these contracts,

a precise estimate of the deviation from optimality is available.

2σ(because α = 0 for this value of ρ).

• Contrary to the optimal portfolio which depends on model parameters

via the constant w introduced in Proposition 4.1, the contracts (BF)

and (RR) are model independent and only the choice of the contract

(BF or RR) and the sign of the risk reversal may depend on model

parameters. This means that these contracts are to some extent robust

to the arbitrageur using a wrong parameter, which is important since

in practice the volatility of volatility and the correlation are difficult to

estimate.

• The instantaneous profit PBFof the contract (BF) does not depend

on the true value of the correlation ρ. This means that (BF) is a pure

convexity trade, and it is clear from the above that it has the highest

instantaneous profit among all such trades. Therefore, the contract

(BF) should be used if the sign of the correlation is unknown.

• When b → 0,α → 1, and in this case (RR) is the optimal strategy.

5 Simulation study in the SABR model

In this section, we consider the case when, under the pricing probability, the

underlying asset price follows the stochastic volatility model known as SABR

19

Page 20

model. This model captures both the volatility of volatility and the corre-

lation effects and is analytically tractable. The dynamics of the underlying

asset under Q is

dSt= ˜ σtSβ

d˜ σt=˜b˜ σtd˜W2

t(

?

t

1 − ˜ ρ2d˜ W1

t+ ˜ ρd˜ W2

t)

(19)

(20)

To further simplify the treatment, we take β = 1, and in order to guarantee

that S is a martingale under the pricing probability [28], we assume that the

correlation coefficient satisfies ˜ ρ ≤ 0. The true dynamics of the instantaneous

implied volatility are

d˜ σt= b˜ σtdW2

t,

(21)

and the dynamics of the underlying under the real-world measure are

dSt= σtSt(

?

1 − ρ2dW1

t+ ρdW2

t).

(22)

The SABR model does not satisfy some of the assumptions of section 2,

in particular because the volatility is not bounded from below by a positive

constant.Nevertheless, it provides a simple and tractable framework to

illustrate the performance of our strategies. A nice feature of the SABR

model is that rather precise approximate pricing formulas for vanilla options

are available, see [16] and [23], which can be used to compute the optimal

strikes of Proposition 3.3. Alternatively, one can directly compute the first

order correction (with respect to the volatility of volatility parameter˜b) to

the Black-Scholes optimal values using perturbation analysis. We now briefly

outline the main ideas behind this approach.

First order correction to option price

option price P satisifies the following pricing equation,

In the SABR model, the call/put

∂P

∂t+S2˜ σ2

2

∂2P

∂S2+

˜b2˜ σ2

2

∂2P

∂σ2+ S˜ σ2˜b˜ ρ∂2P

∂S∂σ= 0

with the appropriate terminal condition. We assume that˜b is small and look

for approximate solutions of the form P = P0+˜bP1+O(˜b2). The zero-order

term P0corresponds to the Black Scholes solution:

∂P0

∂t

+S2˜ σ2

2

∂2P0

∂S2= 0

The first order term satisfies the following equation:

∂P1

∂t

+S2˜ σ2

2

∂2P1

∂S2+ S˜ σ2˜ ρ∂2P0

∂S∂σ= 0

20

Page 21

and can be computed explicitly via

P1=˜ σ2˜ ρ(T − t)

2

S∂2P0

∂S∂σ

First order correction to optimal strikes

Define

f = (L −˜L)P

where we recall that

(L −˜L)P =S2(σ2− ˜ σ2)

2

∂2P

∂S2+b2σ2−˜b2˜ σ2

2

∂2P

∂σ2+ S(σ2bρ − ˜ σ2˜b˜ ρ)∂P2

∂S∂σ

For the general case, we are not assuming anything on the smallness of the

parameters of the true model. Now to the first order in˜b, we can expand f

as,

f ≈ f0+˜bf1

where,

f0=S2(σ2− ˜ σ2)

2

and,

∂2P0

∂S2+b2σ2

2

∂2P0

∂σ2+ Sσ2bρ∂P2

0

∂S∂σ

f1=S2(σ2− ˜ σ2)

2

∂2P1

∂S2+b2σ2

2

∂2P1

∂σ2+ Sσ2bρ∂2P2

1

∂S∂σ− S˜ σ2˜ ρ∂2P2

0

∂S∂σ

Define the expression to be maximized from Proposition 3.3:

F =w2f1− w1f2

w1+ w2

where wi=∂Pi

wi

∂σ. These vegas are also expanded in the same manner: wi≈

1. Now to first order in˜b, we can expand F as,

0+˜bwi

F ≈˜F := F0+˜b(F1+ F3F0)

(23)

where

F0=w2

0f1

w1

0f1

0− w1

0+ w2

1+ w2

0f2

0

0

1f1

w1

F1=w2

0− w1

0+ w2

0f2

1− w1

1f2

0

0

and

F3=w1

1+ w2

w1

1

0+ w2

0

.

21

Page 22

0.10.20.30.4b?

1.39

1.40

1.41

1.42

1.43

k1

1st order perturbation

Black Scholes

SABR model

0.10.20.30.4b?

1.055

1.056

1.057

1.058

1.059

1.060

k2

1st order perturbation

Black Scholes

SABR model

Figure 2: Optimal Strikes for the set of parameters σ = .2, S = 1,b = .3,ρ =

−.3, ˜ ρ = −.5,t = 1, as a function of the misspecified˜b ∈ [.01,.4].

We would like to represent the optimal strike K∗which maximizes˜F as

the sum of the strike K0which maximizes F0plus a small correction of order

of˜b:

K∗≈ K0+˜bK1.

The optimal strike K∗is the solution of the equation,

∂˜F

∂K(K∗) = 0 ⇐⇒

∂˜F

∂K(K0) +˜bK1∂2˜F

∂K2(K0) ≈ 0.

Using the representation (23), the fact that K0is the maximizer of F0and

keeping only the terms of order up to one in˜b, we get

∂F1

∂K(K0) + F0∂F3

∂K(K0) + K1∂2F0

∂K2(K0) = 0

which gives the following correction to the optimal strike:

K1= −

∂F1

∂K(K0) + F0∂F3

∂2F0

∂K2(K0)

∂K(K0)

Figure 2 compares this first-order correction to the two optimal strikes

to the zero-order Black-Scholes value and the precise value computed using

Hagan’s implied volatility asymptotics [16], for different values of the volatil-

ity of volatility parameter˜b. Given that already the difference between the

zero-order correction and the precise value is not that big, our first order

correction appears acceptable for reasonably small values of˜b.

Numerical results

our strategies on a simulated example. We assume that a trader, who is

aware both about the true and the misspecified parameters, invests in our

In this section we demonstrate the performance of

22

Page 23

strategy. We simulate 10000 runs2of the stock and the volatility during 1

year under the system specified by (19)–(22), assuming misspecification of

the correlation ρ and the volatility of volatility b but not of the volatility

itself (σ = ˜ σ). The initial stock value is S0 = 100 and initial volatility is

σ0= .1. The interest rate is assumed to be zero. During the simulation, at

each rebalancing date, the option portfolio is completely liquidated, and a

new portfolio of options with the same, fixed, time to maturity and desired

optimal strikes is purchased.

We illustrate the impact of different characteristics of the strategy on the

overall performance. In each of the graphs, we plot two groups of curves:

the cross-marked ones correspond to misspecified model, and the diamond-

market ones correspond to the setting when the model parameters are equal

to the true ones. In each group, the three curves correspond (from lower to

upper) to the 25% quantile, the median and the 75% quantile computed from

the 10000/100000 simulated trajectories.

• Impact of correlation misspecification. Figure 3 (left) shows the per-

formance of portfolios using options with 1 month to maturity. The

true parameters are ρ = −.8,b = .2. The misspecified or the market

parameters are ˜ ρ = −.2,˜b = .2.

• Impact of volatility of volatility misspecification. Figure 3 (right) shows

the performance of portfolios using options with 1 month to maturity.

The true parameters are ρ = −.8,b = .2. The misspecified or the

market parameters are ˜ ρ = −.8,˜b = .1.

• Impact of the options’ maturity. Figure 4 shows the difference in per-

formance of portfolios using options with 1 month and 6 months to

maturity. The true parameters are ρ = −.8,b = .2. The misspeci-

fied or the market parameters are ˜ ρ = −.2,˜b = .1. As predicted by our

analysis in the Black-Scholes case, the performance is better for options

with longer time to maturity and higher intrinsic value.

• Impact of the rebalancing frequency. Figure 5 illustrates the effect

of the total number of rebalancings on portfolio performance. The

true parameters are ρ = −.8,b = .2. The misspecified or the market

parameters are ˜ ρ = −.2,˜b = .1. The average profit is roughly the same

for both the rebalancing frequencies but the variance is higher for lower

rebalancing frequencies.

• Impact of trader using different parameters from the true ones. Figure 6

shows the performance of portfolios using options with 1 month to ma-

2For the figure which includes transaction costs, 100000 simulation runs are done.

23

Page 24

Figure 3: The evolution of portfolios using options with 1 month to maturity

for only correlation misspecification (left) and only volatility of volatility

misspecification (right).

turity. The misspecified or the market parameters are ˜ ρ = −.2,˜b = .1.

The true historical parameters are ρ = −.8,b = .2 but the trader esti-

mates these parameters with an error and uses the values ˆ ρ = −.6,ˆb =

.15 instead. The average profit increases with better estimation of mis-

specification in parameters.

• Impact of transaction costs. In all our analysis till now, we have ne-

glected the transaction costs. In our last test, we include a bid ask-fork

of 0.45% in implied volatility terms for every option transaction. More

precisely, if we denote by Pmidthe option price given by the model

and by σmidthe corresponding implied volatility, the price that our

trader must pay to buy an option corresponds to the implied volatility

of σmid+ 0.225% and the price she receives for selling an option cor-

responds to the implied volatility of σmid− 0.225%. This corresponds

roughly to the bid-ask interval observed for the most liquid options on

the S&P 500 index. The true parameters are ρ = −.9,b = .9. The

misspecified or the market parameters are ˜ ρ = −.1,˜b = .1. The evo-

lution of the portfolio performance with 32 rebalancing dates per year

using options with 1 month remaining to maturity is plotted in Figure

7. With the parameter values that we have chosen, it seems that even

in the presence of moderate transaction costs, our strategy produces

a positive pay-off in at least 75% of cases. Whether this will be the

case in real options markets is a more difficult question, which requires

an extensive empirical study. We plan to address this issue in future

research.

24

Page 25

time in years

0.2 0.30.40.61.00.50.9

Portfolio value

0.10.0

0.6

0.5

0.4

0.3

0.2

0.1

0.0

-0.1

0.80.7

misspecified

not misspecified

Figure 4: Impact of the options’ maturity: The evolution of portfolios using

options with 1 month (left) and 6 months to maturity with 256 rebalancing

dates.

Figure 5: Impact of the rebalancing frequency: Evolution of the portfolio

using options with 1 month to maturity and 128 rebalancing dates. Compare

with Figure 4, left graph.

25

Page 26

Figure 6: Impact of using a different set of parameters: Evolution of the

portfolio for a different set of parameters than the true ones for options with

1 month to maturity and 256 rebalancing dates.

Figure 7: The effect of transaction costs on the performance of the arbitrage

strategy.

26

Page 27

Acknowledgements

We would like to thank the participants of the joint seminar of Société

Générale, Ecole Polytechnique and Ecole des Ponts et Chaussées, and espe-

cially L. Bergomi (Société Générale) for insightful comments on the previous

version of this paper.

This research of Peter Tankov is supported by the Chair Financial Risks

of the Risk Foundation sponsored by Société Générale, the Chair Derivatives

of the Future sponsored by the Fédération Bancaire Française, and the Chair

Finance and Sustainable Development sponsored by EDF and Calyon. This

research of Rudra Jena is supported by the Chair Financial Risks of the Risk

Foundation sponsored by Société Générale.

AProof of Lemma 2.1

First, let us briefly recall some results on fundamental solutions of parabolic

PDE [21]. Let

L

?

x,t,∂

∂x,∂∂t

?

u :=∂

∂t+1

2

n

?

i,j=1

aij(x,t)

∂2u

∂xi∂xj

+

n

?

i=1

bi(x,t)∂u

∂xi

+ c(x,t).

Assumption A.1. There is a positive constant µ such that

n

?

i,j=1

aij(x,t)ξiξj? µ|ξ|2

∀(x,t) ∈ Rn× [0,T], ξ ∈ Rn.

Assumption A.2. There exists α ∈ (0,1) such that the coefficients of L are

bounded and Hölder continuious in x with exponent α and Hölder continuous

in t with exponentα

2, uniformly with respect to (x,t) in Rn× [0,T].

The fundamental solution of the parabolic second-order equation with

operator L is the function Γ(x,t,ξ,T) which satisfies

L

?

x,t,∂

∂x,∂∂t

?

Γ(x,t,ξ,T) = δ(x − ξ)δ(t − T),t ≤ T.

Under A.1 and A.2, the operator L admits a fundamental solution Γ with

|Dr

tDs

xΓ(x,t,ξ,T)| ≤ c(T − t)−n+2r+s

2

exp

?

−C|x − ξ|

T − t

?

(24)

27

Page 28

where r and s are integers with 2r + s ≤ 2, t < T and c,C are positive.

Consider now the Cauchy problem,

Lu(x,t) = f(x,t),

u(x,T) = φ(x),

(x,t) ∈ Rn× [0,T),

x ∈ Rn,

where f is Hölder continuous in its arguments, φ is continuous and these

function satisfy reasonable growth constraints at infinity. Then the solution

to this problem can be written as

u(x,t) =

ˆT

t

dτ

ˆ

RndξΓ(x,t,ξ,τ)f(ξ,τ) +

ˆ

RndξΓ(x,t,ξ,T)φ(ξ).

Let us now turn to the proof of Lemma 2.1. We discuss the results for

the put options. The result for calls follows directly by put-call parity. As

described in Section 2, the put option price P(S,y,t) solves the PDE (5).

Let x = logS

Kand p(x,y,t) := P(Kex,y,t). Then p solves the PDE

∂p

∂t+ Ap = 0,p(x,y,T) =K(1 − ex)+.

(25)

where

Ap =1

2σ2

?∂2p

∂x2−∂p

∂x

?

+ a∂p

∂y+1

2b2∂2p

∂y2+ ρbσ∂2p

∂x∂y

The quantities a,b,σ,ρ correspond to market misspecified values, but in this

appendix we shall omit the tilda and often drop the explicit dependence on

t and y in model parameters to simplify notation.

Therefore, the option price can be written as

p(x,y,t) =

ˆ

dz

ˆ

dvΓ(x,y,t,z,v,T)K(1 − ez)+,

(26)

where Γ(x,y,t,z,v,T) is the fundamental solution of (25). Since the coef-

ficients of A do not depend on x, Γ(x,y,t,z,v,T) ≡ Γ(0,y,t,z − x,v,T).

Coming back to the original variable S, we have

P(S,y,t) =

ˆ

dz γ(y,t,z − logS

K,T)K(1 − ez)+

(27)

where we write

γ(y,t,z,T) :=

ˆ

dvΓ(0,y,t,z,v,T).

28

Page 29

Decay of the gamma

∂2P

∂S2

By direct differentiation of (27), we get

∂2P

∂S2=K

S2γ(y,t,logK

S,T)

(28)

and it follows from (24) that

????

∂2P

∂S2

????≤

CK

S2|T − t|

1

2exp

?

−clog2 K

T − t

S

?

.

(29)

This proves the required decay properties, and the continuity of

follows from (24).

∂2P

∂S2 also

Decay of the vega

∂P

∂y

We denote U(S,y,t) :=

∂P(S,y,t)

∂y

and u(x,y,t) :=

∂p(x,y,t)

∂y

solutions of parabolic PDE [20, Corollary 2.4.1], we conclude that the deriva-

tives

ferentiated term by term with respect to y, producing

. Using the regularity of coefficients and local regularity results for

∂3p

∂x2∂y,

∂3p

∂x∂y2and

∂3p

∂y3exist, and therefore the operator (26) may be dif-

∂u

∂t+ A1u = −σσ′

?∂2p

∂x2−∂p

∂x

?

,

(30)

where

A1=A + (ρbσ)′∂

∂x+ bb

′ ∂

∂y+ a′.

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

∂2p

∂x2−∂p

∂x= Kγ(y,t,−x,T)

(31)

so from (24) we get

????

∂2p

∂x2−∂p

∂x

????≤

CK

(T − t)

1

2exp

?

−cx2

T − t

?

.

(32)

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

29

Page 30

continuity) and the bound (32), we can show that the solution to (30) is

given by

u(x,y,t) =

ˆT

t

dr

ˆ

R2dz dvΓ1(x,y,t,z,v,r)σ(v)σ′(v)

?∂p

∂z−∂2p

∂z2

?

(z,v,r).

Using the boundedness of σ and σ′, the bound on the fundamental solution

and (32), and integrating out the variable v, we get

|u(x,y,t)| ≤

ˆT

t

dr

ˆ

dz

CK

(T − r)

1

2(r − t)

1

2exp

?

−

cz2

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−

T−t

(33)

c log2 S

T−t

K

(34)

from which the desired decay properties follow directly.

Decay of

differentiate equation (30) with respect to x:

∂2P

∂S∂y

We denote w(x,y,t) =

∂u

∂xand W(S,y,t) =

∂U

∂S=

w

Sand

∂w

∂t+ A1w = −σσ′

?∂3p

∂x3−∂2p

∂x3

?

.

From (31) and (24),

????

∂3p

∂x3−∂2p

∂x2

????≤

CK

T − texp

?

−cx2

T − t

?

.

(35)

Similarly to the previous step we now get:

|w(x,y,t)| ≤

ˆT

t

dr

ˆ

dz

CK

(T − r)(r − t)

1

2exp

?

−

cz2

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

Se−

c log2 S

T−t

K

(36)

30

Page 31

Decay of

equation (30) with respect to y:

∂2P

∂y2

We denote v(x,y,t) =∂u

∂y, V (S,y,t) =∂U

∂yand differentiate

∂v

∂t+ A2v = −σσ′∂

∂y

?∂2p

∂x2−∂p

∂x

?

− a′′u − (ρbσ)

′′w

where

A2=A1+ bb

′ ∂

∂y+ (ρbσ)

′ ∂

∂x+ a′+ (bb′)′.

Once again, from (31) and (24),

????

∂

∂y

?∂2p

∂x2−∂p

∂x

?????≤

CK

T − texp

?

−cx2

T − t

?

.

Using this bound together with (36) and (33) and proceding as in previous

steps, we complete the proof.

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