commodities and energy markets supplementary notes: basic

Commodities and Energy Markets SupplementaryNotes: Basic Valuation

Princeton RTG summer school in Financial Mathematics

Presenters: Michael Coulon and Glen Swindle

17 April 2013

c© Glen Swindle: All rights reserved

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Introduction

Forwards versus Futures

Options on Forwards vs Futures

Black 76

2 / 26


Measures

Assumption: Given a reference asset, there is a uniqueequivalent measure under which the price of any tradedsecurity discounted by the reference asset is a martingale.

Money-market measure E [·]:- The equivalent martingale measure with Mt as the reference asset

(numeraire).

- Standard spot rate accrual: dMt = rtMtdt.

- Mt+δt is previsible at time t since Mt+δt = Mt(1 + rtδt)

T-forward measure ET :

- Reference asset is the zero-coupon bond: B(t,T ) = Ehe−

R Tt rsds |Ft

i.

- Particularly useful for derivatives on forward contracts.

3 / 26


Measures

Consider a payoff VT (FT -measurable)

- For the money-market measure:

V0 = M0E

»VT

MT

–= E

he−

R T0 rsdsVT

i- For the T -forward measure:

V0 = B0ET

»VT

BT

–= B0ET [VT ]

Deterministic interest rates =⇒ E and ET are identical.

4 / 26


A Fact About Forward Prices

Zero-price of entry can be written as:

0 = E[(F (T ,T )− F (t,T )) e−

R Tt rsds |Ft

].

which implies:

F (t,T ) =E[e−

R Tt rsdsF (T ,T )|Ft

]B(t,T )

5 / 26


Key Fact: Forward prices are martingales under the T -forwardmeasure.

Denoting the value of a forward contract established at time tfor delivery at time T by Vs for any t ≤ s ≤ T , we know that:

(1) VtB(t,T )

is a martingale under the T -forward measure.

(2) Vt = 0.

Therefore:

0 =Vt

B(t,T )= ET

[VT

B(T ,T )| Ft

]

Using the fact that VT = F (t,T )− F (T ,T ) establishes themartingale property.

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Futures Contracts

A futures contract is a margined forward contract.

- Each contract is marked to market on a daily basis with the change in

value reflected in the balance of the customers margin account.

- Margining occurs through the exchange that supports the contract.

- Margining requirements vary between exchanges.

Key Points:

- Margining means that the value of a futures contract is zero at the end of

each day.

- Forward and futures prices are in general different due to the potential

difference in cash flows.

7 / 26


Forward and futures prices are coincident if interest rates aredeterministic

Forward price: F (t, t); futures prices F (t,T ).

The following two strategies require zero initial investment:

- A long forward position initiated at time t will have a terminal payoff of

F (T ,T )− F (t,T ).- Maintaining α(s) futures contracts for s ∈ [t,T ] has a payoff (ignoring

transaction costs) of: Z T

tα(s) e

R Ts rududFs

Choosing α(s) = e−R T

srudu and observing that F (T ,T ) = F (T ,T ),

it follows that F (t,T ) = F (t,T ).

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Forwards versus FuturesFutures contracts are martingales in the money-market measure

Mark-to-market (margining) implies that the value is reset to zeroat the next margining time t + δ:

0 = E [B(t, t + δt)Vt+δ− |Ft ]

where Vt+δ− is the value of the position just before t + δ.

This implies that:

0 = E[B(t, t + δt)

(F (t + δt,T )− F (t,T )

)|Ft

]Since B(t, t + δt) is Ft measurable at time t, the martingale

property follows.

Given the terminal value we have:

F (t,T ) = E [F (T ,T )|Ft ]

Note: The same argument applies to any cash margined derivative

contract.9 / 26

Basic Valuation

More General Relationship Between Forwards and Futures

Recall these two facts:

- Forward Prices:

F (0,T ) =E[e−

R T0

rsdsF (T ,T )]

B(0,T )

- Futures Prices:

F (0,T ) = E [F (T ,T )] .

It follows that:

F (0,T )−F (0,T ) =1

B(0,T )

nEhe−

R T0 rsdsF (T ,T )

i−E

he−

R T0 rsds

iE [F (T ,T )]

o

10 / 26

Basic Valuation


This can be written as:

F (0,T )− F (0,T ) =1

B(0,T )cov

[e−

R T0 rsds ,F (T ,T )

].

Intuition:

- Suppose that the covariance between rates and price is positive.

- Then the margin account for a long futures contract tends to be credited

when rates are high, and debited when rates are low.

- This means that the futures price should be higher than the corresponding

forward price.

- Note that positive correlation between rates and prices results in a

negative covariance above.

Fact: If rates are uncorrelated with the underlying commodityprices, forward prices are identical to futures prices.

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Basic Valuation


Does this matter?

The relationship can be written as:

F (0,T )

F (0,T )− 1 = −cov

[e−

R T0 rsds

B(0,T ),F (T ,T )

F (0,T )

].

The following figure shows a scatter of:

- The forward ratio F (T ,T )F (T−1,T )

versus

- The following proxy for the discount ratio

h1 + r (12)

i 12Ym=1

"1 +

r(1)m

12

#−1

where r(n)m is the n-month USD LIBOR rate at the beginning of month m.

for each contract month spanning Jan92 to Dec10.

12 / 26

Basic Valuation


The correlation is nontrivial (increasing rates tending to beassociated with increasing WTI prices), but the covariance isvery small.

0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 1.03 1.0350

0.5

1

1.5

2

2.5

Discount Factor Ratios

Forw

ard

Ratio

s

Forward Ratios Versus Discount Ratios (1992−2011)

Correlation: −0.32 Covariance: −0.001

Rates Increasing

13 / 26

Basic Valuation

An American feature of an option on a forward contract is trivial

Assume the case of a call.

At any time t, the option holder can exercise into a long forward

contract, of time-T value F (t,T )− K .

Alternatively, the holder could short an (ATM) forward contract andhold the option to expiration:

V (T ) =

{F (t,T )− F (T ,T ), if F (T ,T ) < KF (t,T )− K , otherwise

which dominates the payoff had the holder exercised at time t.

14 / 26

Basic ValuationAn American feature of an option on a forward contract is trivial

5 6 7 8 9 10 11 12 13 14 15−6

−4

−2

0

2

4

6

8

Forward Price

Payo

ffEarly Exercise of a Call

Swap Payoff

Put PayoffCall Payoff

Intrinsic Value

15 / 26

Basic Valuation

An American feature of an option on a forward contract is trivial

This result is true for any convex payoff f (·) of the forwardprice:

- By Jensen’s inequality we have:

f (F (t,T )) = f (ET [F (T ,T ) | Ft ]) ≤ ET [f (F (T ,T )) | Ft ]

where we have used the fact that the forward price is an ET -martingale.

- The first term is the undiscounted value of immediate exercise at time t;

the last term is the undiscounted value of the option—each at time t.

- It follows that such options are never optimal to exercise early.

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Basic Valuation

Options on Futures

Commodities options traded on exchanges are options onfutures.

Example: A call option gives the holder the right to acquireupon exercise:

- The futures contract

- A cash balance of the difference between the futures price and option

strike.

Mechanics vary by exchange.

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Basic Valuation

Options on Futures: Upfront premium (“Equity-style”)

Traded on CME or NYMEX.

Early exercise provision of such options is nontrivial.

Upon exercise of a call at time t, the value to the holder is Ft,T −Kwhere:

- t is the time of exercise;

- T and K are the option expiration and strike respectively.

If the option is in-the-money, as interest rates increase but

everything else (including the futures price) stays constant, the

immediate payoff and the forward value of the option do not change.

Since the latter needs to be discounted to time t, we can easily

imagine a situation when the value of the American option is strictly

higher than that of its European analog.

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Basic Valuation

Options on Futures: Margined options (“Futures-style”)

American options traded on ICE are marked to market just as for

futures contracts.

They are in fact futures contracts on[Ft,T − K

]+

.

By the same argument as for futures if Pt is the time-t price:

Pt = E [PT | Ft ]

The premium of such an option is not paid upfront; rather the

buyer will pay the option price at the time of exercise/expiration.

The value to the buyer upon exercise of a call option is

Ft,T − K − Pt

19 / 26

Basic Valuation

Options on Futures: Margined options

Since the option payoff is convex, it follows from Jensen’s inequalitythat:

(E[FT ,T | Ft

]− K )+ ≤ E

[(FT ,T − K )+ | Ft

]= E [PT | Ft ] = Pt

which means the value at immediate exercise is nonpositive.

Consequently these options are never optimal to exercise early.

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Basic Valuation

The Basic Option Valuation Framework: Black 76

The classical model for valuation of futures optionspostulates:

- A constant risk-free rate r .

- Under the risk-neutral measure the forward price is a geometric Brownianmotion (GBM)

dFt

Ft= σdBt

which is trivially integrated to give

Ft = F0 e−12σ2t+σBt

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Basic Valuation


The usual arbitrage argument proceeds by invoking Ito’sformula.

A buyer of the option will experience the following variation ofits value Vt = V (t,Ft):

dVt =∂V

∂tdt +

∂V

∂FtdFt +

1

2

∂2V

∂F 2t

(dFt)2

22 / 26

Basic Valuation


To hedge this, the option holder will go short a number ∆ of futurescontracts, so that portfolio variation is:

dVt −∆dFt =

(∂V

∂t+

1

2σ2F 2

t

∂2V

∂F 2t

)dt +

(∂V

∂Ft−∆

)dFt

Choosing ∆ = ∂V /∂Ft , the instantaneous variation of the portfolio

becomes deterministic.

By no arbitrage, its growth should then equal the carry of the initialcost under the risk-free rate, which result in:

∂V

∂t+

1

2σ2F 2

t

∂2V

∂F 2t

= rV

with boundary data V (Te ,FTe ,T ) for a European option payoff.

23 / 26

Basic Valuation


For a standard European option expiring at Te on a forwardcontract with delivery at time T ≥ Te

- For example, a call has boundary condition is

V (Te ,FTe ,T ) = d(Te ,T ) maxˆFTe ,T − K , 0

˜.

Assuming constant interest rates the result is the Black ’76 formulas for thevalue of a call:

C(t,F ) = e−r(T−t) (FΦ(d1)− KΦ(d2))

and a put:

P(t,F ) = e−r(T−t) (KΦ(−d2)− FΦ(−d1))

with

d1,2 =ln( F

K)± 1

2σ2(Te − t)

σ√

Te − t

where Φ is the c.d.f. of the standard normal distribution.

24 / 26

Basic Valuation


In the risk neutral world, a futures contract has zero cost ofcarry.

This is equivalent to a stock paying dividends at a rate equalto the risk free rate.

Black ’76 formula can be obtained from the Black-Scholesformula for a dividend paying stock, when the dividend andrisk free rates are equal.

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Basic Valuation

Physical versus Risk Neutral

Mean-reversion of forward prices is one aspect commoditiesfolklore.

Suppose that in the physical measure:

dFt = µ(t,Ft)dt + FtσdBt

where for example we could have

µ(t,Ft) = −β(L− Ft)

.

The same arguments above apply as hedging eliminates thedrift.

The only exception to this would be if the drift was singularenough to violate absolute continuity.

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