Transcript Callable Agencies - Incisive Media Plc

See the Disclosure Appendix for the Analyst
Certification and Other Disclosures.
Pricing and Risk Management of
Commodity Exotics
July 9, 2008
Yann Coatanlem
Managing Director
Multi-Asset Quantitative Analysis
What drives product innovation?
 Ultimate taylorisation: long tenors, global/local caps/floors,
KI/KO, quanto, trigger based on different asset class, deferred
premium, etc
 Need for a good mix of underlyings, bringing diversification:
static or dynamic allocation – basket vs. Rainbow/Himalaya-type
 Need for option with good upside but low cost…
-> client doesn’t want to pay the full vanilla price and is ready to
forget some downside risk
 Clients want transparent trading strategies, which can be
packaged with an option overlay: principal guaranti,
leveraging/deleveraging mechanism (ala CPPI)
- Example: Momentum strategies (« algorithmic trading »…)
Page 2
Citigroup Global Markets Inc.
Types of underlyings
Futures / Indices (S&P GSCI, DJ AIG, Bespoke) based on:
 Oil products
 Base/Precious Metals
 Agricultural Commodities (grains: soybeans, corn, wheat;
live cattle, pork bellies, orange juice, etc)
 Soft Commodities (Cotton, Sugar, Coffee and Cocoa)
 Natural Gas
 Carbon Emissions
 Coal
 Freight
Page 3
Citigroup Global Markets Inc.
Himalaya Option
 Note with yearly coupons determined by performance of
basket of underlying assets
 First coupon is net return of best-performing asset in
basket. This asset is then removed from the basket.
 Second coupon is net return of best-performing asset
remaining in basket, which is then removed from basket,
etc.
 Local/global caps and floors can apply
Page 4
Citigroup Global Markets Inc.
Himalaya Option
 With global floor on total performance, we are long
correlation (similar to a basket option)
 Without caps/floors, Himalaya options are a sum of BestOf options, and therefore we are short correlation
 In the general case, correlation sensitivity can switch from
positive to negative near a fixing date, even with small
changes in underlying prices
Page 5
Citigroup Global Markets Inc.
Himalaya Option
 Gamma and cross-gamma can be significant
 Risk near fixing date is similar to that of barrier options
 Practical hedging: use Principal component risk rather
than asset delta risk
Page 6
Citigroup Global Markets Inc.
Range Accrual with TARN condition
 Pays a specified amount provided that commodity price is
within a pre-specified or re-settable range
 Knocks out if cumulative payments exceed a specified cap
 Or knocks out if any / all / best-of / worst-of of a basket
exceed some predetermined level
Page 7
Citigroup Global Markets Inc.
Pricing Platform
 Flexibility: new payoff functions should be easy to
implement
 Use a scripting language: proprietary, SLANG, Python,
LUA, etc
 Traders don’t want to wait several days for, say, a global
cap feature on an existing product…
 In fact, traders should be able to change a payoff without
help from quants or IT groups!
Page 8
Citigroup Global Markets Inc.
Pricing Platform
 Scalability: the pricing framework should allow for any
number of factors or underlyings
 Computation time needs to be proportional to the number of
factors
 High-dimensional, efficient Monte-Carlo simulation
becomes a must…
Page 9
Citigroup Global Markets Inc.
Multi-Asset Derivatives Modelling
Citigroup Global Markets Inc.
Ornstein-Uhlenbeck Models
 Why use 2 or more factors?
- Front futures contract tend to trade at a higher volatility than
back contracts
- There is de-correlation along the futures curve
 Example 1: Gibson & Schwartz (1990):
dX t  (rt   t  12  12 )dt   1dWt1

2
d


K
(



)
dt


dW
 t
t
2
t

1
2
E
(
dW
dW
t
t )  dt

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Citigroup Global Markets Inc.
Ornstein-Uhlenbeck Models
 Example 2 with stochastic interest rates:
Casassus and Collin-Dufresne (2005)
drt  ( t  K r rt )   r dWt r


d


(
K
(
t
)

K
r

K


K
X
)
dt


dW
 t
0
r t
 t
X t

t

2
X
1
dX

(
r




)
dt


dW
t
t
X
X
t
2
 t
Page 12
Citigroup Global Markets Inc.
Stochastic Volatility Models
Why using a stochastic volatility model?
 Un-spanned stochastic volatility: Cf. Collin-Dufresne and
Goldstein (2000) Do bonds span the Fixed Income markets?
 Stationary shape of the forward skew: better than using local
volatility models
Page 13
Citigroup Global Markets Inc.
Stochastic Volatility Models
 Affine models are convenient: it is not rare to obtain closed-form
solution for characteristic functions
 Milstein does an ok job in general but cutting negative variance
can sometimes create a significant bias
 An alternative to Milstein: QE scheme (Cf. Leif Andersen, 2006,
Efficient simulation of the Heston stochastic volatility model)
Page 14
Citigroup Global Markets Inc.
Stochastic Volatility Models
 Calibration of a 9-parameter stochastic volatility model on Asian
Crude Oil Straddle Options (implied BS vol residuals):
3/23/2007
4/30/2007
5/31/2007
6/30/2007
9/30/2007
12/31/2007
3/31/2008
6/30/2008
9/30/2008
12/31/2008
3/31/2009
6/30/2009
9/30/2009
3/31/2010
3/31/2011
3/31/2012
3/31/2013
0.75
0.85
0.95
1
1.05
1.15
1.35
2.15%
0.47%
0.67%
1.03%
1.05%
0.85%
0.43%
0.35%
0.53%
0.36%
0.26%
0.16%
-0.20%
-0.40%
0.83%
-0.19%
-0.17%
0.10%
-0.03%
-0.16%
-0.44%
-0.37%
-0.12%
-0.21%
-0.22%
-0.23%
-0.45%
-0.55%
-0.16%
-0.47%
-0.20%
-0.07%
0.02%
-0.05%
-0.28%
-0.13%
0.16%
0.13%
0.17%
0.21%
0.05%
-0.27%
-0.01%
-0.27%
0.03%
0.01%
-0.02%
-0.07%
-0.28%
-0.08%
0.26%
0.25%
0.33%
0.39%
0.27%
-0.04%
0.05%
-0.18%
0.14%
-0.09%
-0.12%
-0.18%
-0.38%
-0.17%
0.27%
0.28%
0.36%
0.44%
0.32%
-0.05%
-1.10%
-0.64%
0.09%
-0.21%
-0.34%
-0.46%
-0.69%
-0.50%
-0.04%
-0.02%
0.07%
0.15%
0.06%
-0.27%
1.53%
0.92%
1.13%
0.50%
0.36%
0.16%
-0.17%
-0.08%
0.28%
0.21%
0.22%
0.24%
0.04%
-0.42%
-0.84%
-1.37%
-0.87%
-1.29%
-0.52%
-0.89%
-0.24%
-0.55%
-0.19%
-0.51%
-0.36%
-0.66%
-0.58%
-0.92%
Source: Citigroup
Page 15
Citigroup Global Markets Inc.
Stochastic Volatility Models
 Typical Monte-Carlo time steps for a model with stochastic
volatility will be one month or less
 Path-dependent / Bermudan payoffs: use the LongstaffSchwartz method
 Quanto adjustment is the same as in deterministic vol case.
Under the domestic risk neutral measure:
 dSt f
~f
f
f
f

(
r


(
t
)

)
dt


d
W
 f
t
t
t
t
S
 t
 (t )   f , X  (t )
X

Page 16
Citigroup Global Markets Inc.
Beyond Gibson-Schwartz and Heston
 Helyette Geman (2000):
dSt  k ( Lt  ln S t ) S t dt  Vt dWt1

 dLt
2


dt


dW

t
2
t
L
 t
dV  b(c  V )dt   V dW 3
t
3
t
t
 t
This is an extension of the Eydeland and Geman model (1998):
L(t) was a fixed level: not so clear for oil in a bull cycle…
Page 17
Citigroup Global Markets Inc.
Beyond Gibson-Schwartz and Heston
 Seasonality model: Richter and Sorensen (2002):
dPt  Pt [(rt   t )dt  e  (t ) vt dWt1 ]

 (t )
2
d


(

(
t
)


)
dt

e

v
dW
 t
t

t
t

3
dvt  (  vt )dt   v vt dWt
K
with
 (t )   0   ( k cos(2kt)   k* sin(2kt))
k 1
K
and
 (t )   ( k cos(2kt)   k* sin(2kt))
k 1
Page 18
Citigroup Global Markets Inc.
Beyond Gibson-Schwartz and Heston
 Calibration of a 28-parameter stochastic volatility seasonal
model on NYMEX Natural Gas futures options:
 Residuals = Market implied BS vol – Model implied BS vol
Page 19
Citigroup Global Markets Inc.
10-Jun-08
28-Jul-08
26-Aug-08
28-Oct-08
21-Nov-08
24-Dec-08
27-Jan-09
24-Feb-09
26-Mar-09
27-Apr-09
26-May-09
25-Jun-09
28-Jul-09
26-Aug-09
25-Sep-09
27-Oct-09
23-Nov-09
28-Dec-09
26-Jan-10
23-Feb-10
26-Mar-10
27-Apr-10
25-May-10
25-Jun-10
27-Jul-10
26-Aug-10
27-Sep-10
26-Oct-10
23-Nov-10
26-Jan-11
23-Feb-11
28-Mar-11
26-May-11
27-Jun-11
26-Jul-11
26-Aug-11
27-Sep-11
26-Oct-11
23-Nov-11
26-Jun-12
27-Nov-12
Page 20
60%
1.63%
1.89%
2.47%
0.12%
0.19%
-0.87%
1.12%
-4.12%
-1.74%
-1.41%
-0.57%
-1.44%
-0.64%
-1.74%
-1.71%
-0.18%
-1.72%
0.07%
0.23%
-1.42%
-1.65%
-1.48%
-1.40%
-1.45%
-1.28%
-0.94%
-0.72%
0.37%
-1.23%
-1.77%
-0.41%
-0.22%
0.32%
0.53%
-0.22%
0.60%
0.04%
1.08%
0.21%
1.76%
75%
-1.04%
-1.92%
1.29%
0.66%
1.66%
0.14%
0.94%
-1.63%
0.68%
0.25%
-0.27%
0.79%
1.28%
0.35%
-1.15%
-0.12%
-1.21%
0.23%
0.46%
-0.74%
-0.94%
-0.69%
-0.71%
-0.78%
-0.61%
-0.31%
0.10%
0.93%
-0.63%
-1.22%
0.15%
0.30%
0.82%
1.00%
0.24%
1.02%
0.45%
1.47%
0.11%
1.13%
85%
-2.28%
-1.77%
0.61%
0.64%
1.70%
0.52%
1.45%
-0.34%
0.63%
-0.39%
-0.24%
0.48%
0.94%
0.75%
-0.96%
-0.36%
-0.94%
0.23%
0.38%
-0.28%
-0.51%
-0.30%
-0.34%
-0.42%
-0.28%
-0.11%
0.44%
1.15%
-0.52%
-1.07%
0.44%
0.56%
1.01%
1.15%
0.38%
1.11%
0.49%
1.53%
-0.05%
0.77%
95%
-1.30%
-0.70%
0.52%
0.64%
1.20%
0.12%
1.11%
-0.50%
0.38%
-0.14%
0.04%
0.24%
0.62%
0.37%
-1.28%
-0.69%
-0.96%
-0.11%
-0.17%
-0.09%
-0.35%
-0.18%
-0.24%
-0.35%
-0.24%
-0.21%
0.21%
0.91%
-0.57%
-1.19%
0.18%
0.25%
0.69%
0.82%
0.02%
0.72%
0.10%
1.10%
-0.39%
-0.05%
100%
-0.63%
-0.02%
0.65%
0.83%
0.77%
-0.51%
0.31%
-0.56%
0.31%
0.05%
0.16%
0.13%
0.47%
0.18%
-1.33%
-0.74%
-0.65%
0.17%
0.04%
-0.10%
-0.40%
-0.23%
-0.32%
-0.43%
-0.33%
-0.29%
0.27%
0.65%
-0.41%
-1.03%
-0.04%
0.01%
0.44%
0.55%
-0.26%
0.44%
-0.04%
0.80%
-0.76%
-0.16%
105%
0.50%
0.85%
0.25%
0.22%
-0.01%
-0.85%
-0.33%
-0.59%
0.14%
0.06%
0.14%
0.03%
0.31%
-0.11%
-1.31%
-0.96%
-0.34%
0.45%
0.30%
0.11%
-0.32%
-0.11%
-0.15%
-0.24%
-0.15%
-0.09%
0.46%
0.40%
-0.25%
-0.89%
-0.20%
-0.22%
0.33%
0.28%
-0.55%
0.14%
0.06%
0.49%
-0.97%
-0.19%
115%
1.60%
0.87%
0.54%
0.23%
-0.64%
-0.28%
-0.47%
-0.62%
-0.19%
0.00%
-0.03%
-0.20%
-0.02%
-0.50%
-1.28%
-1.39%
0.25%
0.99%
0.80%
0.70%
0.23%
0.42%
0.33%
0.20%
0.26%
0.28%
0.83%
-0.11%
0.04%
-0.60%
0.11%
-0.67%
0.59%
-0.27%
-1.10%
-0.44%
0.26%
-0.14%
-1.23%
-0.08%
125%
2.16%
-0.28%
1.06%
0.64%
-0.72%
-0.81%
-0.84%
-0.23%
-0.56%
-0.07%
-0.15%
-0.49%
-0.26%
-0.60%
-1.28%
-1.48%
0.82%
1.50%
1.28%
1.27%
0.76%
0.92%
0.80%
0.62%
0.65%
0.63%
1.17%
-0.65%
0.32%
-0.34%
0.41%
-0.40%
0.83%
-0.83%
-1.47%
-1.05%
0.45%
-0.79%
-1.37%
0.01%
140%
2.99%
0.64%
2.18%
1.12%
-2.17%
-2.13%
-1.88%
0.61%
-0.95%
-0.22%
-0.20%
-0.70%
-0.28%
-0.44%
-0.94%
-0.95%
1.62%
2.21%
1.95%
2.07%
1.50%
1.63%
1.46%
1.22%
1.19%
1.12%
1.65%
-1.31%
0.72%
0.03%
0.83%
-0.04%
1.17%
-1.70%
-1.18%
-1.70%
0.71%
-1.77%
-1.61%
0.14%
Source: Citigroup
Citigroup Global Markets Inc.
Monte-Carlo Greeks
 Finite Differences
- At the very least, re-use same samples when bumping
parameters…
 Pathwise differentiation
- switching derivative and expectation operators
- doesn’t work for non continuous payoffs
Page 21
Citigroup Global Markets Inc.
Monte-Carlo Greeks
 Likelihood Ratio Method
- Broady / Glassermann, 1996
- works for both non-continuous payoffs and path-dependent
options
- not easy to implement in the context of generic parsers or
stochastic vol…
Page 22
P


 ( S ( )) ( S )dS

 
 ( S ,  )
   (S )
dS


1  
   ( S )
 ( S , )dS

 

Citigroup Global Markets Inc.
Model Calibration
Citigroup Global Markets Inc.
Model Calibration
 Cross-sectional calibration: least Square Optimization Problem
  arg min  i Pi
N
~

mod el
i 1
( )  Pi

Market 2
 Use straddle opposed to calls or puts
 Weights should be a function of Bid-Ask spread:
good proxy = vega x volatility points
Page 24
Citigroup Global Markets Inc.
Model Calibration
 Local Minima: transform a least square problem into a “more
convex” problem in  (set of parameters) in order to:
1. get closer to uniqueness
2. ensure stability of solution over time
 Example of quasi-convex problem: Baysian Fit:
  arg min  i Pi
N
~

i 1
mod el
( )  Pi

Market 2
 j  j
 
 
j 1 
j
M




2
 How to determine trade-off between accuracy of calibration (MRSE)
and stability (“entropy” term)?
Page 25
Citigroup Global Markets Inc.
Model Calibration
 Optimal trade-off: Morozov discrepancy principle (Cf. Rama Cont
and Peter Tankov, Financial Modelling with Jump Processes,
Chapman, 2004):
N

  Sup :    i Pi mod el ( ,  )  Pi Market
i 1

*

   P
2
N
i 1
i
i
Bid
2
 Pi Ask 


In practice  is an increasing function of  so the solution can be
found by a simple search function – like Newton
Page 26
Citigroup Global Markets Inc.
Model Calibration
 Are we fitting too many parameters?
 Use Singular Value Decomposition
 If rank(Jacobian) < card() => “co-linearity” in parameters:
keep some parameters constant, i.e. estimated by Maximum Likelihood
Page 27
Citigroup Global Markets Inc.
How to estimate correlations?
 What correlations? For which factors?
 Is there a term structure of correlation? Can we imply it from vanilla
basket options?
 Historical estimation: how? What time window?
 Correlation regimes: should we use a regime switching/Markov
chain model?
 What is the impact of modelling a stochastic correlation?
 Historical vs. risk neutral parameters
 Extreme events: common jumps across asset classes?
Page 28
Citigroup Global Markets Inc.
Pricing an Himalaya with stochastic correlation
 Asset dynamics (multivariate lognormal):
dSi  rS i dt   i Si dWt i
dWt i dWt j  ij (t )dt
 Constant equicorrelation (Benchmark): all off-diagonal entries of
correlation matrix are the same:
ij (t )  
 Stochastic equicorrelation:
dij (t )  dt   (   t )dt   t (1  t )dWt

 Both models are calibrated to the same ATM basket options
Page 29
Citigroup Global Markets Inc.
Pricing an Himalaya with stochastic correlation
Numerical illustration: Sensitivity to stochastic correlation parameters
20.00%
15.00%
10.00%
0.9
5.00%
0.5
0.00%
0.1
10%
30%
50%
70%
90%
Volatility of correlation, 
110%
Mean,

Source: Citigroup
Figure: Percentage price difference for stochastic correlation, as a
function of the vol-of-correlation and mean correlation, relative to the
benchmark (with    ). Correlation mean reverts slowly (  1).
Page 30
Citigroup Global Markets Inc.
Estimating Skew when no market data available…
Joint maximum likelihood of risk-neutral and physical parameters and
risk-premia: Cf. Ait-Sahalia and Kimmel ( 2005) Maximum likelihood
estimation of stochastic volatility models:
 application to general diffusion-based stochastic volatility models
 closed form expansions of transition functions
 approximated volatility state variables using short term implied ATM
vols
 closed-form solution likelihood function (and asymptotic standard
errors)
Page 31
Citigroup Global Markets Inc.
Estimating Skew when no market data available…
Example: Risk neutral and physical parameters in the Heston model:
 Under the risk neutral measure, the dynamics are
 (1   2 )Y
 st  rt  d t  12 Yt 
t
d   
 dt  
0
Yt   ' ( 'Yt ) 

 Yt  W1 (t )
d  Q 
 Yt  W2 (t )
Q
 Under the physical measure
 (1   2 )Y
st  at  bYt 
t
d   
dt  

0
Yt   (  Yt )

 Yt  W1 (t )
d  P 
 Yt  W2 (t )
P
 Assuming an affine risk premium
  [1 (1   2 )Yt , 2 Yt ]'
   2 
2
We have at  rt  d t , b  1 (1   )  2   12 ,    '2 ,   
 '
 

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Citigroup Global Markets Inc.
Hedging an Exotic Book
Citigroup Global Markets Inc.
Hedging an Exotic Book
 All Greeks matter – although not always at the same time…
 Often overlooked: bucketed risk, cross-gamma, volga, vonna,
“charm” and… numerical stability
 Use your hedging portfolio as Monte-Carlo control variate: better
than hedging Monte-Carlo Greeks with Closed-form solution Greeks!
 Before hedging your Greeks check which ones are relevant in
explaining your P&L!
 Validating your PAA: backtesting and stress-testing are important
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Citigroup Global Markets Inc.
Hedging an Exotic Book
 Model-based hedging is an optimization problem: match a desired
risk profile – usually at the second order – while minimizing
transaction costs
 At the same time, maximize some relative value objective if one
considers that the market is rich/cheap vis-à-vis the model
 Alternatively: use a Conditional VaR optimization based on a
statistical model of the market variables (curves, volatility surfaces,
etc)
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Citigroup Global Markets Inc.
Disclosure
Disclosure Appendix A1
ANALYST CERTIFICATION
I, Yann Coatanlem, hereby certify that all of the views expressed in this research report accurately reflect my personal views about any and all of the subject issuer(s) or securities. [I(We)] also certify that no part of
my compensation was, is, or will be directly or indirectly related to the specific recommendation(s) or views in this report.
Other Disclosures
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Disclosure (Continued)
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