Best Practice Modeling for
Equity Structured Products
Dilip B. MadanRobert H. Smith School of Business
University of Maryland
CARISMA Event
7 City Learning
London
June 28 2007
OUTLINE
• Description of Models and their properties.
• Results of Market Calibration.
• Model Prices for Equity Structured Products.
• Spot and Option Risks to be Hedged in EquityStructured Products.
Description of Models and their Properties
• There are two broad classes of models used incalibrating the prices of options and on occasion,
forward starting options.
— One dimensional Markov models in which the
future evolution of the stock price at any time
depends on just the level of the current stock
price.
— Stochastic volatility models in which the volatil-
ity of stock has an independent component to
its evolution.
• The sample paths of prices may be continuous,continuous with occasional jumps in the price, or
purely discontinuous in that the uncertain com-
ponent is just the sum of all the discrete jumps
or moves.
— The continuous sample path models may be
seen as ones with an uncertain component
made up of infinitesimally small jumps.
• I will describe two one dimensional Markov mod-els, one that has continuous sample paths and the
other that is purely discontinuous. These are
— (LV ) The Local Volatility Model.
— (LL) The Local Levy Model.
• For the class of stochastic volatility models, I willdescribe the structure of 9 models. These are
— (HSV ) Heston Stochastic Volatility.
— (SV J) Merton Jump Diffusion with stochas-
tic volatility.
— (V GSA) Stochastic Volatility for Levy Processes.
— (SV DNE) Jump Diffusion with stochastic volatil-
ity and Double Negative Exponential Jumps.
— (SV V G) Jump Diffusion with stochastic volatil-
ity and Variance Gamma Jumps.
— (SV CGMY ) Jump Diffusion with stochastic
volatility and CGMY jumps.
— Stochastic volatility and stochastic jump ar-
rival for DNE,V G,CGMY or the models
∗ (SV ADNE), (SV AV G), (SV ACGMY ).
Local Volatility Model
• For all models we focus attention on the uncer-tainty, the drift or growth rate for pricing purposes
is the interest rate net of the dividend yield or the
net finance cost.
• The local uncertainty for a local volatility model isa zero mean Normally distributed random variable
with a variance that is an unknown function of the
stock price and calendar time.
— The variance is not constant as it is for Black
Scholes.
∗ But is given by the functionσ2(S, t)
if at time t the stock price is at level S.
• Dupire (1994) showed how one may recover thisfunction from the prices of traded call options of
all strikes K and maturities T.
• Specifically we have the formula in terms of callprices C(K,T ) that
σ2(K,T ) =2
K2CKK(CT + ηC + (r − η)KCK) .
• Different strategies are used for interpolating callprices across the strike and maturity ranges to
recover the local volatilities.
• A well known problem with local volatility is that
the market skew is calibrated completely by the
dependence of volatility on the spot price.
— As a result we get a sharp dependence with
the consequence that for low spots there is
a substantial probability of getting to a high
spot but not the other way around,
∗ So over time, volatilities and skews drop aswe observed earlier.
• This makes the model unsuited for cliquet struc-tures that are highly dependent on forward skews.
Local Levy Models
• Recently, Carr, Geman, Madan and Yor (2004)generalized local volatility to allow for local un-
certainty to be modeled by a Levy distribution
that can accomodate asymmetry directly by al-
lowing for the arrival rate of negative moves at a
higher rate than the correspondingly sized posi-
tive move.
• In this way some of the skew is hard wired into
the process with a residual part being modeled
by the dependence of local quadratic variation on
the spot.
• In a local Levy model, the local variance is mod-eled by the speed at which the Levy process is run-
ning and we have a local space dependent speed
function
a(S, t).
• Carr, Geman, Madan and Yor (2004) show how
to recover this speed function from traded option
prices.
• Specifically we have the formula
•a(K,T ) =
b (ln(K), T )
K2CKK
where we call the function b(ln(K), T ) the log
speed function and it may be recovered from op-
tion prices written as functions of the log strike
c(k, T ) = C(ek, T ).
• using
Z ∞−∞
b(y)ψe(k − y)dy = (cT + ηc+ (r − η) ck)
where the function ψe is known in terms of the ar-
rival rate of jumps of the localizing Levy process.
• An implementation of local Levy on SPX for
20040706 shows that forward volatilities and skews
maintain their levels and shapes as evidenced by
the following forward implied volatility curves.
Heston Stochastic Volatility
• Stochastic volatility models are important for man-aging the exposure to volatility of volatility forproduts with volgamma. The most basic stochas-tic volatility is the Heston model in which the lo-cal variance of the stock is mean reverting with
long term variance = η
mean reversion rate = κ
— and local variance is itself stochastic with alocal volatility of
vol of vol = λ
— with a correlation between the volatility andstock movements of
correlation = ρ.
— the initial level of variance is also to be esti-mated
initial volatilty = v0
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.250.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36Six month Forward Start Implied Volatilities in Local Levy on SPX 20040706
Strike
Impl
ied
Vol
atili
ty
Six months out
2 years out3 years out
4 years out
Figure 1:
• The sample paths are continuous and in this ab-sence of jumps the shorter maturity options are
poorly calibrated.
Merton Jump Diffusion with stochastic volatility
• Here we add to the Heston stochastic volatilitymodel jumps in prices that arrive at rate
jump arrival rate = λJ
with a jump size that is normally distributed with
mean and variance
jump size mean = µJ
jump size variance = σ2J
• This type of jump may estimate a model witha large negative mean jump with small volatility
and arrival rate and hence is viewed as reflecting
a fear of a market crash.
Stochastic Volatility for Levy Processes
• Purely discontinuous Levy processes parsimoniouslycalibrate option prices at each maturity well.
• An example is the three parameter variance gammaprocess obtained on evaluating Brownian motion
with drift θ and variance σ2 at a random time
given by a gamma process with unit mean and
variance ν.
• The process is purely discontinuous and can alsobe viewed as the difference of a gamma process
for the price up ticks and an independent gamma
process for the price down ticks. In this view the
parameters are
jump arrival rate
mean up tick
mean down tick
• For calibrating across maturity we introduce sto-chastic volatility by using the Heston process for
the speed at which time is running for the vari-
ance gamma process.
• The resulting process is called V GSA.
SVDNE, SVVG, SVCGMY
• Here we keep the structure of Merton jump diffu-sion with stochastic volatility but alter the jump
activity to come from
• DNE
— exponentially distributed positive and negative
jumps with
arrival rates = λp, λn
mean size = βp, βn
• V G
— gamma processes for up and down ticks with
V G parameters σ, ν, θ or arrival rates and two
mean jump sizes.
• CGMY
— positive jumps of size x have arrival rate
cpe−Mx
x1+Yp
— negative jumps of size −x have arrivale rate
cne−Gx
x1+Yn
Stochastic Jump Arrival Rates
• In all the above models, the arrival rate of jumpsis insensitive to the stochastic volatility. It isonly the continuous component or the infinites-imal jumps that respond to an increase in volatil-ity.
• In the stochastic arrival rate class of models welet the jump arrival rate have a linear response tothe stochastic volatility as well.
• We add this response to the SV DNE,SV V G, SV CGMYmodels and hence add two parameters for the re-sponse of the positive and negative jump arrivalrates to volatility to get the models SV ADNE,SV AV G, SV ACGMY. These new parametersare
sensitivity of positive arrival rate = sp
sensitivity of negative arrival rate = sn
Results of Calibration on SPX for 20040706
• Both local volatility and local Levy are nonpara-metric models and there is considerable experi-ence in calibrating local volatility.
• Here I focus attention on local Levy.
• We estimate for 20040706 the CGMY modelon option data for the first maturity exceeding
one month. The estimated parameters values forG,M,Y were
G = 5;M = 13;Y = .5
• We used these values for the localizing Levy processand obtained the log speed and speed functions.
• I present a graph of the resulting speed and logspeed functions
−0.4
−0.2
0
0.2
0.4 0
0.2
0.4
0.6
0.8
1
0
5
10
15
20
25
30
35
time
logspeedmspx20041214
logreturn
5
10
15
20
25
30
Figure 2:
Remarks on Levy Speed Function
• The Speed lifts sharply at the short maturity downside as opposed to the up side.
• At the back end the Speed movement is relatively
damped.
• This structure is comparable to what one is ac-customed to seeing for local volatility, except now
one is only purchasing part of the smile from the
spot depenence of speed or volatility.
• A considerable part has been hard wired by the
choice of an asymmetric localizing Levy process
with G = 5,M = 13 and 2% down moves coming
with a 16% greater frequency than 2% up moves.
HSV Calibration
• The results for these calibrations used 108 optionswith maturities ranging from one month to a year.
The APE = 3.25%.
initial volatility = 16.14%
long term volatility = 20.77%
mean reversion = 3.8057
vol of vol = .9147
correlation = −.6601
— ∗ There is an initial upward drift in volatil-ity. Mean reversion leads to convergence in
just under three months. The volatility of
volatility is quite substantial. Correlation is
negative.
SVJ Calibration
• For SV J or Merton jump diffusion with stochas-
tic volatility the structure is quite different in the
long term volatility. The APE = 4.38%.
initial volatility = 15.29%
long term volatility = 165.53%
mean reversion = .00618
vol of vol = .2751
correlation = −.6701
— to which we add a
jump arrival rate = .7291
mean jump size = −4.14%jump volatility = 24.99%
∗ There is a long term positive drift in volatil-
ity with the stock subject to occasional down
drifts that however, have a substantial volatil-
ity.
VGSA Calibration
• The APE = 2.67% .Stochastic volatility struc-
ture Parameter estimates are (C = 116.8068, G =
86.84,M = 114.536, κ = 5.5686, η = 65.7855, λ =
42.7730) we quote in volatility terms as
initial volatility = 15.62%
long term volatility = 11.72%
mean reversion = 5.5686
vol of vol = .3662
— and the jump structure is
M = 114.536
G = 86.84
∗ The calibration is characteristically very good.Mean reversion leads to convergence in a
few months. We have a slight down drift
in volatility to begin with. The jump moves
are on the small side with a slight skew to-
wards the negative moves.
SVDNE
• The APE = 3.08%. Stochastic volatility struc-
ture
initial volatility = 15.29%
long term volatility = 89.29%
mean reversion = .0211
vol of vol = .2711
correlation = −.6691
— and the jump structure is
arrival rate positive = 1.8319
mean positive jump = 8.73%
arrival rate negative = 2.0431
mean negative jump = 9.81%
∗ There is a long term upward drift in volatil-
ity. The jumps are sizable but not very fre-
quent.
SVVG
• The APE = 4.41%. Stochastic volatility struc-ture
initial volatility = 15.26%
long term volatility = 62.82%
mean reversion = .0398
vol of vol = .2695
correlation = −.6701
— and the jump structure is
σ = .5553
ν = 8.2277
θ = −.1619
• We may instead consider the jump quadratic vari-ation in vol terms and the size/direction premia.
volatility = 72.39%
direction premium = 2.12%
size premium = 4.21%
SVCGMY
• The APE = 3.08%. Stochastic volatility struc-ture
initial volatility = 15.26%
long term volatility = 18%
mean reversion = 5.0896
vol of vol = .6760
correlation = −.7
— and the jump structure is
cp = 3.5724
M = 16.7449
Yp = .4243
cn = 4.8817
G = 6.2530
Yn = .1650
• The quadratic variation of the positive jumps is27.39% and those of the negative jumps is 56.39%.
SVADNE
• The APE = 2.86%. Stochastic volatility struc-
ture
initial volatility = 14.69%
long term volatility = 16.63%
mean reversion = 3.3142
vol of vol = .5203
correlation = −.5842
— and the jump structure is
positive arrival rate = 4.3001
positive sensitivity = 3.4077
mean size = 10.05%
negative arrival rate = 3.1290
negative sensitivity = .000666
mean size = 20.31%
SVAVG
• The APE = 2.86%. Stochastic volatility struc-
ture
initial volatility = 14.69%
long term volatility = 16.64%
mean reversion = 3.31136
vol of vol = .5201
correlation = −.5849
— and the jump structure is
cp = .1418; sp = 4.1069;M = 2.1069
cn = .1438; sn = .1271;G = .9034
∗ Slight upward volatility drift. Mean rever-sion in a few months. More sensitivity of up
jumps to increases in volatility than down
jumps.
SVACGMY
• The APE = 2.86%. Stochastic volatility struc-
ture
initial volatility = 14.69%
long term volatility = 16.64%
mean reversion = 3.3114
vol of vol = .5200
correlation = −.5843
— and the jump structure is
cp = .000265; sp = 3.9523;M = 20.2194;Yp = .513
cn = .1944; sn = .2156;G = 4.8494;Yn = .1159
∗ The fit is second best after VGSA. Up movesare more sensitive to volatility moves. Jump
sizes are reduced relative to SVAVG.
Common Features of Models
• All models consistently estimate the initial volatil-ity around 15%.
• There is considerable agreement in the estimateof correlation at around −.67.
• Mean reversion is either absent or a few months.
• Estimates of the volatility of volatility are variableacross models.
• The jump structures are quite varied.
700 800 900 1000 1100 1200 1300 14000
10
20
30
40
50
60
70
80Fit of SVACGMY to SPX 20040706
Strike
Opt
ion
Pric
e
o option price data+ model price
Figure 4:
Sample of Fit for SVACGMY
The Products Priced by the Models
• We priced six products for all 11 models. Theseare
— Locally Floored and Globally Capped 4 year
monthly monitored Arithmetic Cliquet with three
local floors at −10,−7.5,−5 and global capsat 10, 20, 30, 40, and 50.
— Locally capped and Globally Floored 4 year
monthly monitored Arithmetic Cliquet with three
local caps at 5, 7.5, 10 and global floors at
−50,−40,−30,−20 and −10.
— 4 year monthly monitored Arithmetic Swing
cliquet with strikes 5, 6, 7, 8, 9 and 10 and three
caps at 80, 90 and 100.
— 4 year monthly monitored Arithmetic Reverse
Swing Cliquet with local caps at 5, 10, 15, 20, 25.
— 4 year monthly monitored Arithmetic Swing
Cliquet with Lock In and no caps, for strikes
5, 6, 7, 8, 9 and 10.
— TARN with down barriers at 50 to 70 in steps
of 2, cancellation at levels 110, 120, 130, 140
at 6%, 12%, 18%, 24% and long an at-the-money
call in the absence of cancellation or the down
event.
Model Pricing
• For the Locally-Floored and Globally capped cli-quets we have for each model 15 prices for three
floors five caps.
• The prices rise as we raise the caps five times andthen fall for the raised floor but lowest cap.
• We graph these 15 prices for all models.
Locally Floored Globally Capped Cliquet Model
Rankings
• For this product the average contract prices inrank order are
SVADNE 27.22SVCGMY 26.33LL 24.46SVJ 22.91SVDNE 18.14HSV 17.03SVAVG 16.60LV 16.26SVVG 14.94SVACGMY 11.99VGSA 8.56
0 5 10 15−10
0
10
20
30
40
50Locally Floored and Globally Capped Cliquet
local levy
local vol
vgsa
hsv
svj
svdne
svvg
svcgmy
svadne
svavg
svaccgmyy
Figure 5:
Locally Capped and Globally Floored Cliquet
• Here we three local caps and five global floors.
• The value rises as we raise the floors and falls aswe get to the next lowest cap with a higher floor.
• We observe that some models like SV J are really
exposed to the absence of the local floor and have
values stretching down to meet the global floor.
Locally Capped and Globally Floored
• The model rankings and average prices areVGSA 7.88LL 4.15LV 2.34HSV 2.19SVACGMY -1.0SVAVG -8.99SVVG -15.91SVCGMY -16.28SVDNE -17.41SVADNE -23.35SVJ -28.51
0 5 10 15−50
−40
−30
−20
−10
0
10
20Locally Capped and Globally Floored Cliquet
local levylocal vol
vgsa
hsv
svj
svdne
svvg
svcgmy
svadne
svavg
svaccgmyy
Figure 6:
Capped Swing Cliquet
• We have six strikes and three caps.
The value drops as we raise the strikes and rises as we
raise the cap.
Capped Swing Cliquet
• The model rankings and average prices areSVADNE 89.76SVCGMY 88.72SVJ 87.69SVDNE 74.33SVVG 67.05SVAVG 58.42LL 48.04LV 35.20HSV 35.03SVACGMY 29.11VGSA 13.13
0 2 4 6 8 10 12 14 16 180
10
20
30
40
50
60
70
80
90
100Swing Cliquet
local levy
local vol
vgsa
hsv
svj
svdne
svvg
svcgmy
svadne
svavgsvaccgmyy
Figure 7:
Capped Reverse Swing Cliquet
• We have five caps for the reverse swing cliquet.
• The value rises as we raise the cap.
Reverse Swing Cliquet
• The model rankings and average prices areVGSA 596.31SVACGMY 559.27HSV 551.03SVAVG 544.22LL 541.07SVVG 519.39LV 513.61SVDNE 481.55SVADNE 428.47SVCGMY 393.60SVJ 137.21
5 10 15 20 250
200
400
600
800
1000
1200Reverse Swing Cliquet
Cap
Reve
rse S
win
g C
liquet
local levy
local vol
vgsa
hsv
svj
svdne
svvg
svcgmy
svadne
svavg
svacgmy
Figure 8:
Uncapped Swing Cliquet with Lock In
• We have six strikes with value falling as we raisethe strike.
• The Lock In gives substantial value to the con-tract.
• The Lock In also exaggerates the model differ-ences.
• We drop SV J as the value is too exagerated.
Swing Cliquet with Lock In
• The model Rankings and average prices areSVADNE 1788SVVG 1671SVAVG 1129SVCGMY 1052SVDNE 895LL 555HSV 344SVACGMY 336LV 298VGSA 177
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100
200
400
600
800
1000
1200
1400
1600
1800
2000Swing Cliquet with Lock In
Strike
Sw
ing C
liquet W
ith L
ock
In
vgsa locvolsvacgmyhsv loclevy
svdnesvcgmy
svavg
svvg svadne
Figure 9:
Trigger Autocancellable Redeemable Note
• We have 11 down barrier strikes with value risingas we raise the strike and limit the loss experi-
enced.
• Again we drop SV J.
Trigger Autocancellable Redeemable Note
• The model prices and rankings areLL 70.62SVCGMY 69.49LV 67.61SVADNE 67.16SVDNE 64.96SVAVG 63.35HSV 59.96VGSA 58.16SVVG 57.24SVACGMY 53.27
50 52 54 56 58 60 62 64 66 68 7045
50
55
60
65
70
75Trigger Autocancellable Redeemable Note
Strike
TA
RN
PR
ICE
local Levy svcgmylocal vol
svadne
svdne svavg hsv
vgsa
svvgsvacgmy
Figure 10:
Model Rank Correlations
• The rankings of the 10 models, excluding SV J,
across the six products are
LFGC LCGF S RS SWLIN TARNLL 3 2 6 5 6 1LV 7 3 7 7 9 3VGSA 10 1 10 1 10 8HSV 5 4 8 3 7 7SVDNE 4 9 3 8 5 5SVDVG 8 7 4 6 2 6SVCGMY 2 8 2 10 4 2SVADNE 1 10 1 9 1 4SVAVG 6 6 5 4 3 9SVACGMY 9 5 9 2 8 10
• The model rank correlations between the productsare
LFGC LCGF S RS SWLIN TARNLFGC 1 -.56 .78 -.76 .59 .68LCGF -.56 1 -.87 .72 -.81 -.10S .78 -.87 1 -.90 .87 .52RS -.76 .72 -.90 1 -.60 -.72SWLIN .59 -.81 .87 -.60 1 .19TARN .68 -.10 .52 -.72 .19 1
• We expect that quite naturally when hedging toacceptability, one would quite often be using dif-
ferent models for different products.
Volatility Options
• We now consider the one year capped volatility
options that pays at the end of the year
Max
252 TX
t=1
Min
Ãln(St/St−1)2
T,cap2
252
!1/2 − k, 0
• We present a graph of the valuations from 9 cal-
ibrated models of this contract.
Risks in Equity Structured Products
• Each model when simulated for the evaluation ofthe price may also be used to detect the nature
of the risk in the product, at least as this is seen
from the perspective of the model.
• For each forward date one may use the simulatedcash flows to construct the forward spot slide us-
ing a kernel estimator for the value function.
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
capped volatility option prices
strike
cap
pe
d v
ola
tility
op
tion
price
vgsa
local Levy
Local Volatility
hsv,svadne,svdne
svavg,svacgmy
svcgmy
Figure 11:
• Say we are interested in the value of the struc-tured product at forward time n when the spot is
at p. Let cs be the simulated cash flow on path s
and let ps be the price of the underlying on path
s at time n. The kernel estimator for the value at
n when the spot is at p, V (p, n) is
V (p, n) =Xs
ws(p, ps)cs
where the weights ws(p, ps) are a normalization
of the sequence
exp
Ã−µp− ps
2h
¶2!
for a prespecified band width h.
Forward Spot Slides and Option Trades
• We illustrate here for the locally floored, globallycapped (LFGC), locally capped globally floored
(LCGF ) and swing (S) cliquet under the model
SV ACGMY.
• The top graph is the spot slide for one month out,six months out and one year out.
• The bottom graph plots the difference in value
functions and gives the option trade needed to
hedge the change in the value function.
Implicit Hedge Costs
• We see that for LFGC we have the forward risk of
acquiring some out of the money puts and selling
some out of the money calls.
— An increase in the skew will make this trade
more expensive and hence the strong skew
delta and gamma of this position.
• In contrast, the swing cliquet has us buying outof the money calls and puts.
— Hence the exposure to volgamma in this case.
85 90 95 100 105 110 11510
11
12
13
14
15
16Forward Spot Slides for LFGC
85 90 95 100 105 110 115−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6Forward Option Trades for LFGC
One Month Out
Six Months Out
One Year Out
One Month Trade
Six Month Trade
Figure 12:
85 90 95 100 105 110 115−12
−10
−8
−6
−4
−2Forward Spot Slides for LCGF
85 90 95 100 105 110 115−1.5
−1
−0.5
0
0.5
1Forward Option Trades for LCGF
One Month Out
Six Months Out
One Year Out
One Month TradeSix Months Trade
Figure 13:
85 90 95 100 105 110 11536
37
38
39
40
41
42
43Forward Spot Slides for Swing Cliquet
85 90 95 100 105 110 115−0.5
0
0.5
1
1.5
2
2.5Forward Option Trades for Swing Cliquet
One Month Out
Six Months Out
One Year Out
One Month Trade
Six Month Trade
Figure 14:
Conclusion
• Equity Structured Products are a desirable way ofrisk positioning for the investing public.
• Model contingent hedging and valuation is feasi-ble for the supplier.
• It is a consequence of hedging to acceptabilitythat different models be used to support the val-
uation and hedging of different products.
— In fact even for the same product, different
models are the support for the bid and the
ask price.