andrea roncoroni
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Energy and Commodity Asian-Style Options underSeasonal Data and Stochastic Volatility
Andrea RoncoroniESSEC Business School, Paris - Singapore
Practical Quantitative Analysis in Commodities
June 17-18, 2010London, UK
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Commodity Price Modelling
Model construction focuses on:
1 Primitives= Input state variables! should be quantities with:
Reliable observations;
Economic signicance.2 Structural elements = Form of drift, volatility, jump, if any
! should be identied using statistical analysis of historical data andthen tted to observed prices.
3 Driving noise terms = Number (&nature) of noise terms
!should be assessed based on historical price analysis (e g , exam
of the trajectorial properties of price paths, Principal ComponentsAnalysis, jump ltering).
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Modelling Frameworks
Frameworks: We identify four classes of arbitrage free models forcommodity prices according to selection of primitives:
1 [SC] Spot Price-Convenience Yield Models (Gibson-Schwartz (1990))! primitives = spot price + instantaneous spot convenience yield;
2
[FD] Forward Curve Models (Reisman (1991), Jamshidian (1991))! primitive = forward price curve;3 [FC] Forward Convenience Yield Models (Cortazar-Schwartz (1994))
! primitives = spot price + instantaneous fwd convenience yield;4 [SP] Spot Price Models (Black (1976))
! primitive = spot price (deterministic convenience yield)Roncoroni, A., Commodity Price Models, in: Cont et al., Encyclopedia ofQuantitative Finance, Wiley (forthcoming).
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Stylized Facts and Market Price Information
Principle! Commodity derivatives should be priced using modelsreproducing:
1 Stylized facts about underlying price dynamics:
Mean reversion characterizing spot price dynamics,Time and stochastic patterns aecting historical price volatility,Jump-like price behavior and non normal returns.
2 Market price information available at the valuation time:
Market quoted forward and futures prices,Volatility surfaces (liquid option quotes).
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Stylized Facts I: Empirical Evidence
2 4 6 8 10 12
0.16
0.18
0.2
0.22
0.24
0.26
Month
Std.
Dev.
Corn Historical Volatility, 1980-2009
2 4 6 8 10 12
0.18
0.2
0.22
0.24
0.26
Month
Std.
Dev.
Soybean Historical Volatility, 1980-2009
2 4 6 8 10 12
0.2
0.21
0.22
0.23
0.24
Month
Std.
Dev.
Wheat Historical Volatility, 1980-2009
2 4 6 8 10 12
0.4
0.5
0.6
0.7
Month
Std.
Dev.
HH Gas Historical Volatility, 1990-2007
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Stylized Facts II: Empirical Evidence
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Market Price Information I: Empirical Evidence
0 0.5 162
64
66
68
Light,Sweet CrudeOil (Nymex 1-3-2007)
Maturity (years)
0 0.5 17
8
9
10
HH Natural Gas (Nymex 1-3-2007)
Maturity (years)
0 0.5 11.7
1.8
1.9
2Heating Oil (Nymex 1-3-2007)
Maturity (years)
0 1 2 3 43200
3400
3600
3800
4000Corn (CBOT 1-12-2006)
Maturity (years)
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Market Price Information II: Empirical Evidence
0.20.40.60.8
11.2
0.8
1
1.2
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Moneyness
Smile curve implied by Crude Oil Futures Options on July 7, 2009
Maturity
ImpliedVol
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Commodity Asian-Style Options
Discrete monitoring = Prices are monitored every time units.
Underlying variable = Price average ni=0iSi.
Name Weight j Average Avgn
Standard arithmetic 1/(n+1) (n+1)1
n
i=0SiVolume weighed Vj/iVi (kVk)
1
ni=0ViSi
Cash ows:
Fixed strike Floating strike
max fAvgn K, 0g max fAvgn Sn, 0g
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Articles
Analytical Pricing of Commodity Asian-Style Options under DiscreteMonitoring (with G.Fusai,M.Marena). JBF32(10), 2033-2045, 2008
Analytical pricing (up to FT) of arithmetic average options on:
dSt=tStdt+ tpStdWt+dJt (VC-SQRT-J)
(Variants: CC-SQRT:const.coe.+dJ=0, CV-SQRT:const.vol.+dJ=0; SC-SQRT:seas.coe.+dJ=0, C-SQRT-J:const.par.)
Control Variates for Asian-Style Options under Seasonality, Stochastic
Volatility and Jumps (with G.Fusai, M.Marena). WP, ESSEC, 2009
Analytical pricing (up to FT) of geometric average options on:
dlg St = (t mt vt/2) dt+ pvtdW1
t +dJ1
t (SV-JJ)dvt = (t vt) dt+ pvtdW2t +dJ2t.
and use as control variable for pricing arithmetic average options. (Variants: JD:v=const., SV=dJ1=dJ2=0, SV-J:dJ2=0)
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Transform-Based Option Pricing (Carr-Madan (1999))
Pay-o(including call and xed-strike Asian):
max f0,YT kg ! , k= constant,Y 0.
1 Vanilla call
!YT =Sn, = 1,k=K;
2 Fixed-strike Asian! YT = Si, = 1/(n+1),k=(n+1) K.Laplace transformof the option price wrt strike k:
L:call price=funct.of strike k
z }| {C0,Y0(T, k) !Laplace transf.=funct.of
z }| {L [C] () , Z +0
ekC0,Y0(T, k) dk.
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Transform-Based Option Pricing (Carr-Madan (1999))
Laplace transform:
L [C] () AF price= erT0BB@E0
heYT
i2
+E0[YT]
1
2
1CCA
.
Laplace inversion! Option price:
C0,Y0(T, k)=erTL1 "
L [YT] ()()2 # (k)+E0[YT] k! .
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Arithmetic Average Options under a CV-SQRT Model
Constant volatility square-root dynamics:
dSu=(ru cu) Sudu+ pSudWu, starting at: S0 =x.
Curve tting: Set rtct=t;
Input! Fwd prices observed for maturities up to option expiration.Problem! Find tsuch that the spot model ts fwd prices.Solution!F0,T = E0(ST)=xexp
RT0 sds i:
T=T ln F0,T.
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Main Result (Fusai-Marena-Roncoroni (2008))
Theorem: MgfSt! Mgf (nal price, arithm.avg.price):
v0,x(n,;, ) , E0e[Sn+jSj]
=e0(;,)x,
where j(;,) satises the recursive equation:
j(;, )= A;j+1(;,)
+ j, for j=n 1 ! 0,
n(,,)= + n (starting condition),
with A (;)=e(rc)/ 1+ 2 e(rc) 1 /2 (r c) .
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Pricing Formula
Price Fixed-strike Arithmetic Asian-style option price:
V = ert
1
2p1
Z al+p1al
p1e
n+1 K(n+1)
v0,x(n,; 0,)
2 d
+e(rc)(n+1) 1(er 1) (n+1)xK(n+1)
!.
Extensions:
1 Mean reversion + Time-varying volatility;2 Time-varying drift + Jumps.
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Geometric Average Options under a CC-SV-JJ Model
Constant coe. stoch.vol. double jump model:
dlg St = (r c mt vt/2) dt+ pvtdW1t +dJ1t (SV-JJ)dvt = ( vt) dt+ pvtdW2t +dJ2tdt = CovdW(1), dW(2) ,1 =2,Ji
i.i.d.
NPay-o (Geometric Asian-style call):
Cg
(T,K
) max
8>>>>>:0,
n
k=0Sk!
1n+1
geometric avg.=:YT
K
9>>>=>>>; .
Andrea Roncoroni Commodity Asian-Sytle Options
M i R l (F i M R i (2009))
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Main Result (Fusai-Marena-Roncoroni (2009))
Theorem: Cf (lg St, vt)!Cf log(geom.avg.priceYT):
0,x,v(n,; u)= E0eiuY
T
=eiux+1(u;n,)v+1(u;n,),
where j and jsatisfy the recursive equations (j : n 1 ! 1):
j(u; n,)= D(n j+1)/(n+1) , ij+1(u; n,) ; ,starting at:
n(u; n,)= D(u/(n+1) , 0;) ;
j(u; n,)= j+1(u; n,)+C
(n j+1)/(n+1) u, ij+1(u; n,) ;+J(nj+1)/(n+1) u, ij+1(u; n,) ; ,
starting at:
n(u; n,)=C(u/(n+1) , 0;)+J(u/(n+1) , 0;),
and D,C, Jare given in analytic form.
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T 2 C i M h d f h M k M d l
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Test 2: Comparison to Methods for the Market Model
Methods for geometric Brownian motion dynamics- Geman and Yor (1993): Laplace trans.inv. with cont.monitoring
- Turnbull and Wakeman (1991) approximation of the lognormal price distribution
Comparative modelSQRT:CallSRmod.(SQRT)= CallBSmod.(GBM)
K GBM Option Prices in the GBM case SR Option price (SQRT)Inv.Lap. Logn,
90 0.1 15.39763 15.39906 0.97411 15.39890
110 0.1 1.41362 1.41080 1.02356 1.41070
90 0.5 19.30572 19.55391 4.86178 19.37724
110 0.5 10.07128 10.18997 5.11247 10.06599
1 SQRT accurately approx.GBM quotes, yet SQRT! real time val.2 SQRT lies between the two approx.quotes (but for deep OTM)
Andrea Roncoroni Commodity Asian-Sytle Options
T t 3 I l di Q t d F d C
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Test 3: Including a Quoted Forward Curve
Methods Compute prices with at and market fwd curves
n=5
K Flat Non Flat %Di
-0.05 0.14 0.15 4.04
0 0.13 0.13 3.99
0.05 0.12 0.12 3.95
n=250
K Flat Non Flat %Di
-0.05 0.15 0.17 9.73
0 0.14 0.15 9.78
0.05 0.13 0.14 9.82
1 Monitoring frequency" ; price discrepancy between consideringand discarding the quoted forward curve
#2 This is important in commodity/energy markets where fwd curvesoften display seasonality
Andrea Roncoroni Commodity Asian-Sytle Options
T t 4 I l di S l V l tilit
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Test 4: Including Seasonal Volatility
Step I Compute historical avg.vol. GBM for each month
Step II Conv.GBM! SR :GBMSpot=SRp
Spot
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
GBM 0.17 0.14 0.16 0.17 0.20 0.22 0.27 0.22 0.19 0.18 0.16 0.13
SR 10.2 8.59 9.74 10.2 12.4 13.8 16.3 13.7 11.4 11.2 9.80 8.40
Step III Building 3 vols
Flataverage vol.! (a) :2(a)R
T
01
F0,sds=
RT
02s
F0,sds
Flatimplied vol.!
(b)
matching a benchmark option (ATM Asianwith a 5-period monitoring)Time varyinghistorical market volatility structure
Andrea Roncoroni Commodity Asian-Sytle Options
Test 4: Including Seasonal Volatility
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Test 4: Including Seasonal Volatility
Results
K/S(t) n a =11.21%Di
V(a )-V(nf)b=10.84
%Di
V(b)-V(nf)Non Flat Vol.
0.9 12 460.72 0.59 458.62 0.13 458.03
1 12 196.08 3.02 191.37 0.54 190.34
1.1 12 54.71 9.03 50.80 1.23 50.18
0.9 1000 472.95 0.66 470.81 0.20 469.85
1 1000 206.46 3.35 201.67 0.96 199.76
1.1 1000 60.28 10.08 56.16 2.55 54.76
1 Important price dierences2 This eect is rather signicant for OTM options3 Method 2) method 1), but requires option price observation
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Test 5: Variance Reduction
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Test 5: Variance Reduction
Description: Evaluate an arithmetic average option using:
Naive Monte Carlo,Geometric control variate,Normal antithetic variables.
Control variate:
bCV = arith avgg(X) + estim.by simulationCov (g(X) , f (X))Var (f (X)) 0B@geo avgf (X)geo opt.price!analyticz }| {
E (f (X))1CA .
Antithetic:
bAV := 1n
n
i=1
gXN1i ,N2i +gXN1i , N2i 2
.
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Test 5: Variance Reduction
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Test 5: Variance Reduction
Results: SV Model! Convergence and std.errors:#ratio or method/ n.simul.! 100,000 200,000 300,000 400,000 500,000MC/Antithetic Variable 2.84707 2.84326 2.84170 2.84381 2.84391MC/Control Variate 46.39385 46.08531 45.77696 45.77019 45.88352
MC/
Antithetic+Control V 59.69837 59.20087 58.81804 58.83800 58.94951
Arithmetic Naive Monte Carlo(x 0.01)
5.16737 5.17290 5.18414 5.18125 5.18159
Arithmetic Control Variate 5.19153 5.19143 5.19156 5.19147 5.19137
Arithmetic Antithetic Variable 5.17596 5.17491 5.17556 5.17965 5.18035
Arithmetic Antithetic+Control 5.19130 5.19129 5.19138 5.19130 5.19126
1 Variance reduction is dramatic with control variate;2 Control variate leads to fast convergence.
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Conclusion
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Conclusion
We price arithmetic Asian-style options under realisticassumptions:
1 Averages arediscretelymonitored! Real-world practice2 SQRT Model! Analytic pricing formulae3
SV-JJ Model! Eective control variate4 The underlying dynamics exhibitstylized behavioral features:Time varying volatility! Seasonal price vol.Jumps! Spikes and non-normal returns
5 Market information is accounted for using:
Time varying drift!
Fitting the quoted fwd curve/price trendStochastic volatility + jumps! smile tting (to be conducted)
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The Author
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The Author
Andrea Roncoroni is Professor of Finance at ESSEC Business School (Paris - Singapore) and regular Lecturer at Bocconi University
(Milan), He holds a BS in Economics from Bocconi University (Italy), an MS in Mathematics from the Courant Institute of Mathematical
Sciences (New York) and PhD's in Applied Mathematics and Finance from the University of Trieste (Italy) and University Paris Dauphine
(France), respectively. His research interests cover Energy Finance, Financial Econometrics and Derivative Structuring. He has consulted
for private companies (e.g., Gaz de France, Edison Trading, EGL, Dong Energy) and lectured for public institutions (e.g., International
Energy Agency, Central Bank of France, Italian Stock Exchange). He regularly writes on academic journals and has recently published
"Implementing Models in Quantitative Finance: Methods and Cases" (with G.Fusai), edited by Springer-Verlag in 2008.
E-mail: [email protected]
Web page: http://www45.essec.edu/faculty/andrea-roncoroni
Andrea Roncoroni Commodity Asian-Sytle Options