commodities, derivatives on futures, and multiscale ......commodities, derivatives on futures, and...

Commodities, Derivatives on Futures, and MultiscaleVolatility Models

Jorge Zubelli1

joint work with Jean-Pierre Fouque2 and Yuri F. Saporito2

2Department of Statistics & Applied ProbabilityUniversity of California - Santa Barbara

1Institute for Pure and Applied MathematicsRio de Janeiro, Brazil

September 27, 2014

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Prologue

Long term cooperation agreement with PETROBRAS

Cooperation with BMF & BOVESPA

Math Finance professional Master’s program

Importance of Commodities and Futures in Brazilian economy

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Prologue

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Prologue

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Prologue

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Prologue

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Prologue

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Prologue

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United States Wellhead Prices

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Term Structure

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Seasonality in Term Structure

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Seasonality in Storage

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Commodities comprise over 23% of the traded assets in the market;

The spot commodity is generally perishable.

Every futures contract is born with an expiration date.

Every futures investment really depends upon the futures termstructure.

Mean reversion is an important feature of commodity prices.

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IntroLiterature

Classical Black model

One and Two-Factor Models: Schwartz, Gibson - Schwartz, Schwartz- Smith

H. Geman: Commodities and Commodity Derivatives: Energy, Metalsand Agriculturals

R. Carmona and M. Coulon: A Survey of Commodity Markets andStructural Models for Electricity Prices

Huge literature

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IntroLiterature

Huge literature

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IntroLiterature

Huge literature

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IntroLiterature

Huge literature

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IntroLiterature

Huge literature

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Multiscale Stoc Vol. Model for Derivatives on Futuresjt work w/ J-P Fouque and Y. Saporito. To appear at Int. Journal Theoretical & AppliedFinance

Motivation

The simplicity of the model of Black (1976) is not sufficient toprovide a good understanding of the modern futures financial market.

The most restrictive assumption in the Black model is the constancyof the volatility.

Many models have been proposed (e.g. Gibson Schwartz, SchwartzSmith, HJM, etc). However, the pricing of option on futures may beextremely complicated and requires long numerical simulations.

We seek approximate solutions in the context of multiscale analysisproposed by Fouque, Papanicolaou, Sircar, Solna [1].

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Motivation

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Motivation

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Motivation

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Motivation

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Method

write the SDE for the future Ft,T with all coefficients depending onlyon Ft,T . This means we will need to invert the future prices of V inorder to write Vt as a function of Ft,T ;

consider the pricing partial differential equation (PDE) for a Europeanderivative on Ft,T .

The coefficients of this PDE will depend on the time-scales of thestochastic volatility of the asset in a complicated way. Useperturbation analysis to treat such PDE by expanding the coefficients;

determine the first-order approximation of derivatives on Ft,T as it isdone in [1].

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Upshot

we do not rely on the Taylor expansion of the payoff function toderive the first-order approximation

our first-order correction is a substantial improvement of earlierperturbative work.

we present a simple calibration procedure of the market groupparameters. The simple expression of our first-order correction is oneof the reasons such calibration procedure is possible.

the essential aspect of our method is that we consider the future priceFt,T as the variable, and not spot price Vt .

So, since the future price is a martingale (as opposed to the V ),better formulas for the Greeks of the 0-order term are available.

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Vt denotes the asset value and it will be modeled by an exp-OU withmultiscale stochastic volatility (under a risk-neutral measure Q):

Time Scales: ε� T � 1/δ

Vt = eUt ,

dUt = κ(m − Ut)dt + η(Y εt ,Z

δt )dW

(0)t ,

dY εt =

εα(Y ε

t )dt +1√εβ(Y ε

t )dW(1)t ,

dZ δt = δc(Z δ

t )dt +√δg(Z δ

t )dW(2)t .

Correlation: dW(0)t dW

(i)t = ρidt, i = 1, 2, dW

(1)t dW

(2)t = ρ12dt

Future prices: Ft,T = EQ[VT | Ft ], 0 ≤ t ≤ T

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Vt = eUt ,

δt )dW

(0)t ,

dY εt =

εα(Y ε

t )dW(1)t ,

dZ δt = δc(Z δ

t )dt +√δg(Z δ

t )dW(2)t .

(i)t = ρidt, i = 1, 2, dW

(1)t dW

(2)t = ρ12dt

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Vt = eUt ,

δt )dW

(0)t ,

dY εt =

εα(Y ε

t )dW(1)t ,

dZ δt = δc(Z δ

t )dt +√δg(Z δ

t )dW(2)t .

(i)t = ρidt, i = 1, 2, dW

(1)t dW

(2)t = ρ12dt

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First Steps

Goal: to apply singular and regular perturbation techniques to priceoptions on Ft,T .

(i) Write a SDE for Ft,T .

(ii) Expand the coefficients of the PDE for options on Ft,T .

(iii) Determine the first-order approximation for options on Ft,T .

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First Steps

Goal: to apply singular and regular perturbation techniques to priceoptions on Ft,T .

(i) Write a SDE for Ft,T .

(ii) Expand the coefficients of the PDE for options on Ft,T .

(iii) Determine the first-order approximation for options on Ft,T .

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Remarks about Ft,T

EQ[VT | Ut = u,Y εt = y ,Z δ

t = z ] = hε,δ(t, u, y , z ,T ) =

= h0(t, u, z ,T ) +√εh1,0(t, u, z ,T ) +

√δh0,1(t, u, z ,T ) + · · ·

h0(t, u, z ,T ) = exp

{m + (u −m)e−κ(T−t) +

η2(z)

(1− e−2κ(T−t)

η2(z) = 〈η2(·, z)〉,

h1,0(t, u, z ,T ) = g(t,T )V3(z)∂3h0

∂u3(t, u, z ,T ),

h0,1(t, u, z ,T ) = f (t,T )V1(z)∂3h0

∂u3(t, u, z ,T )..

Important Remark: h0(t, u, z ,T ), h1,0(t, u, z ,T ) and h0,1(t, u, z ,T ) are independent of y .

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Remarks about Ft,T

t = z ] = hε,δ(t, u, y , z ,T ) =

= h0(t, u, z ,T ) +√εh1,0(t, u, z ,T ) +

√δh0,1(t, u, z ,T ) + · · ·

h0(t, u, z ,T ) = exp

{m + (u −m)e−κ(T−t) +

η2(z)

(1− e−2κ(T−t)

η2(z) = 〈η2(·, z)〉,

h1,0(t, u, z ,T ) = g(t,T )V3(z)∂3h0

∂u3(t, u, z ,T ),

h0,1(t, u, z ,T ) = f (t,T )V1(z)∂3h0

∂u3(t, u, z ,T )..

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Remarks about Ft,T

t = z ] = hε,δ(t, u, y , z ,T ) =

= h0(t, u, z ,T ) +√εh1,0(t, u, z ,T ) +

√δh0,1(t, u, z ,T ) + · · ·

h0(t, u, z ,T ) = exp

{m + (u −m)e−κ(T−t) +

η2(z)

(1− e−2κ(T−t)

η2(z) = 〈η2(·, z)〉,

h1,0(t, u, z ,T ) = g(t,T )V3(z)∂3h0

∂u3(t, u, z ,T ),

h0,1(t, u, z ,T ) = f (t,T )V1(z)∂3h0

∂u3(t, u, z ,T )..

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Remarks about Ft,T

t = z ] = hε,δ(t, u, y , z ,T ) =

= h0(t, u, z ,T ) +√εh1,0(t, u, z ,T ) +

√δh0,1(t, u, z ,T ) + · · ·

h0(t, u, z ,T ) = exp

{m + (u −m)e−κ(T−t) +

η2(z)

(1− e−2κ(T−t)

η2(z) = 〈η2(·, z)〉,

h1,0(t, u, z ,T ) = g(t,T )V3(z)∂3h0

∂u3(t, u, z ,T ),

h0,1(t, u, z ,T ) = f (t,T )V1(z)∂3h0

∂u3(t, u, z ,T )..

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Remarks about Ft,T

We can write

h0(t, u, z ,T ) = E[V T | Ut = u],

{V t = eUt ,

dUt = κ(m − Ut)dt + η(z)dW(0)t ,

η2(z) = 〈η2(·, z)〉.

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SDE for Ft,T

Since Ft,T is a Q-martingale:

dFt,T =∂hε,δ

∂u(t,Ut ,Y

εt ,Z

δt ,T )η(Y ε

t ,Zδt )dW

(0)t +

+1√ε

∂hε,δ

∂y(t,Ut ,Y

εt ,Z

δt ,T )β(Y ε

t )dW(1)t +

+√δ∂hε,δ

∂z(t,Ut ,Y

εt ,Z

δt ,T )g(Z δ

t )dW(2)t .

The coefficients depend on Ut instead of Ft,T .

We need to invert hε,δ with respect to u and write the expansion of thisinverse:

Hε,δ(t, ·, y , z ,T ) = (hε,δ(t, ·, y , z ,T ))−1

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SDE for Ft,T

dFt,T =∂hε,δ

∂u(t,Ut ,Y

εt ,Z

δt ,T )η(Y ε

t ,Zδt )dW

(0)t +

+1√ε

∂hε,δ

∂y(t,Ut ,Y

εt ,Z

δt ,T )β(Y ε

t )dW(1)t +

+√δ∂hε,δ

∂z(t,Ut ,Y

εt ,Z

δt ,T )g(Z δ

t )dW(2)t .

Hε,δ(t, ·, y , z ,T ) = (hε,δ(t, ·, y , z ,T ))−1

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SDE for Ft,T

dFt,T =∂hε,δ

∂u(t,Ut ,Y

εt ,Z

δt ,T )η(Y ε

t ,Zδt )dW

(0)t +

+1√ε

∂hε,δ

∂y(t,Ut ,Y

εt ,Z

δt ,T )β(Y ε

t )dW(1)t +

+√δ∂hε,δ

∂z(t,Ut ,Y

εt ,Z

δt ,T )g(Z δ

t )dW(2)t .

Hε,δ(t, ·, y , z ,T ) = (hε,δ(t, ·, y , z ,T ))−1

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SDE for Ft,T

Since h0(t, u, z) is invertible, so is hε,δ at least for small ε and δ. If wechoose H0, H1,0, H0,1 to be

(i) H0(t, ·, z ,T ) = (h0(t, ·, z ,T ))−1,

(ii) H1,0(t, x , z ,T ) = − h1,0(t,H0(t, x , z ,T ), z ,T )

∂u(t,H0(t, x , z ,T ), z ,T )

(iii) H0,1(t, x , z ,T ) = − h0,1(t,H0(t, x , z ,T ), z ,T )

∂u(t,H0(t, x , z ,T ), z ,T )

we have

Hε,δ(t, x , y , z ,T ) = H0(t, x , z ,T ) +√εH1,0(t, x , z ,T )+

+√δH0,1(t, x , z ,T ) + O(ε+ δ).

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SDE for Ft,T

dFt,T = ψε,δ1 (t,Ft,T ,Yεt ,Z

δt ,T )η(Y ε

t ,Zδt )dW

(0)t +

+1√εψε,δ2 (t,Ft,T ,Y

εt ,Z

δt ,T )β(Y ε

t )dW(1)t +

+√δψε,δ3 (t,Ft,T ,Y

εt ,Z

δt ,T )g(Z δ

t )dW(2)t

ψε,δ1 (t, x , y , z ,T ) :=∂hε,δ

∂u(t,Hε,δ(t, x , y , z ,T ), y , z ,T ),

ψε,δ2 (t, x , y , z ,T ) :=∂hε,δ

∂y(t,Hε,δ(t, x , y , z ,T ), y , z ,T ),

ψε,δ3 (t, x , y , z ,T ) :=∂hε,δ

∂z(t,Hε,δ(t, x , y , z ,T ), y , z ,T ),

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Options on Ft,T

Fix a maturity S < T and a non-path dependent payoff ϕ.

The no-arbitrage price of this vanilla European option on Ft,T is givenby

Pε,δ(t, x , y , z ,T ) = EQ[e−r(S−t)ϕ(FS ,T ) | Ft,T = x ,Y εt = y ,Z δ

t = z ]

and then Lε,δPε,δ(t, x , y , z ,T ) = 0,

Pε,δ(S , x , y , z ,T ) = ϕ(x),

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PDE for Ft,T

Lε,δ =1

2(ψε,δ2 )2β2(y)

∂x2+ ψε,δ2 β2(y)

∂x∂y

+1√ε

(ρ1ψ

ε,δ1 ψε,δ2 η(y , z)β(y)

∂x2+ ρ1ψ

ε,δ1 η(y , z)β(y)

∂x∂y

2(ψε,δ1 )2η2(y , z)

∂x2− r ·+

+√δ

(ρ2ψ

ε,δ1 ψε,δ3 η(y , z)g(z)

∂x2+ ρ2ψ

ε,δ1 η(y , z)g(z)

∂x∂z

2(ψε,δ3 )2g2(z)

∂x2+ ψε,δ3 g2(z)

∂x∂z

(ρ12ψ

ε,δ2 ψε,δ3 β(y)g(z)

∂x2+

+ ρ12ψε,δ3 β(y)g(z)

∂x∂y+ ρ12ψ

ε,δ2 β(y)g(z)

∂x∂z+ ρ12β(y)g(z)

∂y∂z

2β2(y)

∂y2+ α(y)

2g2(z)

∂z2+ c(z)

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PDE for Ft,T

Denote by ψk,i ,j the term of order i in√ε and order j in

√δ in the

expansion of ψε,δk :

ψε,δk = ψk,0,0 +√εψk,1,0 +

√δψk,0,1 +

√ε√δψk,1,1 + · · ·

For example, we can compute

ψ1,0,0(t, x ,T ) = e−κ(T−t)x .

Since h0, h1,0 and h0,1 do not depend on y ,

ψ2,0,0 = ψ2,1,0 = ψ2,0,1 = 0.

Notation: Dk = xk ∂

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PDE for Ft,T

Lε,δ =1

εL0 +

1√εL1 + L2 +

√εL3 +

√δM1 +

εM3 + · · ·

2β2(y)

∂y2+ α(y)

∂yL1 = ρ1e

−κ(T−t)η(y, z)β(y)D1∂

L2 =∂

2e−2κ(T−t)η2(y, z)D2 − r· −

2e−2κ(T−t)∂φ

∂yβ2(y)D1

L3= (ψ2,3,0β2(y) + ρ1ψ1,2,0η(y, z)β(y))

∂x∂y−

−ρ11

2e−3κ(T−t)∂φ

∂y(y, z)η(y, z)β(y)D2

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Perturbation - P0

Define the Black operator

LB(σ) =∂

2σ2x2 ∂

∂x2− r ·

σ(t, y , z ,T ) = e−κ(T−t)η(y , z).

L2 = LB(σ(t, y , z ,T ))− 1

2e−2κ(T−t)

∂yβ2(y)D1

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Perturbation - P0

Carrying on the formal perturbation arguments, we choose P0 to solveLB(σ(t, z ,T ))P0 = 0,

P0(S , x , z ,T ) = ϕ(x)

and then

P0(t, x , z ,T ) = PB(t, x , σ2t,S(z ,T ))

σ2t,S(z ,T ) =1

S − t

tσ2(s, z ,T )ds = η2(z)

(e−2κ(T−S) − e−2κ(T−t)

2κ(S − t)

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A Remark about P0

The Feynman-Kac’s representation tells us

P0(t, x , z ,T ) = E[e−r(S−t)ϕ(F S,T ) | F t,T = u],

dF t,T = F t,T e−κ(T−t)η2(z)dW(0)t

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Perturbation - Pε1,0 and Pδ

The first-order correction for P0 is given by the following combination ofGreeks of P0:

Pε1,0(t, x , z ,T ) = (S − t)V ε

3 (z)λ3(t, S ,T , κ)(D2+

+ D1D2)PB(σt,S(z ,T )).

Pδ0,1(t, x , z ,T ) = (S − t)V δ

0 (z)(λ0(t, S ,T , κ)D2+

+ λ1(t, S ,T , κ)D1D2)PB(σt,S(z ,T )).

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Accuracy Theorem

Theorem

We assume

(i) Existence and uniqueness of the SDEs for fixed (ε, δ).

(ii) The process Y 1 with infinitesimal generator L0 has a unique invariantdistribution and is mean-reverting.

(iii) The function η(y , z) is smooth in z and such the solution φ to therelated Poisson equation is at most polynomially growing.

(iv) The payoff function ϕ(x) and its derivatives are smooth and bounded.

Pε,δ = PB(σt,S(z ,T )) + Pε1,0 + Pδ

0,1 + O(ε+ δ).

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Some Remarks

In order to compute the first order price approximation, we only needthe group market parameters (κ, η2(z),V δ

0 (z),V ε3 (z)).

Model independence: the approximation is independent of the choiceof α, β,c, g .

This approximation can be seen as a correction of the Black optionprice with effective volatility σt,S(z ,T ) using some of its Greeks.

Regularization arguments can be used to prove the accuracy of theapproximation for non-smooth payoff like the Call option payoff.

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Some Remarks - Path Dependent Options

The group market parameters can be direct calibrated to liquidoptions and used for pricing other derivatives to the same level ofapproximation. Functional Ito Calculus can be used to prove thisassertion:

Bruno Dupire, “Functional Ito Calculus” (2009),http://ssrn.com/abstract=1435551

One should consider the functional space and time derivatives, ∆t

and ∆x .

We have the same interpretation for the first-order approximation:

the zero-order term will be the option price when the volatility is equalto η(z);the first-order correction will be a combination of (path-dependent)Greeks of the zero-order term with the same parameters(κ, η2(z),V δ

0 (z),V ε3 (z)).

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Advantages of this Approach

Another approach would be to consider the Taylor expansion of the payoffϕ around the zero-order term of the future price approximation, h0. Theapproach we presented today has the following advantages:

(i) Requires less regularity of the payoff function ϕ.

(ii) Allows direct calibration of the group market parameters to calloption prices.

(iii) Considers the right underlying asset (the future contract).

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Numerical Example - Call Option

Parameter Value

r 0.01T 2 year

F0,T ∈ [50, 70]S 1 yearsK 60κ 0.4η(z) 0.2

V ε3 (z) -0.001

V δ0 (z) 0.0008

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Calibration example

The calibration procedure requires only simple regressions (comparewith [2])

data considered were Black implied volatilities of call and put optionson the crude-oil future contracts on October 16th, 2013. On this day,533 implied volatilities are available.

organized as follows: for each future contract (i.e. for each maturityTi ), there is one option maturity T0ij and 41 strikes Kijl .

contractual specifications, the option maturity is roughly one monthbefore the maturity of its underlying future contract (i.e.Ti ≈ T0ij + 30).

The future prices are shown in Figure 1. (no seasonality for WTI)The calibration of our model to all the available data is shown in Figure3and in Table 1.

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Calibration example

We show in Figures 2 and 3 the implied volatility fit for differentmaturities, where the solid line is the model implied volatility and thecircles are the implied volatilities observed in the market.The shortest maturities implied volatility curves are on the leftmost threadand the maturity increases clockwise.The calibrated group market parameters are given in Table 1.It is important to notice that V ε

3 (z) and V δ0 (z) are indeed small and hence

these parameters are compatible with our model.

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Calibration example

Parameter Value (maturities greater than 90 days) Value (all)

κ 0.1385 0.30853η(z) 0.21967 0.23773

V ε3 (z) -0.00017637 -0.00011823

V δ0 (z) -0.012656 -0.007633

Table: Calibrated Parameters

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Calibration example

Figure: Future prices on October 16th, 2013.

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Figure: Market (circles) and calibrated (solid lines) implied volatilities for optionson crude-oil futures with maturity greater than 90 days.

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Calibration example

Figure: Market (circles) and calibrated (solid lines) implied volatilities for optionson crude-oil futures using all data available.

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Conclusions

We have derived an efficient way to approximate prices of options onfutures in the context of exp-OU process with multiscale stochasticvolatility.

In the same lines of the Equity case, the group market parameters canbe direct calibrated to liquid options and used for pricing otherderivatives to the same level of approximation.

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References

Jean-Pierre Fouque et al., “Multiscale Stochastic Volatility for Equity,Interest Rate, And Credit Derivatives”, Cambridge University Press(2011)

S. Hikspoors and S. Jaimungal, Asymptotic Pricing of CommodityDerivatives for Stochastic Volatility Spot Models, Appl. Math. Finance15 (2008) 449–447.

M.C. Chiu, Y.W. Lo, and H.Y. Wong.Asymptotic Expansion for Pricing Options on Mean-Reverting Assetswith Multiscale Stochastic Volatility.Operations Research Letters, 39:289–295, 2011.

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Name at least 4 things all these pics have in common...

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Name at least 4 things all these pics have in common...ANSWER!

1 Math Finance Talk

2 Mostly nice people (Research in Options RiO Meeting)

3 Nice place in Brazil

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1 Math Finance Talk

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1 Math Finance Talk

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1 Math Finance Talk

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Thank you Jean-Pierre!

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commodities, derivatives on futures, and multiscale ......commodities, derivatives on futures, and...

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