Arbitrage Opportunities in Misspecified Stochastic Volatility Models

8/6/2019 Arbitrage Opportunities in Misspecified Stochastic Volatility Models

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Arbitrage opportunities in misspecifiedstochastic volatility models

Rudra P. Jena Peter Tankov

cole Polytechnique - CMAP

Modeling and Managing Financial Risks, Paris, 12th Jan

2011

Rudra P. Jena - [email protected] misspecification of SV models


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Outline

1. Motivation/ Introduction/Background

2. Set up of the problem

3. General solution

4. The Black Scholes case

5. The Stochastic Volatility case: A perturbative approach

6. Numerics/Results



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IntroductionThe motivation

Widely documented phenomenon of option mispricing.Given set of assumptions on the real-world dynamics of an

asset, the European options on this asset are not efficientlypriced in options markets.[Y-Ait Sahaliya et. al, Bakshi et. al]



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Introduction

Discrepancies between the implied volatility and historicalvolatility levels

Nov 1997 May 20090

10

20

30

40

50

60

Realized vol (2month average)

Implied vol (VIX)

Substantial differences between historical and option-basedmeasures of skewness and kurtosis [Bakshi et. al] have been

documented.Rudra P. Jena - [email protected] misspecification of SV models


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Background

Misspecification studied extensively in Black Scholesmodel with misspecified volatility

El Karoui, Jeanblanc & ShreveWe address the question of Misspecified stochastic volatility

models.



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Background



models. Financial engineering folklore !misspecified correlation a risk reversalmisspecified volatility of volatility a butterfly spread



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Background



models. Financial engineering folklore !misspecified correlation a risk reversalmisspecified volatility of volatility a butterfly spread

We concentrate on arbitrage strategies involving

underlying asset

liquid European options.


S i


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SettingReal world dynamics

Under real-world probability P, the underlying price S follows astochastic volatility model

dSt/St = tdt + (Yt)12t dW

1t + (Yt)tdW

2t

dYt = atdt + btdW2t ,

: R (0,) is a Lipschitz C1-diffeomorphism(y) > 0 for all y

R; , a, b > 0 and

[

1, 1] areadapted(W1, W2) is a standard 2-dimensional Brownian motion.


S i


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Setting

Process t, which represents the instantaneous volatility usedby the options market for all pricing purposes.We assume that t = (Yt)

dYt = atdt + btdW2t , (1)

where at and bt > 0 are adapted. : R (0,) is a Lipschitz C1-diffeomorphism with0 <

(y)

0 for all y

R;


S tti


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SettingThe Market model dynamics

Assumptions,Another probability measure Q, called market or pricingprobability

All traded assets are martingales under Q

The interest rate is assumed to be zeroUnder Q, the underlying asset and its volatility form a2-dimensional Markovian diffusion:

dSt/St = (Yt)12(Yt, t)dW

1t + (Yt)(Yt, t)dW

2t

dYt = a(Yt, t)dt + b(Yt, t)dW2t ,

a, b and are deterministic functions.


S tti C td


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Setting Contd.

Suppose that a continuum of European options for allstrikes and at least one maturity, quoted in the market.

The price of an option with maturity date T and pay-offH(ST) of St, Yt and t:

P(St, Yt, t) = EQ[H(ST)|Ft].

For every such option, the pricing function P belongs to theclass C2,2,1((0,) R [0, T)) and satisfies the PDE

aP

y+ LP = 0,

where we define

Lf = ft

+S2(y)2

22f

S2+

b2

22f

y2+ S(y)b

2f

Sy.



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Under our assumptions any such European option can beused to complete the Q-market.(Romano, Touzi)

And price satisfies

P

y> 0, (S, y, t) (0,) R [0, T).

The real-world market may be incomplete in our setting.


Deca Properties of the Greeks


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Decay Properties of the Greeks

Lemma

Let P be the price of a call or a put option with strike K andmaturity date T. Then

limK+

P(S, y, t)

y= lim

K0P(S, y, t)

y= 0,

limK+

2P(S, y, t)y2

= limK0

2P(S, y, t)y2

= 0,

and limK+

2P(S, y, t)

Sy= lim

K02P(S, y, t)

Sy= 0

for all(y, t) R [0, T). All the above derivatives arecontinuous in K and the limits are uniform in S, y, t on anycompact subset of(0,) R [0, T).


Sketch of the Proof


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Sketch of the Proof

The option price satisfies,

aP

y+ LP = 0,

Differentiate w.r.t. y and S,

Use Feynman Kac representation to relate the variousgreeks to the fundamental solutions of pde.

Using the classical bounds for fundamental solutions ofparabolic equations.


Formulation of the Problem


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Formulation of the Problem

The arbitrage problem is set up from the perspective of a trader,

Who knows market is using misspecified modelWants to construct a strategy to benefit from thismisspecification.

The first step,

sets up a dynamic self financing delta and vega-neutralportfolio Xt with zero initial value.at each date t, a stripe of European call or put options witha common time to expiry Tt.t(dK) : quantity of options with strikes between K and

K + dKt of stockBt of cash.

|t(dK)| = 1Rudra P. Jena - [email protected] misspecification of SV models

Formulation contd


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Formulation contd.

The value of the resulting portfolio is,

Xt =

PK(St, Yt, t)t(dK) tSt + Bt,

The dynamics of this portfolio is given by,

dXt =

t(dK)

LPKdt + P

K

SdSt +

PK

ydYt

tdSt

where,

Lf = ft

+S2t (Yt)

2

22f

S2+

b2t2

2f

y2+ St(Yt)btt

2f

Sy



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choose,t(dK)

PK

y= 0,

t(dK)

PK

S= t

to eliminate the dYt and dSt terms.

The resulting portfolio is risk free.The portfolio dynamics reduces to,

dXt =t(dK)LPKdt,



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Now we can write down the risk free profit from modelmisspecification as,

dXt =

t(dK)(L L)PK

dt.

At the liquidation date T,

XT

=T

0

t(dK)(L L)P

K

dt,

where,

(L

L)PK =

S2t (2t 2(Yt))

2

2PK

S2+

(b2t b2t )2

2PK

y2

+ St(tbtt (Yt)btt) 2PK

Sy


The problem in a Nutshell


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The problem in a Nutshell

The trader needs to maximize this aribtrage profit.Taking advantage of arbitrage opportunity to the followingoptimisation problem,

Maximize Pt =

t(dK)(L L)PK

subject to|t(dK)| = 1 and

t(dK)

PK

y= 0.

ANSWER: Spread of only two options is sufficient to solvethis problem.


General Result


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General Result

Proposition

The instantaneous arbitrage profit is maximized by

t(dK) = w1t K1t

(dK) w2t K2t (dK),

whereK(dK) denotes the unit point mass at K , (w1t , w

2t ) are

time-dependent optimal weights given by

w1t =

PK2y

PK1y +

PK2y

, w2t =

PK1y

PK1y +

PK2y

,

and(K1t , K2t ) are time-dependent optimal strikes given by

(K1t , K2t ) = arg max

K1,K2

PK2

y (L L)PK1 PK

1

y (L L)PK2

PK1

y +PK2

y

.


Sketch of Proof


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Sketch of Proof

The proof is done in two steps,

First show that the optimization problem is well-posed, i.e.,

the maximum is attained for two distinct strike values.show that the two-point solution suggested by thisproposition is indeed the optimal one.


The Black Scholes case


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The Black Scholes case

The misspecified model is the Black-Scholes with constant

volatility (but the true model is of course a stochasticvolatility model).

In the Black-Scholes model (r = 0):

P

= Sn(d1)T = Kn(d2)T,2P

S= n(d1)d2

,

2P

2=

Sn(d1)d1d2

T

,

where d1,2 = m

T

T

2 , m = log(S/K) and n is the standardnormal density.



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Proposition

Let b = = 0. The optimal option portfolio maximizing theinstantaneous arbitrage profit is described as follows:

The portfolio consists of a long position in an option withlog-moneyness m1 = z1

T 2T2 and a short position in

an option with log-moneyness m2 = z2

T 2T2 , wherez1 and z2 are maximizers of the function

f(z1, z2) =(z1 z2)(z1 + z2 w0)

ez21/2 + ez

22/2

with w0 =(bT+2)

b

T.

The weights of the two options are chosen to make theportfolio vega-neutral.

We define by Popt the instantaneous arbitrage profit realized by

the optimal portfolio.


Proof


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Proof

Substituting the Black-Scholes values for the derivatives ofoption prices,change of variable z = m

T+

T

2 ,the function to maximize w.r.t. z1, z2 becomes:

n(z1)n(z2)

n(z1) + n(z2)

b

T

2(z21 z22 )

bT

2(z1 z2) (z1 z2)

,

from which the proposition follows directly.


Role of Butterflies and Risk reversals: Part 1


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Role of Butterflies and Risk reversals: Part 1

Proposition

Let b = = 0, and define byP

opt the instantaneous arbitrage

profit realized by the optimal strategy.

Consider a portfolio (RR) described as follows:

If bT/2 + 0buy 12 units of options with log-moneyness

m1 = T

2T2 , or, equivalently, delta value

N(1) 0.16selling 12 units of options with log-moneyness

m2 =

T 2T2 , or, equivalently, delta value N(1) 0.84.if bT/2 + < 0 buy the portfolio with weights of theopposite sign.

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

problem under the additional constraint that it is

-antisymmetric.


Part 2


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Part 2

Proposition

Consider a portfolio (BB) consisting in

buying x0 units of options with log-moneyness

m1 = z0

T 2T , or, equivalently, delta valueN(z0) 0.055, where z0 1.6 is a universal constant.buying x0 units of options with log-moneyness

m2 = z0T 2T , or, equivalently, delta valueN(z0) 0.945selling1 2x0 units of options with log-moneynessm3 = 2T2 or, equivalently, delta value N(0) = 12 .

The quantity x0 is chosen to make the portfolio vega-neutral,that is, x0 0.39.Then, the portfolio (BB) is the solution of the maximization

problem under the additional constraint that it is-symmetric.


Part 3


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Part 3

PropositionDefine byPRR the instantaneous arbitrage profit realized by theportfolio of part 1 and byPBB that of part 2. Let

=

|bT + 2

||bT + 2|+ 2bK0Twhere K0 is a universal constant, defined below in the proof,

and approximately equal to0.459. Then

PRR Popt and PBB (1 )Popt.


Sketch of Proof


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S etc o oo

The maximization problem can be reduced to,

maxSb2

T

2

z2n(z)t(dz) Sb(bT/2 + )

zn(z)t(dz)

subject to n(z)t(dz) = 0, |t(dz)| = 1.Observe that the contract (BB) maximizes the first term whilethe contract (RR) maximizes the second term. The values forthe contract (BB) and (RR) are given by

PBB = Sb2T

2ez20/2, PRR = Sb|bT/2 + |

2e 12 .



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therefore

PRRPBB + PRR =

|bT + 2||bT + 2|+ 2bK0

T

with K0 = e12

z202 .

Since the maximum of a sum is always no greater than the sumof maxima, Popt PBB + PRR


Remarks


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Risk reversals are never optimal and butterflies are notoptimal unless = bT2 .Nevertheless, risk reversals and butterflies are relativelyclose to being optimal, and have the additional advantageof being independent from the model parameters, whereas

the optimal claim depends on the parameters.This near-optimality is realized by a special universal riskreversal (16-delta risk reversal in the language of foreignexchange markets) and a special universal butterfly

(5.5-delta vega weighted buttefly).When b 0, 1, In this case RR is nearly optimal.


Stochastic Volatility Model


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y

A simple stochastic volatility model, which captures all thedesired effects, the SABR = 1 .

The dynamics of the underlying asset under Q is

dSt = tSt(

1 2dW1t + dW2t ) (2)dt = btdW

2t (3)

The true dynamics of the instantaneous implied volatility is

dt = btdW2t , (4)

and the dynamics of the underlying under the real-worldmeasure is

dSt = tSt(

1 2dW1t + dW2t ). (5)


First order correction


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Call option price C satisifies the following pricing equation,

C

t + S2

2

2C

S2 +

b2

2

2C

2 + Sb

2C

S = 0

stochastic volatility is introduced as a perturbation b = .Look for asymptotic solutions of the form,

C = C0 + C1 + 2

C2 + O(3

)Here C0 corresponds to the leading Black Scholes solution.

C0t

+ S222C0S2

= 0

The first leading order to satisfies the following equationneglecting the higher order terms O(2),

C1 =2(T t)

2S

2C0S


Perturbation Results


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0.1 0.2 0.3 0.4 0.5 0.6

v

1.32

1.34

1.36

1.38

1.40

k1

0.1 0.2 0.3 0.4 0.5 0.6v

1.0290

1.0295

1.0300

1.0305

1.0310

1.0315

Figure: Optimal Strikes for the set of parameters = .2,S = 1, b = .3, = .3, = .5, t = 1, as a function of themisspecified b

[.01, .4].


Numerical Example and ConclusionsNumerical Setup


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Numerical Setup

The trader is aware about the misspecification.

Stock price = 100 and volatility = 0.1

Real world parameters: b = .8, =

.5

Market or pricing parameters: b = .3, = .7Demonstration for only one month options.

Results are shown for 100 trajectories of the stock andvolatility.



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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.1

0.0

0.1

0.2

0.3

0.4

0.5

time (in yrs)

Portfolio

value

misspecifiednot misspecified

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

time (in yrs)

Portfolio

value

misspecifiednot misspecified

Figure: The evolution of portfolios using options with 1 month

Left: The true parameters are =

.2, b = .1. The

misspecified or the market parameters are = .3, b = .9.include a bid ask-fork of 0.45% in implied volatility terms forevery option transaction.The evolution of the portfolioperformance with 32 rebalancing dates.



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Thank You

This research is part of the Chair Financial Risksof the RiskFoundationsponsored by Socit Gnrale, the Chair

Derivatives of the Futuresponsored by the Fdration BancaireFranaise, and the Chair Finance and Sustainable

Development sponsored by EDF and Calyon.


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Arbitrage Opportunities in Misspecified Stochastic Volatility Models