seasonal stochastic volatility and correlation together with ......•corn - futures •corn -...

EMLYON Quant 12 Workshop 1 / 38

Seasonal Stochastic Volatility and Correlation together with theSamuelson Effect in Commodity Futures Markets

Lorenz Schneider

EMLYON Business School

Bertrand Tavin

EMLYON Business School

26-27 November 2015, Lyon

Introduction

Introduction

• Introduction

Market Data

Seasonal StochasticVolatility Model

Calibration Results withthe One-Factor Model

Calendar SpreadOptions

Literature


Introduction

Introduction

• Introduction

Market Data




Literature


• Recently, several volatility indices for agricultural markets have

been introduced on the Chicago Board Options Exchange

(CBOE):

• in 2011, the Corn Volatility Index (CIV) and Soybean Volatility

Index (SIV);• and in 2012, the Wheat Volatility Index (WIV).

• Modelling stochastic volatility allows to calibrate to option volatility

smiles and skews typically seen in futures option markets.

• A good model should also account for the Samuelson effect that

the volatility of nearby futures prices is higher than that offar-away futures prices.

Introduction

Introduction

• Introduction

Market Data




Literature


In this presentation, we will:

• Show and discuss some agricultural market data;

• introduce a seasonal stochastic volatility model;• discuss various seasonality functions;

• show that the model can be fitted very well to the market;

• discuss the pricing of calendar spread options;

• give evidence of the influence of seasonality on prices and

implied correlations of such options.

Market Data

Introduction

Market Data• Wheat - VolatilitySurface• Corn - VolatilitySurface• Soybean - VolatilitySurface• Raw Sugar - VolatilitySurface




Literature


Wheat - Volatility Surface

Introduction





Literature


Corn - Volatility Surface

Introduction





Literature


Soybean - Volatility Surface

Introduction





Literature


Raw Sugar - Volatility Surface

Introduction





Literature


Seasonal Stochastic VolatilityModel

Introduction

Market Data


• A SeasonalStochastic VolatilityModel• The SeasonalityFunction• The SeasonalityFunction• Examples ofSeasonality Functions

• The JointCharacteristic Function• The Transform of theSeasonality Function



Literature


A Seasonal Stochastic Volatility Model


The futures price F (t, Tm) at time t, t ≤ Tm, follows the SDE

dF (t, Tm) = F (t, Tm)n∑

j=1

e−λj(Tm−t)√

vj(t)dBj(t),

F (0, Tm) = Fm,0.

The processes vj , j = 1, ..., n, are CIR/Heston square-root stochastic

variance processes

dvj(t) = κj (θj(t)− vj(t)) dt+ σj

√

vj(t)dBn+j(t),

vj(0) = vj,0.

with time-dependent, deterministic mean-reversion levels θj(t).We assume

〈dBj(t), dBn+j(t)〉 = ρjdt,

and that otherwise the Brownian motions are independent of each other.

The Seasonality Function


We want to include discontinuous seasonality functions in our framework:

• Let T = {ti, i = 1, ...} be a set of times having only finitely many points inevery bounded interval.

• Let Z = {0 ≤ t1 < t2 < ... < ti < ...} be the partition of R+0 defined by

T .

• Let the seasonality function θ : R+0 → R

+ be piecewise continuous with

respect to Z .

• Assume that it is bounded from below and above by positive constants θmin

and θmax.



We compare two processes v (seasonal) and v (non-seasonal), which are

given by the SDEs

dv(t) = κ (θ(t)− v(t)) dt+ σ√

v(t)dB(t), (1)

dv(t) = κ (θmin − v(t)) dt+ σ√

v(t)dB(t), (2)

with identical parameters κ > 0, σ > 0 and initial conditions

0 < v(0) = v0 ≤ v(0) = v0.



It is well known that (2) has a unique strong solution. The following result

describes the solution to (1).

Proposition 1 Assume that the seasonality function θ is piecewise continuous

w.r.t. the partition Z of R+0 , and bounded by positive constants θmin and

θmax, i.e. for all t ≥ 0, 0 < θmin ≤ θ(t) ≤ θmax. Let the processes v and v

be given by (1) and (2), respectively. Then:

1. The process (1) has a unique strong solution with continuous sample

paths.

2. P [vt ≤ vt, ∀t ≥ 0] = 1.

3. If the Feller condition σ2 < 2κθmin is satisfied for θmin, then the processv is strictly positive.

Examples of Seasonality Functions

Introduction

Market Data






Literature


• Sinusoidal pattern, with a, b > 0 and t0 ∈ [0, 1[:

θ(t) = a+ b cos (2πω (t− t0)).

• Exponential-sinusoidal pattern, with a, b > 0 and t0 ∈ [0, 1[:

θ(t) = a exp (b cos (2πω (t− t0))).

A similar parametric form is used in [1].

• Sawtooth pattern, with a, b > 0 and t0 ∈ [0, 1[:

θ(t) = a+ b (t− t0 − ⌊t− t0⌋) .

This is an example of a discontinuous seasonality function.

Examples of Seasonality Functions

Introduction

Market Data






Literature


• Triangle pattern, with a, b > 0 and t0 ∈ [0, 1[:

θ(t) = a+ b

∣

∣

∣

∣

1

2− (t− t0 − ⌊t− t0⌋)

∣

∣

∣

∣

.

• Spiked pattern, with a, b > 0 and t0 ∈ [0, 1[:

θ(t) = a+ b

(

2

1 + |sin(π(t− t0))|− 1

)2

.

This parametric form is applied to electricity markets in [3] tomodel the time varying intensity of a jump process.

The Joint Characteristic Function


Proposition 2 The joint characteristic function φ at time T ≤ T1, T2 for the

log-returns X1(T ), X2(T ) of two futures contracts with maturities T1, T2 isgiven by

φ(u) =n∏

j=1

exp

(

−iρj

σjfj,1(u, 0)

(

vj(0) + κj θj,T

)

)

· exp (Aj(0, T )vj(0) +Bj(0, T )) ,

with

fj,1(u, t) =2

∑

k=1

uke−λj(Tk−t), fj,2(u, t) =

2∑

k=1

uke−2λj(Tk−t),

qj(u, t) = iρjκj − λj

σjfj,1(u, t)−

1

2(1− ρ2j )f

2j,1(u, t)−

1

2ifj,2(u, t),

θj,T =

∫ T

0θj(t)e

λjtdt.

The Joint Characteristic Function


The functions Aj : (t, T ) 7→ Aj(t, T ) and Bj : (t, T ) 7→ Bj(t, T ) satisfy the

two differential equations

∂Aj

∂t− κjAj +

1

2σ2jA

2j + qj = 0

∂Bj

∂t+ κjθj(t)Aj = 0

with conditions

Aj(T, T ) = iρj

σjfj,1(u, T )

Bj(T, T ) = 0

Note that only the differential equation for Bj , but not the one for Aj , depends

on the seasonality function θj . We give closed-form expressions for the

functions Aj in terms of Kummer functions in [5].

The Transform of the Seasonality Function

Introduction

Market Data






Literature


• The integrals

θj,T =

∫ T

0θj(t)e

λjtdt

only depend on the specification of the seasonality functions θjand the maturity T .

• Therefore, their value can be calculated once and then stored,

avoiding recalculations during repeated calls to the characteristic

function.

• We show in [6] how to calculate these integrals analytically for the

sinusoidal, sawtooth and triangle patterns.• If θ is a constant function, then the joint characteristic function

given above is the same as the non-seasonal one given in [5].

Calibration Results with theOne-Factor Model

Introduction

Market Data


Calibration Results withthe One-Factor Model• The One-FactorModel (n = 1)

• Wheat - Futures• Wheat - VolatilityTerm-Structure

• Corn - Futures• Corn - VolatilityTerm-Structure

• Soybean - Futures

• Soybean - VolatilityTerm-Structure

• Raw Sugar - Futures

• Raw Sugar - VolatilityTerm-Structure


Literature


The One-Factor Model (n = 1)

Introduction

Market Data










Literature


Futures and Variance processes:

dF (t, Tm)

F (t, Tm)= e−λ(Tm−t)

√

v(t)dB1(t)

dv(t) = κ (θ(t)− v(t)) dt+ σ√

v(t)dB2(t)

with correlation 〈dB1(t), dB2(t)〉 = ρdt.

Seasonality function:

θ(t) = a+ b cos(2πω(t− t0))

Wheat - Futures

Introduction

Market Data










Literature


0 0.2 0.4 0.6 0.8 1 1.2 1.4500

510

520

530

540

550

560

570

580

590

600

T (years)

March 2015

Wheat - Volatility Term-Structure

Introduction

Market Data










Literature


0 0.2 0.4 0.6 0.8 1 1.2 1.40.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

T (years)

March 2015

MarketModel

Corn - Futures

Introduction

Market Data










Literature


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8350

360

370

380

390

400

410

420

430

440

450

T (years)

March 2015

Corn - Volatility Term-Structure

Introduction

Market Data










Literature


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0.15

0.2

0.25

0.3

0.35

T (years)

March 2015

MarketModel

Soybean - Futures

Introduction

Market Data










Literature


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6900

910

920

930

940

950

960

970

980

990

1000

T (years)

March 2015

Soybean - Volatility Term-Structure

Introduction

Market Data










Literature


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

T (years)

March 2015

MarketModel

Raw Sugar - Futures

Introduction

Market Data










Literature


0 0.5 1 1.5 2 2.52

4

6

8

10

12

14

16

18

T (years)

March 2015

Raw Sugar - Volatility Term-Structure

Introduction

Market Data










Literature


0 0.5 1 1.5 2 2.50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

T (years)

March 2015

MarketModel

Calendar Spread Options

Introduction

Market Data




• Calendar SpreadOptions (CSO)

• Pricing CSOs withCaldana and Fusai’sMethod• Comparing No, Weak,and Strong Seasonality

• Calendar SpreadOption Prices andSeasonality

• CorrelationTerm-Structure andSeasonality

Literature


Calendar Spread Options (CSO)

Introduction

Market Data








Literature


• Calendar spread options are popular options in commodities

markets.

• Corn, Soybean and Wheat CSOs are traded on the Chicago

Mercantile Exchange (CME).

• In February 2015 the Minneapolis Grain Exchange (MGX)introduced North American Hard Red Spring Wheat (HRSW)

CSOs for trade on the CME.

• CSOs in agricultural grain markets are studied in [4] with a joint

Heston model for the two underlying futures contracts. The

variance process has a constant mean-reversion level, andtherefore does not display seasonality.

• In the following, we show the effects volatility seasonality can

have on CSO prices.

Calendar Spread Options

Introduction

Market Data








Literature


• Let K (which is allowed to be negative) denote the strike of the

option.

• Payoff of a calendar spread call option CSC :

(F (T, T1)− F (T, T2)−K)+

= max (F (T, T1)− F (T, T2)−K, 0) ,

• Payoff of a calendar spread put option CSP :

(K − (F (T, T1)− F (T, T2)))+

= max (K − (F (T, T1)− F (T, T2)) , 0) .

• We use the 2-factor model together with Caldana and Fusai’smethod [2] to price these options.

Pricing CSOs with Caldana and Fusai’s Method


Let Φ(u) = exp(

i∑2

k=1 uk lnF (0, Tk))

· φ(u) be the joint c.f. of

lnF (T, T1) and lnF (T, T2). The price of a calendar spread option call with

maturity T and strike K is given as

CSC(K,T, T1, T2) =

(

e−δk−rT

π

∫ +∞

0e−iγkΨ(γ; δ, α)dγ

)+

,

where

Ψ(γ; δ, α) =ei(γ−iδ) ln(Φ(0,−iα))

i(γ − iδ)· [Φ ((γ − iδ)− i,−α(γ − iδ))

− Φ(γ − iδ,−α(γ − iδ)− i)−KΦ(γ − iδ,−α(γ − iδ))],

α =F (0, T2)

F (0, T2) +K, k = ln(F (0, T2) +K).

The parameter δ controls an exponential decay term.

Comparing No, Weak, and Strong Seasonality

Introduction

Market Data








Literature


θj(t) = aj + bj cos(2πωj(t− t0j)), j = 1, 2

parameter Case 1 Case 2 Case 3

v01 0.10 0.10 0.10v02 0.04 0.04 0.04λ1 2.00 2.00 2.00λ2 0.50 0.50 0.50κ1 0.80 0.80 0.80κ2 0.80 0.80 0.80σ1 1.20 1.20 1.20σ2 0.90 0.90 0.90ρ1 −0.25 −0.25 −0.25ρ2 −0.25 −0.25 −0.25a1 0.25 0.25 0.25a2 0.10 0.10 0.10b1 0.00 0.15 0.35b2 0.00 0.00 0.00t01 7/12 7/12 7/12t02 7/12 7/12 7/12

Calendar Spread Option Prices and Seasonality


Case 1: no seasonality Case 2: moderate seasonality Case 3: strong seasonality

T = T1 T2 K = 0 K = 10 K = 0 K = 10 K = 0 K = 10

0.33 0.83 2.5993 0.4951 3.0931 0.6822 3.4190 0.83510.58 1.08 3.4672 0.9188 3.7314 1.1026 3.9252 1.24000.83 1.33 4.5229 1.6366 4.3690 1.5636 4.2429 1.51201.08 1.58 4.6534 1.8404 4.5533 1.7709 4.4773 1.71871.33 1.83 4.1480 1.6234 4.4534 1.7695 4.6612 1.88431.58 2.08 4.4875 1.7783 4.6726 1.9165 4.8091 2.01971.83 2.33 5.2374 2.3126 5.0666 2.2079 4.9289 2.13122.08 2.58 5.2422 2.3840 5.1193 2.2922 5.0262 2.22332.33 2.83 4.6788 2.0671 4.9321 2.1982 5.1052 2.29922.58 3.08 4.8929 2.1509 5.0521 2.2717 5.1692 2.36172.83 3.33 5.5345 2.6146 5.3604 2.4988 5.2210 2.4132

The CSOs in this example are all for six months, i.e. T2 − T1 is half a year.Note that seasonality can lead to increasing or decreasing prices.

Correlation Term-Structure and Seasonality

Introduction

Market Data








Literature


This effect can also be seen in the change in implied correlations

calculated from CSO prices. Case 1: no seasonality, Case 2:

moderate seasonality, Case 3: strong seasonality.

0 0.5 1 1.5 2 2.50.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

T

Literature

Introduction

Market Data




Literature

• Literature


Literature

Introduction

Market Data




Literature

• Literature


[1] Janis Back, Marcel Prokopczuk, and Markus Rudolf. Seasonal stochasticvolatility: Implications for the pricing of commodity options. ICMA CentreDiscussion Papers in Finance, June 2011.

[2] Ruggero Caldana and Gianluca Fusai. A general closed-form spread optionpricing formula. Journal of Banking and Finance, 37(12):4893–4906, December2013.

[3] Helyette Geman and Andrea Roncoroni. Understanding the fine structure ofelectricity prices. Journal of Business, 79(3):1225–1261, 2006.

[4] Adam Schmitz, Zhiguang Wang, and Jung-Han Kimn. Pricing and hedgingcalendar spread options on agricultural grain commodities. In Proceedings ofthe NCCC-134 Conference on Applied Commodity Price Analysis, Forecasting,and Market Risk Management. St. Louis, MO, 2013.

[5] Lorenz Schneider and Bertrand Tavin. From the Samuelson volatility effect to aSamuelson correlation effect: An analysis of crude oil calendar spread options.Working Paper, available on ssrn, September 2015.

[6] Lorenz Schneider and Bertrand Tavin. Seasonal stochastic volatility andcorrelation together with the Samuelson effect in commodity futures markets.Working Paper, available on ssrn, June 2015.

seasonal stochastic volatility and correlation together with ......•corn - futures •corn -...

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