schwartz2 factor implementation in r

8/13/2019 Schwartz2 factor Implementation in r

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Commodity Modelling

Schwartz two-factor model

Philipp Erb IDP Winterthur David Lthi Clariden Leu Zrich

July 2, 2009

School ofEngineering

IDP Institute ofData Analysis andProcess Design

Zurich University

of Applied Sciences


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Commodity Modelling 2 / 33

Motivation

Motivation and Context

CTI project between IDP and FRSGlobal

Extending FRSGlobals softwaretool riskpro with commodity models

Schwartz fator model: Basic model

A Hinz model: More advanced model

Implementation of protoype models in R

No commodity models on CRAN so far

Launching commodity model package schwartz97

IDP Winterthur, Clariden Leu Zrich


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Commodity model types

Outline

1 Motivation2 Commodity model types

3 Schwartz two-factor model in a nutshell

Model dynamicsJoint distribution of state variables

Pricing of basic derivatives

Risk measures

4 Using schwartz97

Parameter Estimation

Examples



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Spot price models: E.g. Schwartz factor models

Few parameters

Inflexible term sturcutre of futures prices

Futures price models: E.g. HJM type model by Juri Hinz

Numerous parameters

Flexible term sturcture generic


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Schwartz factor models

One-factor model

Spot price Log-price mean-reverting (Ornstein-Uhlenbeck)

Two-factor model

Spot price Log-normal (Geometric Brownian motion)

Convenience yield Mean-reverting (Ornstein-Uhlenbeck)

Three-factor model

Spot price Log-normal (Geometric Brownian motion)

Convenience yield Mean-reverting (Ornstein-Uhlenbeck)

Interest rate Mean-reverting (Vasicek)

Empirical study: Two-factor model most-parsimonious choice


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Schwartz two-factor model in a nutshell

Outline

1 Motivation

2 Commodity model types


Model dynamicsJoint distribution of state variablesPricing of basic derivativesRisk measures

4 Using schwartz97Parameter EstimationExamples




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Model dynamics: Two-factor model

The P-dynamics:

dSt= ( t) Stdt+ SStdWS

dt= ( t) dt+ dW

dWSdW =dt

The Q-dynamics:

dSt= (r t) Stdt+ SStdWS

dt= [ ( t) ] dt+ dW

dWSdW =dt




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Outline



Model dynamics

Joint distribution of state variables


Risk measures

4 Using schwartz97


Examples




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Log-priceX :=log Sand convenience yield joint normal

Solve SDE under P and Q and compute first two moments: Distribution underP and Q

X

= N

P

=

X

, =

2X XX

2

,

X

= N

Q

=

X

, =

2X XX

2

Basic tool for all computations(pricing, simulation, risk measures, calibration)




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Outline

1 Motivation



Model dynamicsJoint distribution of state variablesPricing of basic derivativesRisk measures

4 Using schwartz97Parameter EstimationExamples




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y g /


Futures

Futures prices:

G(S0, 0, T) =EQ [ST] =e

X+122

X

=S0eA(T)+B(T)0

with

A(T) = r + 1

2

2

2

S

T+ 1

4

2

1 e2T

3

+

+ S

2

1 e

T

2 ,

B(T) = eT

1




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y g /


Futures

Affine term structure of futures prices:

log G(T) =X0+B(T)0+A(T)

Model can be cast in state space form

Both state variables (factors: Xt, t) unobservable

Kalman filter




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European Options



Futures prices log-normal

BS-type formulas for European options on futures



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Outline

1 Motivation





Risk measures

4 Using schwartz97


Examples




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Risk Measures



Futures log-price Gaussian: log G NG, 2G

Losses (negative log-returns) Gaussian

Closed formulas for value at risk (VaR) and expected

shortfall (ES) for futures contracts




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Using schwartz97

Outline1

Motivation2 Commodity model types


Model dynamics



Risk measures

4 Using schwartz97


Examples




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Using schwartz97

Function Overview: Usual d/p/q/r style

If all parameters are known, the transition density is known.

For a fixed point in time, the distribution is defined.

d/p/q/r-state d/p/q/r functions for the bivariate state variables.

simstate Generate trajectories from the state variables.

d/p/q/r-futures d/p/q/r functions for the log-normal futures prices.



U i h 97


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Using schwartz97

Function Overview: Estimation, pricing, risk

fit2factor Fit parameters to data.

pricefutures Calculate futures prices.

priceoption Price a European call or put option written on a futures

contract.

VaR/ES-futures Value-at-risk and Expected-shortfall of futures prices.

filter2factor Filter a futures price series and obtain the state

variables.



U i h t 97


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Using schwartz97

Object Orientation

Problem: Many parameters involvedThe transition density of the state variables depends on 8 parameter:

(s0, 0, , S, , , , )

The transition density of the futures price depends on 10 parameter:(s0, 0, , S, , , , , , r)

Idea: 3 levels of encapsulation

Raw parameters.

schwartz2factor-class

fit.schwartz2factor-class



Using schwartz97


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Using schwartz97

Example: dfutures

dfutures(x, ttm = 1, s0 = 50, delta0 = 0,mu = 0.1, sigmaS = 0.3, kappa = 1, alpha = 0,

sigmaE = 0.5, rho = 0.75, r = 0.05, lambda = 0,

alphaT = NULL, ...)

## S4 method for signature ANY, ANY, schwartz2factor:

dfutures(x, ttm = 1, s0, r = 0.05, lambda = 0,

alphaT = NULL, ...)

## S4 method for signature ANY, ANY, fit.schwartz2factor:

dfutures(x, ttm = 1, s0, ...)



Using schwartz97


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Using schwartz97

Outline

1 Motivation





Risk measures

4 Using schwartz97Parameter Estimation

Examples



Using schwartz97


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Using schwartz97

State Space Representation

Transition equation:

The state variables arelog-spot priceandconvenience yield.

t+1 = dt+Ttt+Htt, t N(0, I2) (4.1)

Measurement equation:

Futures prices are measured. The log-futures price are an affine

function of the log-spot price and the convenience yield.

yt = ct+Ztt+Gtt, t N(0, In) (4.2)



Using schwartz97


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Using schwartz97

State Space Model: Transition Equation

t+1 = dt+Ttt+Htt, t N(0, I2) (4.3)

t+t =

Xt+tt+t

(4.4)

Tt =

1 1

et 1

0 et

(4.5)

dt = 12

2S

t+

1 et

1 e

t (4.6)

HtH

t =

2X(t) X(t)X(t)

2 (t),

(4.7)

where Xt=log St and t=tk+1 tk.



Using schwartz97


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g

State Space Model: Measurement Equation

yt = ct+Ztt+Gtt, t N(0, In) (4.8)

yt=

log Gt(1)

...

log Gt(n)

Zt= 1 B(mt(1))

... ...

1 B(mt(n))

(4.9)

ct=

A(mt(1))...

A(mt(n))

GtG

t=

g211. .

.g2nn

(4.10)

where mt(i) denotes the remaining time to maturityof the i-th

closest to maturity futures Gt(i).



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Commodity Modelling 27 / 33Using schwartz97


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Outline



Model dynamics



Risk measures

4 Using schwartz97


ExamplesIDP Winterthur, Clariden Leu Zrich



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Function fit2factor

> library(schwartz97)

Loading required package: FKF

Loading required package: mvtnorm

> data(futures)

> na.idx copper.fit




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Function fit2factor> copper.fit----------------------------------------------------------------Fitted Schwartz97 2 factor model:

d S_t = S_t * (mu - delta_t) * dt + S_t * sigmaS * dW_1d delta_t = kappa * (alpha - delta_t) * dt + sigmaE * dW_2E(dW_1 * dW_2) = rho * dt

s0 : 109.375364269883delta0: 0

mu : 0.212827842930593

sigmaS: 0.37460428511304kappa : 1.10747274866091alpha : 0.0595560374598207sigmaE: 0.474809676990843rho : 0.813002684142751r : 0.05lambda: 0.0402214875501909alphaT: 0.0232377735577751

----------------------------------------------------------------Optimization informationConverged: TRUEFitted parameters: mu, sigmaS, kappa, alpha, sigmaE, rho, lambda, meas.sd1; (Nulog-Likelihood: 112485.1Nbr. of iterations: 75---------------------------------------------------------------->




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Function filter2factor> filter2factor(copper$futures[!na.idx,], copper$ttm[!na.idx,], copper.fit)

>

0 1000 2000 3000 4000 5000

50

200

350

Index

Spot priceClosest to maturity futures

0 1000 2000 3000 4000 5000

0.1

0.2

0.5

Index

Convenience yield




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Derivatives

> pricefutures(ttm = 3, copper.fit)

[1] 107.1682> priceoption("call", time = 2, Time = 3, K = 110, copper.fit)

[1] 12.33399

>




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Conclusion & Outlook

Conclusion

(Optimal??) trade-off: Statistical tractability vs. sophistication.

Stylized model.

First step towards commodity modelling in R.

Outlook

Implementing term-structure models (Juri Hinz & Max Fehr, 2007).




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References

Rajna Gibson and Eduardo S. Schwartz.

Stochastic convenience yield and the pricing of oil contingent claims.

The Journal of Finance, 45(3):959976, 1990.

Eduardo S. Schwartz.

The stochastic behavior of commodity prices: Implications for valuation andhedging.

Journal of Finance, 52(3):923973, 1997.

Kristian R. Miltersen and Eduardo S. Schwartz.

Pricing of options on commodity futures with stochastic term structures of

convenience yields and interest rates.Journal of Financial and Quantitative Analysis, 33:3359, 1998.


schwartz2 factor implementation in r

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