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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
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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
Misspecification studied extensively in Black Scholesmodel with misspecified volatility
El Karoui, Jeanblanc & ShreveWe address the question of Misspecified stochastic volatility
models. Financial engineering folklore !misspecified correlation a risk reversalmisspecified volatility of volatility a butterfly spread
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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. 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.
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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.
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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;
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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.
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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.
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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).
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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.
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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.
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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
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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.
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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
.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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)
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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
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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].
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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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