Implied Vol Surface

OVDV Equity Volatility Surface

Equity Implied Volatility Surface

Changrong Cui and David Frank

Quantitative Research and Development, Equities Team

June 21, 2011

Version 3.6

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Introduction

This document describes the construction of the equity volatility surfaces shownin the function OVDV. Volatility surface construction involves data filtering, impliedforward and implied dividend calculation, and option data fitting. We discuss eachof these steps in separate sections below.

In the option data fitting step, our method is based on the mixture of lognormaldistributions. At the end of this document we present the connection between thelognormal mixture method and stochastic volatility models.

Data Filtering

We determine whether to use bid/ask or settlement data as follows:

1. For each maturity, the bid/ask data at the maturity is deemed to be good if atleast 10% of the quotes and at least 3 options have valid bid/ask prices.

2. Use bid/ask data to build the surface if at least 10% of maturities or at least3 maturities have good bid/ask data; otherwise use exchange settlement data(”last”).

The data is then filtered as follows:

1. Filter out maturities that are too short. Currently the threshold is 7 days.

2. Eliminate in-the-money options (we use only out-of-the-money options for thesurface except for the implied forward computation).

3. Eliminate last data older than 0.001 years.

4. Eliminate ”bad” price data with too small bid/ask/last prices or too wide bid/askspreads.

5. To ensure monotonicity of prices, we take the longest increasing/decreasingsubsequence from call/put prices at each maturity and throw out the remain-der.

6. At a given maturity, we calculate implied volatilities from mid prices (when usingbid/ask) or from settlement prices. Then we compare each implied volatility tothe median of all implied volatilities. We eliminate those strikes whose impliedvolatilities are outside a certain range. Currently the range is taken as 0.2 to5.0 times the median volatility.

7. After all valid strikes at all maturities have been found, we check the numberof strikes at each maturity. Again, we compare the number of strikes of eachindividual maturity to the median value across all maturities. If the number ofstrikes is below 20% of the median, we filter out this maturity.

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Implied Forwards and Dividends

We compute option implied forwards and dividends as follows. We take as input a listof declared dividends (dates and cash amounts) and a dividend schedule (dates andweights between discrete (cash or proportional) vs. continuous yield). For maturitiesthat have no scheduled dividend between the current maturity and previous maturity,the forward is calculated from the forward at the previous maturity and the yieldcurve. For other maturities (rolling forward), implied dividends and implied forwardsare calculated using the following iterative method:

1. Start with the forward computed from input dividends and the yield curve asan initial guess.

2. If the the options have American-style exercise, for each strike, compute im-plied volatilities from call and put prices, and then compute European call andput prices from implied volatilities.

3. Now using only European call and put prices, find up to 20 put-call pairs start-ing from the forward and moving outwards to both directions. Let {fi} be theforwards, calculated from put-call parity: forward = call - put + strike.

4. Calculate the median fMED of the {fi}. Keep up to 10 forwards closest to themedian, and average those forwards; that is the estimate of the forward.

5. If the original option prices were European, stop here. Otherwise, comparethe implied forward in (4) with the old forward calculated in (1). If the relativeerror is smaller than a given threshold, then stop. Otherwise go back to (1),replacing the original dividends with the newly computed dividends.

Finally, let tn be the last given discrete dividend time. If there are additional optionmaturities left after dividends have been implied at tn then continuous dividend yieldswill be implied at those maturities using the method described in the next section onoption data fitting. If, on the other hand, the last discrete dividend that can be impliedis at time tm < tn, then dividends are extrapolated for t > tm as continuous dividendyield using equivalent yield from discrete dividend in the last one year period, i.e.dividend yield q such that Ftm = Ftm−j

e(r−q)(tm−tm−j) where tm − tm−j ≥ 1.0, Ftm andFtm−j

are forwards. If there is only less than a year’s implied dividends available,then tm−j can go back to historical dividend time.

Use of Discrete Absolute, Discrete Proportional, and Continuous Dividends

For each (discrete) dividend time (measured from “today”) 0 = t0, t1, . . . , tn, tn+1 = Twe are given weights - for cash dividends: ωd

1 , . . . , ωdn; for stock (proportional) div-

idends: ωy1 , . . . , ω

yn and for (continuous) yields: ωq

1 , . . . , ωqn. Those weights deter-

mine the percentage of different dividends at time ti. The sum of all three weights

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ωdi + ωy

i + ωqi = 1. Currently for single name stocks all ωd

i are set to 1 and for indicesall ωq

i are set to 1. In other words, we currently imply only discrete dividends forsingle name stocks and only continuous dividend yields for indices.

Let Fi denote the forward at time ti. We will imply out the dividend amounts in cashdi and in stock yi paid at ti, as well as continuous yield q applying over the period(0, T ). We do this using the following strategy:

1. Imply d̂ from

Ft = F0erT − d̂

n∑i=1

er(T−ti)

2. Imply q from

F0e(r−q)T = F0e

rT − d̂n∑

i=1

ωqi e

r(T−ti)

3. Imply d from

FT = F0e(r−q)T − d

n∑i=1

e(r−q)(T−ti)

4. For i = 1, . . . , n do

(a) Let di = dωdi

ωdi + ωy

i

(b) Let Fi = Fi−1e(r−q)(ti−ti−1) − d

(c) Imply yi from Fi = Fi−1e(r−q)(ti−ti−1) (1− yi)− di

Option Data Fitting - Mixture of Lognormal Distributions

The mixture of lognormal distribution was proposed by [Brigo and Mercurio (2002)]and the basic idea is to assume that the risk-neutral probability density function ofthe spot price at any maturity is a weighted sum of some lognormal density functionswith different means and variances. An obvious advantage over other parametricapproaches is that the density function is positive and therefore we never have callspread artibrage or butterfly arbitrage. Another advantage is that pricing a vanillaEuropean option only needs a few Black-Scholes formula calculations, which is veryaccurate and fast.

To quantify the description, the risk-neutral probability density function of the stockprice at any future time T > 0 is assumed to be in the following form

pdf(T, S) =N∑l=1

pl(T ) · lognormalpdf(S, ξl(T )F (T ),Σl(T ))

where

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• N is the number of lognormals,

• F (T ) is the forward price,

• pl(T ) is the time-dependent weight of the l-th lognormal,

• ξl(T ) is the time-dependent multiplicative means of the l-th lognormal,

• Σl(T ) is the time-dependent deviation of the l-th lognormal.

We require

• pl(T ) > 0, ξl(T ) > 0, Σl(T ) > 0

•∑

l pl(T ) ≤ 1. If the sum of weights is less than 1, then then there is a pointmass at zero which represents a probablity of default. A strictly positive defaultprobability could help to fit a high skew in the low strike range, which is commonin equities market.

•∑

l pl(T )ξl(T ) = 1. The weighted sum of means is the expectation of ST and itmust be equal to the forward price.

Now, the price of a European call option with maturity T and strike K can be writtenas

C(T,K) =N∑l=1

pl(T ) · blsprice(ξl(T )S0, K, r, T,Σl(T )/√T )

and we can get the implied volatility at any point using this call price formula, whereblsprice(S,K, r, T, σ) is the Black-Scholes price of a call option with spot S, strike K,risk-free rate r, time to expiration T and volatility σ.

Figure 1 shows an example of the mixture of two lognormals. The mixed lognormalhas larger density in the middle range and fatter tail as shown in Figure 2. Also, themixed lognormal produces the volatility smile as shown in Figure 3.

It is difficult to have a perfect answer for the choice of N , the number of lognormals.If N is too small, the volatility smile would not have enough flexibility to fit the marketdata; if N is too large, there will be an overfitting issue. In practice, we choose N = 4and use a more rigid structure for the four lognormals when there are not a sufficientnumber of valid options.

We now describe the optimization which finds the mixed lognormal parameters bestfitting the market prices. The target of our non-linear least square problem is Eu-ropean option prices. If the market options are not European-style exercise, thenwe have already converted the market prices to European prices using the method-ology described in the section “Implied Forwards and Dividends”. The optimization

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0 0.5 1 1.5 2 2.5 30

0.5

1

1.5Density Functions

Mixed LognormalLognormal

Figure 1. Density functions for mixed lognormals and one lognormal. N = 2, ξ1 =0.95, ξ2 = 1.05, σ1 = 0.40, σ2 = 0.20, p1 = p2 = 0.5, T = 1, F = 1. The deviation ofthe single lognormal is chosen to match the variance of ST in the mixed lognormal.

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2 2.2 2.4 2.6 2.8 30

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05Density Functions Tail


Figure 2. Density function Tails, Figure 1 enlarged.

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0 0.5 1 1.5 2 2.5 3

0.3

0.32

0.34

0.36

0.38

0.4

0.42Volatility Smile


Figure 3. Volatility Smile Produced by Mixed Lognormal.

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is done maturity-by-maturity, except that if the optimizations among different marketmaturities are done completely independently, there may be calendar arbitrage. Tocontrol that, we use constraints on the lognormal parameters of different maturities,for example, for each l = 1, · · · , N ,

0 < Σl(Tmkt1 ) < Σl(T

mkt2 ) < · · · < Σl(T

mktnmkt

)

With this constraint, we can build an increasing curve Σl(T ) from market maturitypoints. Similarly, we can have the other two classes of curves ξl(T ) and pl(T ),l = 1, · · · , N . Having constructed the curves pl(T ), ξl(T ) and Σl(T ), we can thencompute the surface volatility given any strike and maturity by interpolating or ex-trapolating the curves to determine the parameters at the required point, computingthe weighted sum of Black-Scholes prices, and inverting to get the required impliedvolatility.

Having the mid implied volatility, we calculate the bid/ask volatility spread using linearinterpolation and flat extrapolation in strike dimension and Hermite interpolation andflat extrapolation in maturity dimension.

Connection Between Lognormal Mixture and Stochastic Volatility

We restate a result in [Fouque, Papanicolaou and Sircar (2000)] to show that themixed lognormal model is an approximation to a wide range of stochastic volatilitymodel. Consider a stochastic volatility model

dSt

St

= rdt+ σt(√1− ρ2dWt + ρdZt)

σt = f(Yt)

dYt = µ(t, Yt)dt+ ν(t, Yt)dZt

Then by Ito’s formula,

d(lnSt) = (r − 1

2σ2t )dt+ σt(

√1− ρ2dWt + ρdZt)

Given a path of the second Brownian motion {Zt}, {Yt} is deterministic and so is{σt} and therefore

lnST = lnS0 +

∫ T

0

(r − 1

2σ2t )dt+ ρ

∫ T

0

σtdZt +√

1− ρ2∫ T

0

σtdWt

is normally distributed with mean

lnS0 +

(r − 1

2σ2ρ

)T + ρ

∫ T

0

σtdZt

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and varianceσ2ρT

where

σ2ρ =

1

T

∫ T

0

(1− ρ2)σ2t dt

Let

ξt = exp

(ρ

∫ t

0

σsdZs −1

2ρ2

∫ t

0

σ2sds

)Then ST is continuously-mixed-lognormal distributed, with multiplicative drift ξT and

volatility√σ2ρ at each realization of {Yt}. When the number of lognormals goes to

infinity, the discrete lognormals will converge to this model.

References

[Brigo and Mercurio (2002)] D. Brigo and F. Mercurio (2002), Lognormal-MixtureDynamics and Calibration to Market Volatility Smiles, International Journal ofTheoretical & Applied Finance 5(4), 427-446.

[Fouque, Papanicolaou and Sircar (2000)] J-P. Fouque, G. Papanicolaou, K. R. Sir-car (2000), Derivatives in Financial Markets with Stochastic Volatility, Cam-bridge University Press

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Implied Vol Surface