Andrea Roncoroni

8/12/2019 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

Andrea Roncoroni Commodity Asian-Sytle Options


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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>>>=>>>; .


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)


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


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




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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


Test 5: Variance Reduction


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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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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.


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)


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


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Andrea Roncoroni