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Transcript of 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
1 / 26
Introduction
Forwards versus Futures
Options on Forwards vs Futures
Black 76
2 / 26
Forwards versus Futures
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
Forwards versus Futures
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
Forwards versus Futures
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
Forwards versus Futures
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.
6 / 26
Forwards versus Futures
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
Forwards versus Futures
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 ).
8 / 26
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
More General Relationship Between Forwards and Futures
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.
11 / 26
Basic Valuation
More General Relationship Between Forwards and Futures
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
More General Relationship Between Forwards and Futures
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.
16 / 26
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.
17 / 26
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.
18 / 26
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.
20 / 26
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
21 / 26
Basic Valuation
The Basic Option Valuation Framework: Black 76
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
The Basic Option Valuation Framework: Black 76
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
The Basic Option Valuation Framework: Black 76
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
The Basic Option Valuation Framework: Black 76
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.
25 / 26
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.
26 / 26