implied risk-neutral distribution as a closed-form function of the volatility smile
TRANSCRIPT
Electronic copy available at: http://ssrn.com/abstract=1738965
Implied risk-neutral distribution as a closed-form function of
the volatility smile
Bertrand TAVIN ∗
Universite Paris 1 - Pantheon Sorbonne
January 26, 2011
Abstract
The aim of this paper is to obtain the risk-neutral density of an underlying asset price as a
function of its option implied volatility smile. We derive a known closed form non-parametric
expression for the density and decompose it into a sum of lognormal and adjustment terms.
By analyzing this decomposition we also derive two no-arbitrage conditions on the volatility
smile. We then explain how to use the results. Our methodology is applied first to the pricing
of a portfolio of digital options in a fully smile-consistent way. It is then applied to the fitting
of a parametric distribution for log-return modelling, the Normal Inverse Gaussian.
Keywords : Option pricing, Risk-neutral distribution, Implied volatility smile.
JEL Classification : C14, C52, G13.
Contents
1 Introduction and financial framework 2
2 Derivation of the formula 3
3 Working with the formula 7
4 Applications 12
4.1 Pricing of Digital Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Fitting a Parametric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 Conclusion 17
A Appendix 19
A.1 Proof of 2.9 and 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19A.2 Expression of the log-return density . . . . . . . . . . . . . . . . . . . . . . . . . 20A.3 Computation of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
∗Acknowledgements : I am grateful to my PhD supervisor Prof. Patrice PONCET (ESSEC Business School)for his comments and helpful discussions. I also thank Peter CARR and Jens JACKWERTH for their commentson an earlier version. All remaining errors are mine. Contact : PRISM - EA4101 17 rue de la Sorbonne 75005Paris France
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Electronic copy available at: http://ssrn.com/abstract=1738965
1 Introduction and financial framework
The knowledge of the risk-neutral density of an asset price is of great interest for researchersand market practitioners. It has applications such as option pricing and analysis of marketinformation. Following the seminal result of Breeden and Litzenberger in [7] (Breeden &Litzenberger, 1978) the study of risk-neutral densities is the topic of an important literature.
The great majority of this literature details approaches where risk-neutral density is obtainedfrom option prices. See among others [17] (Jackwerth & Rubinstein, 1996) [2] (Aıt-Sahalia
& Lo, 1998) and [11] (Figlewski, 2010). See also [16] (Jackwerth, 2004) for a detailed reviewof literature and methodologies. In [6] (Bliss & Panigirtzoglou, 2002) authors recommendan intermediary step where prices are converted to implied volatility to be smoothed and thenconverted back to prices for the density to be computed. [21] (Kermiche, 2009) uses thisapproach to construct the evolution over time of the risk-neutral density implied in CAC40options and performs a principal component analysis to study its behavior.
Our approach to compute risk-neutral density is much simpler because it involves only onestep as we consider implied volatility as the available market data for options. Monitoring anoptions market through implied volatility can be seen as more convenient than doing it throughprices. A first reason is that implied volatility can be instantly compared across strikes, whereprices need to be adjusted for the underlying price before being compared. Another reason isthat, on some markets, options are negotiated in terms of implied volatility instead of price. Forexample this convention is in use for over-the-counter currency options. See [22] (Lee, 2005)who also advocates using implied volatility to represent options data.
In this paper we propose a methodology to directly build a risk-neutral density from impliedvolatility data. We take the Breeden-Litzenberger formula, see [7] (Breeden & Litzenberger,1978), as starting point and derive a known result that can be seen as an implied volatility versionof it. This result has already been obtained in [15] (Jackwerth, 2000) and [8] (Brunner &Hafner, 2003).
It is possible to obtain this implied volatility version because there is a one to one relationbetween option price and implied volatility. This one to one relation is Black-Scholes formula, see[5] (Black & Scholes, 1973), or Black’s formula, see [4] (Black, 1976), both seen as functionsof the volatility parameter σ (other parameters and variables held constant).
The paper is organized as follows. In the sequel of section 1 we explain the financial setup. Insection 2 we derive a closed form expression of the underlying asset price’s risk-neutral density asa function of the volatility smile and make some comments on its terms. In section 3 we examinehow to use the formula and provide a numerical example. In section 4 we apply our methodto the pricing of a portfolio of digital options. In section 5 we study how to fit a parametricdistribution for log returns, the Normal Inverse Gaussian (NIG), directly to the volatility smileusing the formula and avoiding the computation of option prices.
We consider a complete financial market in continuous time where an asset S paying nodividends is traded. We denote St its time t price. European vanilla options on S are alsotraded. A strike K, maturity T , vanilla option pays at time T
CT = (ST −K)+for a Call option
PT = (K − ST )+ for a Put option
We consider that a money market and zero-coupon bonds are available for trading. Wedenote n(t) the time t value of the money market account (with n(0) = 1) and B(t, T ) the time
2
t price of the ZC bond maturing at time T . The money market account is an asset earning theinstantaneous risk-free rate, which is assumed to follow a stochastic process rt (t ≥ 0):
n(t) = exp
(∫ t
0
rsds
)
(1.1)
As we suppose that our market is free of arbitrage, there exists a risk-neutral probability Q
equivalent to the true (historical) probability P. This probability is associated with (n(t), t ≥ 0)as numeraire. Asset prices can be written as expectations, under Q, of their discounted payoffs.
B(0, T ) = EQ
[
1
n(T )
]
= EQ
[
exp
(
−∫ T
0
rsds
)]
(1.2)
C∗0 = EQ
[
CT
n(T )
]
= EQ
[
exp
(
−∫ T
0
rsds
)
(ST −K)+
]
(1.3)
P ∗0 = EQ
[
PT
n(T )
]
= EQ
[
exp
(
−∫ T
0
rsds
)
(K − ST )+
]
(1.4)
When interest rates are stochastic it is more convenient to work with the forward-neutralprobability QT associated with T , the maturity of the European options. QT is the probabilityassociated with (B(t, T ), t ≥ 0) as numeraire:
dQT
dQ=
B(T, T )
B(0, T )
n(0)
n(T )=
1
B(0, T )n(T )(1.5)
C∗0 = B(0, T )EQT [CT ] = B(0, T )EQT
[
(ST −K)+]
(1.6)
P ∗0 = B(0, T )EQT [PT ] = B(0, T )EQT
[
(K − ST )+]
(1.7)
For details on this change of probability, see chap. 11 and 19 in [25] (Portait & Poncet,2010). In the sequel, we work with undiscounted option prices, defined as
C0 =C∗
0
B(0, T )= EQT
[
(ST −K)+]
P0 =P ∗0
B(0, T )= EQT
[
(K − ST )+]
(1.8)
2 Derivation of the formula
Considering a vanilla option price as a function of strike and maturity, the Breeden-Litzenbergerformula links φT , the risk-neutral density of ST , with a second order partial derivative of thisfunction. For a fixed T > 0, and ∀k ≥ 0
φT (k) =1
B(0, T )
∂2C∗
∂K2(k, T ) =
1
B(0, T )
∂2P ∗
∂K2(k, T ) (2.1)
3
This formula is key to obtain Dupire’s equation for local volatility models, see [9] (Dupire,1993) or to reveal anticipations of market participants embedded in option prices, see [18](Jondeau & Rockinger, 2000). With undiscounted option prices, C and P , the formulasimplifies to
φT (k) =∂2C
∂K2(k, T ) =
∂2P
∂K2(k, T ) (2.2)
In our stochastic interest rates setting φT is the density of ST under QT the T -forwardneutral probability. When rates are deterministic, the forward and risk-neutral probabilities areidentical, that is QT = Q ∀T ≥ 0.
Whether interest rates are deterministic or not does not matter here. Breeden-Litzenbergerformula gives the underlying density under the relevant pricing measure. So that the price of aEuropean claim v paying h(ST ) at time T can always be written as
v∗0 = B(0, T )
∫ +∞
0
h(x)φT (x)dx = B(0, T )
∫ +∞
0
h(x)∂2C
∂K2(x, T )dx (2.3)
A maturity T , strike K, vanilla option on S can be seen as an option on FT , the forwardcontract written on S and expiring at time T . As ST = FT
T , we have
CT = (ST −K)+=(
FTT −K
)+PT = (K − ST )
+=(
K − FTT
)+
The time t forward price of S is FTt = St
B(t,T ) (with t ∈ [0, T ]). It is a martingale under QT .
Under the assumption of a constant volatility geometric Brownian motion diffusion for F ,Black’s formula holds, see [4] (Black, 1976). The undiscounted call price given by Black’sformula at time t = 0 is
CB (F, σ,K, T ) = FN (d1)−KN (d0) (2.4)
d0 (F, σ,K, T ) =ln F
K
σ√T
− 1
2σ√T d1 = d0 + σ
√T
where N is the cumulative distribution function of a standard Gaussian:
N : x 7−→ N (x) =1√2π
∫ x
−∞
exp
(
− t2
2
)
dt (2.5)
This function is not known in closed form but fast and accurate approximations are available,see for example Chap. 26 in [1] (Abramovitz & Stegun, 1972).
Black’s formula seen as a function of the volatility (σ 7−→ CB (F, σ,K, T )) is strictly increas-ing. Hence from any observed call price it is possible to numerically back out a unique impliedvolatility parameter by inverting this function. Observed option prices are market prices ormodel outputs.
Practitioners represent vanilla options market data as implied volatility because it is easierto interpret and to monitor. For a given maturity, implied volatility as a function of strike isnot constant and often smile shaped or skewed. It is usually called volatility smile. Although
4
this fact invalidates the assumption behind Black’s formula it is still possible to use it to expressobserved prices with a strike dependent volatility parameter
Cobs (K,T ) = CB (F, σ (K,T ) ,K, T ) (2.6)
In the right hand side of (2.6) the strike dependence is twofold. Plugging this representationof observed prices into formula (2.2) should allow us to obtain the risk-neutral density of ST asa function of the volatility smile. The second order derivative of C with respect to K will beexpressed with partial derivatives of Black’s formula with respect to K and σ. Applying twicethe chain rule for partial derivatives leads to
∂C
∂K=
∂CB
∂K+
∂CB
∂σ
∂σ
∂K(2.7)
φT =∂2C
∂K2=
∂2CB
∂K2+ 2
∂2CB
∂σ∂K
∂σ
∂K+
∂2CB
∂σ2
(
∂σ
∂K
)2
+∂CB
∂σ
∂2σ
∂K2(2.8)
Partial derivatives of CB involved in (2.8) are known in closed form. They are similar to theGreeks in Black’s model: Gamma and Vanna wrt strike, Vomma (also called Volga) and Vega.
∂2CB
∂K2=
n (d0)
Kσ√T
∂2CB
∂σ∂K=
n (d0) d1σ
(2.9)
∂2CB
∂σ2=
d0d1
σn (d1)F
√T
∂CB
∂σ= n (d1)F
√T (2.10)
where n = ∂N∂x
is the probability density function of a standard Gaussian.
n : x 7−→ n(x) =1√2π
exp
(
−x2
2
)
Proof. see Appendix A.1
As K 7−→ σ(K,T ) is the smile function, partial derivatives of σ involved in (2.8) make itdependent of the volatility smile, unsurprisingly. Graphically they correspond to the smile slope,convexity and squared slope. It is interesting to remark that only the first two derivatives ofthe smile are needed to express the risk-neutral density. We define now G = G(K,T ) andH = H(K,T ) as the smile slope and convexity (with respect to strike) at point (K,T ).
∂σ
∂K= G
∂2σ
∂K2= H (2.11)
Substituting (2.9), (2.10) and (2.11) in (2.8) we get the density as a closed form non-parametric function of the implied volatility smile
φT =n (d0)
Kσ√T
+ 2Gn (d0) d1
σ+
(
G2 d0d1
σ+H
)
n (d1)F√T (2.12)
This formula is also derived in [15] (Jackwerth, 2000) where it is used to obtain risk aversionfunctions, and in [8] (Brunner & Hafner, 2003) where it is used in the context of a no-arbitrageanalysis of the volatility smile.
5
We now rearrange the above formula to yield the key result of the paper decomposing φT asa lognormal density term f plus two adjustment terms A1 and A2.
φT (k) = f(k) +A1(k) +A2(k) k > 0 (2.13)
Where f is a probability density function, which corresponds to the lognormal density of ST
in Black’s model with a volatility parameter equal to σ0 = σ(F, T ), the ATM implied volatility(an option is said At The Money if struck at F ). A1 is a level adjustment term, accountingfor the difference between ATM implied volatility σ0 and its value at strike. And A2 is also anadjustment term, accounting for the slope and convexity of the smile. These terms are written
f(k) = fLN
(
k; lnF − 1
2σ20T, σ
20T
)
A1(k) =1
k√T
(
1
σ(k)n (d0(F, σ(k), k, T )) −
1
σ0n (d0(F, σ0, k, T ))
)
A2(k) = 2G(k)n (d0) d1σ(k)
+
(
G(k)2d0d1
σ(k)+H(k)
)
n (d1)F√T
The function x 7−→ fLN
(
x;m, s2)
is the density function of a lognormal distribution withparameters m and s2. That is, ∀x > 0
fLN
(
x;m, s2)
=1
xs√2π
exp
(
−1
2
(
lnx−m
s
)2)
Option traders usually analyze the implied volatility smile in three steps, examining wellknown trading strategies. The first step is to check the ATM level, kept as a reference. In termsof options trading it usually corresponds to an ATM straddle1 strategy. The second is to comparethe implied volatility at a specified strike K with the ATM reference. It corresponds to a callspread2 (or put spread) strategy with one option struck at F and the other at K. The third stepis to analyze the overall shape of the volatility smile, its slope and convexity. That is done witha risk reversal strategy3 for the slope and with a butterfly4 for the convexity.
The decomposition (2.13) of the density as a closed form function of the smile is sound andmakes sense financially because it is built exactly on the same three steps.
Pursuing further the analysis of (2.13) we derive no-arbitrage conditions on adjustment terms.For φT to be a proper probability density function it should integrate to 1 on R+ and be positiveeverywhere. That is
∫ +∞
0
φT (k)dk = 1
φT (k) ≥ 0 ∀k ≥ 0
1A long ATM Straddle involves a call and a put, both long and struck at F .2A Call Spread involves two calls, a long and a short, with different strikes.3A Risk Reversal involves a long call, struck higher than F , and a short put, struck lower than F .4A long Butterfly involves two long calls, struck higher and lower than F , and two short calls, struck at F .
6
This constraint on the density function is used in [8] (Brunner & Hafner, 2003). See also[22] (Lee, 2005) who provides no-arbitrage bounds on the slope of the volatility smile.
As f already is a probability density function, it verifies
∫ +∞
0
f(k)dk = 1
f(k) ≥ 0 ∀k ≥ 0
Hence we bring out two no-arbitrage conditions on the volatility smile. The first one is global.The second one is local and holds at any particular point.
∫ +∞
0
(A1(k) + A2(k)) dk = 0 (2.14)
A1(k) +A2(k) ≥ −f(k) ∀k ≥ 0 (2.15)
The particularity of the above two results is to link at the same time the level, slope andconvexity of the volatility smile.
3 Working with the formula
To work with the risk-neutral density formula (2.13) one needs implied volatility values at anypositive strike. Market data for traded options is only available at discrete strike points. Theimplied volatility smile corresponding to real market data is a set of points instead of a continuouscurve. So we need an interpolation/extrapolation engine for the implied volatility : that is atechnique to produce a continuous smile, given a discrete market data set.
We consider three different techniques, which correspond to different approaches to smile gen-eration. Many techniques are acceptable for that purpose. We choose to present and implementthree classical techniques. The first smile generation technique we consider is a parametrizationbased on the implied volatility formula derived in stochastic volatility SABR model, introducedby [14] (Hagan et al., 2002). We take the version of the formula proposed in [24] (Obloj,2008) because it has less occurrence of arbitrage at very low strikes than Hagan’s version. Thisparametrization works with 3 parameters (α0, ρ, v) plus a fixed fourth parameter β.
Under SABR parametrization, implied volatility is written, for K 6= F
σSABR(K,F ) =v ln
(
FK
)
ln
(√1−2ρz+z2+z−ρ
1−ρ
)
(
1 + T
(
1
24
(1− β)2 α20
(FK)(1−β)+
1
4
ρβα0v
(FK)12(1−β)
+2− 3ρ2
24v2
))
(3.1)and for K = F
σSABR(F, F ) =α0
F 1−β
(
1 + T
(
1
24
(1− β)2 α20
F 2(1−β)+
1
4
ρβα0v
F (1−β)+
2− 3ρ2
24v2
))
where
z =v
α0
F (1−β) −K(1−β)
1− β
7
The second smile generation technique is the SVI parametrization proposed in [13] (Gatheral,2004). SVI stands for Stochastic Volatility Inspired. This parametrization is arbitrage free andworks with 5 parameters (a, b, v, ρ,m). Under SVI parametrization, implied volatility is written
σSV I(K,F ) =1√T
√
√
√
√a+ b
(
ρ
(
lnK
F−m
)
+
√
(
lnK
F−m2 + v2
)
)
(3.2)
SABR and SVI parametrization need to be calibrated to the available market smile points. Itis done by minimizing the sum of squared errors on volatility values using a deterministic, gradientbased, optimization routine (Levenberg-Marquardt algorithm). We consider that market data isavailable for a number N of strikes. Calibration is done by solving the following minimizationprograms
P1 : min(α,ρ,v)
N∑
i=1
(σSABR(Ki, T )− σMarket(Ki, T ))2
P2 : min(a,b,v,ρ,m)
N∑
i=1
(σSV I(Ki, T )− σMarket(Ki, T ))2
The last technique we consider is a cubic spline interpolation/extrapolation of the smile,where implied volatility is built as a piecewise polynomial function of degree three. Impliedvolatility at strike K is written
σCS(K) = p1(K)1]0,K1[(K) +
N−1∑
i=1
pi(K)1[Ki,Ki+1[(K) + pN−1(K)1[KN ,+∞[(K) (3.3)
where (pi)N−1i=1 are polynomials of degree three defined as
pi(X) = αi (X −Ki)3+ βi (X −Ki)
2+ δi (X −Ki) + γi
The cubic spline method needs no actual calibration but it is necessary to compute M , thematrix of coefficients. This is done by solving a set of linear equations. For details see Chap 2.in [27] (Stoer and Burlirsch, 1993) and Chap. 3 in [26] (Press et al., 2007).
M =
α1 β1 δ1 γ1...
......
...αi βi δi γi...
......
...αN−1 βN−1 δN−1 γN−1
The advantage of this approach is a perfect fit to the market data points but the drawbackis the possibility of non regularity in high order derivatives. This occurs when data is not very
8
smooth. Although the use of cubic spline, as a smile generation engine, can give rise to numericalinstabilities we present the results obtained with it to illustrate that the choice of parametrizationmethod is crucial because it can seriously affect the results.
Figure 1 shows implied volatility market data points and the smile curves produced withfitted smile generation engines. The considered maturity T is one year and the underlying hasa T -forward value normalized at 100. It can be noted that between data points smile curves areclose to each other, but away from data points differences are noticeable and due to the differentbehaviors of parametrization methods.
0 20 40 60 80 100 120 140 160 180 2000.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
K
σ imp
Market DataSABRSVICubic Spline
Figure 1: Implied Volatility Smile
Figure 2 shows the risk-neutral density obtained using our method (with the different smileengines). Lognormal density corresponding to Black’s model is plotted as a reference. Non-parametric density curves are significantly different from the lognormal one. Compared to itthey are skewed to the left and the cubic spline curve has irregularities around its maximum.
Figure 3 shows the total adjustment term A1 + A2 and the no-arbitrage bound correspond-ing to the local condition (2.15). The three plotted curves are above the no-arbitrage bound,illustrating that the condition is met.
Figures 4 and 5 show the details of adjustment terms A1 and A2 respectively. For cubic splinecurves, we remark that the irregularities come from the shape adjustment term A2.
9
0 20 40 60 80 100 120 140 160 180 2000
0.005
0.01
0.015
0.02
0.025
0.03
k
φ T
Black (with ATM vol)SABRSVICubic Spline
Figure 2: Risk-Neutral Density
0 20 40 60 80 100 120 140 160 180 200−10
−8
−6
−4
−2
0
2
4
6
8x 10
−3
k
A1+
A2
SABRSVICubic SplineNo−arbitrage bound
Figure 3: Total Adjustement Term
10
0 20 40 60 80 100 120 140 160 180 200−3
−2
−1
0
1
2
3
4
5x 10
−3
k
A1
SABRSVICubic Spline
Figure 4: Level Adjustment Term
0 20 40 60 80 100 120 140 160 180 200−10
−8
−6
−4
−2
0
2
4
6
8x 10
−3
k
A2
SABRSVICubic Spline
Figure 5: Shape Adjustment Term
11
4 Applications
Our approach to expressing the risk-neutral density, detailed in sections 2 and 3, is now appliedto concrete cases. The first case we present is the pricing of a portfolio of digital options onS in a fast and fully smile-consistent way. In the second we model the log-return of S usinga parametric distribution, the Normal Inverse Gaussian, and fit it to the non-parametric risk-neutral distribution obtained from the volatility smile.
4.1 Pricing of Digital Options
A European, maturity T , digital call (put) option is an option paying 1 at time T if ST is above(below) K, its strike, and 0 otherwise.
DCT = 1{ST≥K} for a digital callDPT = 1{ST≤K} for a digital put
Digital options are known to be sensitive not only to the implied volatility level at strike butalso to the shape of the volatility smile, see Chap. 17 in [28] (Taleb, 1997). Using a smiledBlack model to price it (a formula with a strike dependent volatility) is not appropriate becausethe whole shape of the smile is not accounted for.,The undiscounted price of a digital call optionDC0 is obtained as
DC∗0 = B(0, T )EQT
[1{ST≥K}
]
= B(0, T )QT ({ST ≥ K}) (4.1)
DC0 = QT ({ST ≥ K}) =∫ +∞
K
φT (k)dk (4.2)
It can also be written directly using (2.3) with h(ST ) = 1{ST≥K}
DC0 =
∫ +∞
0
1{k≥K}φT (k)dk =
∫ +∞
K
φT (k)dk
Our method is particularly suitable for handling large portfolios of digital options because itallows for a systematic pricing, consistent with the full smile. To obtain digital option prices itis necessary to compute the integral in (4.2). A numerical quadrature method is needed becauseφT , while known, has no parametric form. In our numerical implementation we use a Gauss-Kronrod adaptative quadrature method. This method is appropriate because it is robust andallows for a control of error.
Figure 6 below shows the prices of digital calls on S. Results obtained with the 3 differentsmile interpolation techniques are plotted. Prices corresponding to Black’s model (with volatilityinput equals to ATM value) are also shown as a reference. Market data is identical as in Part 3.
As expected, prices are substantially different from prices obtained with Black’s model, evenfor the ATM struck option, because of the smile shape. Error made using Black’s price is notnegligible, it is positive or negative depending on the option’s strike. For this application, ourmethod is little sensitive to the choice of smile interpolation technique. If we except some irreg-ularities, corresponding to cubic spline method, curves are close to each other. And differencesare small if compared with typical bid-offer spreads.
12
To plot figure 6, digital calls were priced (2000 different strikes) with 3 smile interpolationmethods: the computation is done in less than 20 seconds with a Matlab program on a personalcomputer with no parallelization (Dual core CPU at 2.1GHz).
40 60 80 100 120 140 1600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
K
DC
0
Black (with ATM vol)SABRSVICubic Spline
Figure 6: Prices of digital calls
4.2 Fitting a Parametric Distribution
In this second application we model XT , the time T log-return of S defined as
XT = ln
(
ST
S0
)
(4.3)
We first use the results obtained in Part 2 to get a non-parametric density for the log-return. We then introduce the Normal Inverse Gaussian distribution (NIG hereafter) to modelthe distribution of XT .
Considering that φT , the risk-neutral density of ST , is known and given by (2.13), it is possibleto express fnp the non-parametric density of XT as
fnp(x) = S0exφT (S0e
x) x ∈ R (4.4)
Proof. See Appendix A.2
This approach to obtain the risk-neutral density of log-return from the risk-neutral density ofasset price is also used in [21] (Kermiche, 2009) and [11] (Figlewski, 2010). Figure 7 presentsfnp graphs for different smile construction techniques. Density obtained with Black’s model isalso plotted as a reference, it is a Gaussian density.
13
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
x
f np(x
)
Black (with ATM vol)SABRSVICubic Spline
Figure 7: Non-parametric density of log-return
Table 1 presents the values of the first four moments1 of XT , computed using the methodpresented in Appendix A.3 and with the non-parametric density fnp. Moments of XT in Black’smodel are also computed to provide a check on the numerical quadrature method.
Mean Variance Skewness KurtosisSABR -0.0194 0.0421 -1.2714 6.7719SVI -0.0198 0.043 -1.2662 5.7697Spline -0.0193 0.0429 -1.178 5.5898Black -0.0179 0.0357 0 3
Table 1: Moments of XT numerically computed with non-parametric density
To model XT we now use the NIG distribution. It is parametric and we use fnp to determineits parameter set θ = (α, β, µ, δ).
The NIG distribution was introduced in [3] (Barndorff-Nielsen, 1997). It allows for themodelling of fat tails, skewness and kurtosis. For applications of the NIG distribution see [19](Kalemanova, Schmid & Werner, 2007) where authors use it in the context of CDO2 pricing.See also [10] (Eriksson,Ghysels & Wang, 2009) where it is applied to the modelling of therisk-neutral density of an underlying asset price. For details on its properties and implementationsee [20] (Kalemanova & Werner, 2006).
If we suppose the distribution of XT under QT to be a NIG, its density fθ is
1For a definition of Mean, Variance, Skewness and Kurtosis see Appendix A.3.2A CDO (Collateralized Debt Obligation) is a multi-name credit derivative.
14
fθ(x) =δα exp(δγ + β(x− µ))
π√
δ2 + (x− µ)2K1
(
α√
δ2 + (x− µ)2)
x ∈ R (4.5)
γ =√
α2 − β2 and K1 is the second kind modified Bessel function of order 1, see Chap. 9 in[1] (Abramovitz & Stegun, 1972).
We want to fit the NIG distribution of XT , that is the parameter vector θ, according tothe volatility smile considered to be the available market data. It can be done by matchingrisk-neutral moments, computed numerically using fnp. This method takes advantage of therelationship between moments and parameters of the NIG distribution. It is also used, forexample, in [10] (Eriksson,Ghysels & Wang, 2009).
α β µ δ
SABR 9.9596 -5.7498 0.1419 0.2282SVI 68.0358 -62.5623 0.3956 0.1775
Spline 25.9729 -20.5437 0.3107 0.2553
Table 2: NIG parameters (moment matching method)
The results in Table 2 appear to depend on the smile generation technique. This is becausethe computation of moments is rather sensitive to it. In particular the kurtosis is significantlyhigher for the SABR parametrization (see Table 1).
Another approach to fit the NIG parameters is to minimize the distance between the distri-butions, that is the distance between fθ and fnp seen as elements of the set of probability densityfunctions, with fnp fixed and considered to be the true distribution. This method accounts forthe whole distribution but involves an optimization step.
To achieve the fitting we consider three different distance criteria :
the Hellinger distance
DH (fθ, fnp) =
√
1
2
∫ +∞
−∞
(
√
fθ(x)−√
fnp(x)
)2
dx (4.6)
the L2-norm
L2 (fθ, fnp) =
√
∫ +∞
−∞
(fθ(x) − fnp(x))2dx (4.7)
the Relative Entropy (Kullback-Leibler divergence)
DKL (fnp||fθ) =∫ +∞
−∞
fnp(x) ln
(
fnp(x)
fθ(x)
)
dx (4.8)
It can be noted that H and L2 are proper distances. H lies within [0, 1] and L2 is a memberof the Lp − norms family.
The Kullback-Leibler divergenceDKL has been introduced in information theory. DKL(F ||G)quantifies the additional information needed to describe a reference model F , when a model Gis given. It is used in probability and statistics as a non-symmetric measure of dissimilaritybetween distributions.
15
The optimization programs to solve are
P1 : minθ=(α,β,µ,δ)
DH (fθ, fnp)
P2 : minθ=(α,β,µ,δ)
L2 (fθ, fnp)
P3 : minθ=(α,β,µ,δ)
DKL (fnp||fθ)
These are non linear minimization problems for a multivariate and real valued objectivefunction, solved with a downhill simplex algorithm. Values shown in Table 2, obtained bymoment matching, are used as seed to start the algorithm. Although it makes sense to use thisseed, the algorithm is not sensitive to it.
Table 3 shows the distance values associated to the NIG distributions with parameters usedas seed of the algorithm. Tables 4, 5 and 6 show the NIG parameters obtained after the mini-mization, for different distance criteria and smile interpolation techniques.
Hellinger L2 KLSABR 0.0419 0.0915 0.006SVI 0.015 0.0166 0.0015Spline 0.0504 0.1759 0.0289
Table 3: Distance between non-parametric density and NIG density (parameters in Table 2)
α β µ δ
Hellinger 25.2998 -19.6381 0.3044 0.2629L2 26.5032 -20.3435 0.3171 0.2791KL 24.7267 -19.1248 0.3003 0.262
Table 4: Fitted NIG parameters for SABR parametrization
α β µ δ
Hellinger 32.6036 -27.6054 0.3186 0.2127L2 46.0199 -40.9847 0.3512 0.1904KL 31.0953 -26.141 0.3127 0.2142
Table 5: Fitted NIG parameters for SVI parametrization
α β µ δ
Hellinger 30.2305 -25.4527 0.3048 0.208L2 30.209 -26.4908 0.2851 0.1732KL 18.2102 -13.9883 0.2365 0.2132
Table 6: Fitted NIG parameters for cubic spline parametrization
16
Solving the distance minimization problem is fast. To obtain Tables 4, 5 and 6 takes 20seconds. All computations are done with a Matlab program on a personal computer with noparallelization (Dual core CPU at 2.1GHz).
Figure 8 shows the risk-neutral NIG densities of log-return obtained after minimizing theKullback-Leibler divergence for the different smile generation techniques. Local differences ob-served near their maxima illustrate the sensitivity of the results to the choice of the smile tech-nique. Curves are globally similar to each other which confirm the stability of our approach.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
x
f NIG
(x)
SABRSVICubic Spline
Figure 8: Fitted NIG density of log-return (relative entropy criterion)
5 Conclusion
We have derived in this paper a closed form expression of an underlying asset price risk-neutraldensity as a non-parametric function of the volatility smile. The obtained density formula hasbeen decomposed as a lognormal density, corresponding to Black’s model (fitted ATM), plustwo adjustment terms accounting for smile level and shape. In the way we have also underlinedtwo no-arbitrage conditions concerning the volatility smile. The implementation steps of ourapproach have been detailed and numerical results have been provided in the context of practicalapplications. It proves to be simple to implement and to perform well and fast on realistic marketdata.
Our methodology to build risk-neutral density appears to be suitable for industrial as wellas research purposes. For example, a risk management department can use it to control thevaluation of a trading portfolio of European derivatives. It can also be used to obtain marginaldistributions when modelling the joint distribution of several underlying assets.
17
References
[1] Abramovitz, M. and I.A. Stegun (1972). Handbook of mathematical functions, Dover (10thprinting).
[2] Aıt-Sahalia, Y. and A. Lo (1998). Nonparametric Estimation of State-Price Densities Implicitin Financial Asset Prices. The Journal of Finance, Vol. 53, No. 2 (Apr., 1998), pp. 499-547
[3] Barndorff-Nielsen, O.E. (1997). Normal Inverse Gaussian distributions and stochastic volatil-ity modelling. Scandinavian Journal of statistics, Vol. 24 (1997), pp. 113
[4] Black, F. (1976). The Pricing of Commodity Contracts. Journal of Financial Economics, Vol.3, No. 1, pp. 167-179
[5] Black, F. and M. Scholes (1973). The Pricing of Options and Corporate Liabilities. TheJournal of Political Economy, Vol. 81, No. 3 (May - Jun., 1973), pp. 637-654
[6] Bliss, R. and N. Panigirtzoglou (2002). Testing the stability of implied probability densityfunctions. Journal of Banking and Finance, Vol. 26, pp. 381-422
[7] Breeden, D. and R. Litzenberger (1978). Prices of State-Contingent Claims Implicit in OptionPrices. The Journal of Business, Vol. 51, No. 4 (Oct., 1978), pp. 621-651
[8] Brunner, B. and R. Hafner (2003). Arbitrage-free estimation of the risk-neutral density fromimplied volatility smile. The Journal of Computational Finance, 7(1) pp. 75-106
[9] Dupire, B. (1993). Pricing and hedging with smiles. Proceedings of AFFI conference, LaBaule, June 1993
[10] Eriksson, A., Ghysels, E. and F. Wang (2009). The Normal Inverse Gaussian Distributionand the Pricing of Derivatives. The Journal of Derivatives, Vol. 16 (Spring 2009), pp. 23-37
[11] Figlewski, S. (2010). Estimating the Implied Risk Neutral Density for the U.S. MarketPortfolio. In Volatility and Time Series Econometrics: Essays in Honor of Robert F. Engle,Oxford University Press.
[12] Foata, D. et A. Fuchs (2003). Calcul des Probabilites, Dunod (2eme Ed.)
[13] Gatheral, J. (2004). A parsimonious arbitrage free implied volatility parametrization. GlobalDerivatives conference, Madrid 2004.
[14] Hagan, P., D. Kumar, A. Lesniewski and D. Woodward (2002). Managing smile risk. WilmottMagazine July 2002, 84-108
[15] Jackwerth, J. (2000). Recovering Risk Aversion from Option Prices and Realized Returns.The Review of Financial Studies, Vol. 13, No. 2 (Summer, 2000), pp. 433-451
[16] Jackwerth, J. (2004). Option-Implied Risk-Neutral Distributions and Risk Aversion. TheResearch Foundation of the AIMR.
[17] Jackwerth, J. and M. Rubinstein (1996). Recovering Probability Distributions from OptionPrices. The Journal of Finance, Vol. 51, No. 5 (Dec., 1996), pp. 1611-1631
[18] Jondeau, E. and M. Rockinger (2000). Reading the smile: the message conveyed by methodswhich infer risk neutral densities. Journal of International Money and Finance,Vol. 19 (2000),pp. 885-915
18
[19] Kalemanova, A., Schmid, B. and R. Werner (2007). The Normal Inverse Gaussian Distri-bution for Synthetic CDO Pricing. Journal of Derivatives, Spring 2007, Vol. 14, No. 3, pp.80-94
[20] Kalemanova, A. and R. Werner (2006). A short note on the efficient implementation of theNormal Inverse Gaussian distribution. Working Paper
[21] Kermiche, L. (2009). Dynamics of Implied Distributions: Evidence from the CAC 40 OptionsMarket. Finance, 2009, Vol. 30 Issue 2, pp. 63-103
[22] Lee, R. (2005). Implied Volatility: Statics, Dynamics and Probabilistic Interpretation. InRecent Advances in Applied Probability, Springer.
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[24] Obloj, J. (2008). Fine-tune your smile, correction to Hagan et al., Wilmott Magazine,May/June 2008, pp. 102-109
[25] Portait, R. and P. Poncet (2009). Finance de marche, Dalloz (2eme Ed.)
[26] Press, W., Teukolsky, S., Vetterling, W. and B. Flannery (2007). Numerical Recipes, TheArt of Scientific Computing, Cambridge Press (3rd Ed.)
[27] Stoer, J. and R. Burlirsch (1993). Introduction to Numerical Analysis, Springer (2nd Ed.)
[28] Taleb, N. (1997). Dynamic hedging, Managing vanilla and exotic options. John Wiley &Sons Inc, Wiley Finance Editions.
A Appendix
A.1 Proof of 2.9 and 2.10
First recall three usefull identities for greeks calculation
∂d1
∂K=
∂d0
∂K
∂d1
∂σ=
∂d0
∂σ+√T Fn (d1) = Kn (d0)
We then obtain
∂CB
∂K= F
∂N∂d1
∂d1
∂K−K
∂N∂d0
∂d0
∂K−N (d0) = −N (d0)
∂CB
∂σ= F
∂N∂d1
∂d1
∂σ−K
∂N∂d0
∂d0
∂σ= (Fn (d1)−Kn (d0))
∂d0
∂σ+ Fn (d1)
√T = n (d1)F
√T
∂2CB
∂K2= −∂N
∂d0
∂d0
∂K=
n (d0)
Kσ√T
19
∂2CB
∂σ∂K= −∂N
∂d0
∂d0
∂σ= −n (d0)
(
− 1
σ2
ln FK√T
− 1
2
√T
)
=n (d0) d1
σ
∂2CB
∂σ2=
∂n
∂d1
∂d1
∂σF√T =
(
1√2π
(−d1) exp
(
−d212
))
(
− 1
σ2
ln FK√T
+1
2
√T
)
F√T
=d0d1
σn (d1)F
√T
A.2 Expression of the log-return density
Let g be a monotone and differentiable function and X a random variable with density fX . LetY be the random variable defined as Y = g(X). Its density fY can be expressed as
fY : y 7−→ fY (y) =
∣
∣
∣
∣
∂
∂yg−1(y)
∣
∣
∣
∣
fX(
g−1(y))
For details, see Chap. 15 in [12] (Foata & Fuchs, 2003).
To obtain (4.4), all you need to do is to apply the above result with XT seen as a transfor-mation of ST . That is
XT = g(ST )
with
g(x) = ln
(
x
S0
)
x > 0
g−1(y) = S0ey y ∈ R
∂
∂yg−1(y) = S0e
y y ∈ R
so thatfnp(y) = |S0e
y|φT (S0ey) = S0e
yφT (S0ey) y ∈ R
A.3 Computation of Moments
Considering a random variable X with fX its density function. Mean, Variance, Skewness andKurtosis the first four moments are defined, when they exist, as (respectively)
m = E [ X ] v = E
[
(X −m)2]
s = E
[
(
X −m√v
)3]
k = E
[
(
X −m√v
)4]
20
If X is Gaussian, s = 0 and k = 3. Those values are considered as references when analyzingmoments of a distribution.
The computation of (m, v, s, k) relies on the numerical computation of the following integralfor k = 1, 2, 3, 4
Ik = E[
Xk]
=
∫ +∞
0
xkfX(x)dx
Once the Ik are computed for k = 1, 2, 3, 4, moment values are recovered sequencially usingthe following identities
m = I1
v = I2 −m2
s = v−32
(
I3 − 3mv −m3)
k = v−2(
I4 − 4msv32 − 6m2v −m4
)
Proof.
v = E
[
(X −m)2]
= E[
X2 − 2Xm+m2]
= I2 −m2
s = E
[
(
X −m√v
)3]
= v−32E[
X3 − 3X2m+ 3Xm2 −m3]
= v−32
(
I3 − 3m(
v +m2)
+ 2m3)
= v−32
(
I3 − 3mv −m3)
k = E
[
(
X −m√v
)4]
= v−2E[
X4 − 4X3m+ 6X2m2 − 4Xm3 +m4]
= v−2(
I4 − 4m(
sv32 + 3mv +m3
)
+ 6m2(
v +m2)
− 3m4)
= v−2(
I4 − 4msv32 − 6m2v −m4
)
21