Commodities and Energy Markets Supplementary Notes: Basic Valuation

Transcription

1 Commodities and Energy Markets Supplementary Notes: 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

2 Introduction Forwards versus Futures Options on Forwards vs Futures Black 76 2 / 26

3 Forwards versus Futures Measures Assumption: Given a reference asset, there is a unique equivalent measure under which the price of any traded security discounted by the reference asset is a martingale. Money-market measure Ẽ[ ]: - The equivalent martingale measure with M t as the reference asset (numeraire). - Standard spot rate accrual: dm t = r tm tdt. - M t+δt is previsible at time t since M t+δt = M t(1 + r tδt) T -forward measure Ẽ T : - Reference asset is the zero-coupon bond: B(t, T ) = he Ẽ R T t r s ds F t i. - Particularly useful for derivatives on forward contracts. 3 / 26

4 Forwards versus Futures Measures Consider a payoff V T (F T -measurable) - For the money-market measure: - For the T -forward measure:» VT V 0 = M 0 Ẽ = he M Ẽ R i T 0 r s ds V T T V 0 = B 0 Ẽ T» VT B T = B 0 Ẽ T [V T ] Deterministic interest rates = Ẽ and Ẽ T are identical. 4 / 26

5 Forwards versus Futures A Fact About Forward Prices Zero-price of entry can be written as:. 0 = Ẽ which implies: [(F (T, T ) F (t, T )) e R T t r sds F t ] F (t, T ) = Ẽ [e R T t r sds F (T, T ) F t ] B(t, T ) 5 / 26

6 Forwards versus Futures Key Fact: Forward prices are martingales under the T -forward measure. Denoting the value of a forward contract established at time t for delivery at time T by V s for any t s T, we know that: V (1) t is a martingale under the T -forward measure. B(t,T ) (2) V t = 0. Therefore: 0 = V [ ] t B(t, T ) = Ẽ V T T B(T, T ) F t Using the fact that V T = F (t, T ) F (T, T ) establishes the martingale property. 6 / 26

7 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

8 Forwards versus Futures Forward and futures prices are coincident if interest rates are deterministic 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 R T α(s) e s r udu d F s t Choosing α(s) = e R T s r udu and observing that F (T, T ) = F (T, T ), it follows that F (t, T ) = F (t, T ). 8 / 26

9 Forwards versus Futures Futures contracts are martingales in the money-market measure Mark-to-market (margining) implies that the value is reset to zero at the next margining time t + δ: 0 = Ẽ [B(t, t + δt)v t+δ F t] where V t+δ is the value of the position just before t + δ. This implies that: [ ) ] 0 = Ẽ B(t, t + δt) ( F (t + δt, T ) F (t, T ) F t Since B(t, t + δt) is F t measurable at time t, the martingale property follows. Given the terminal value we have: F (t, T ) = Ẽ [F (T, T ) F t] Note: The same argument applies to any cash margined derivative contract. 9 / 26

10 More General Relationship Between Forwards and Futures Recall these two facts: - Forward Prices: - Futures Prices: It follows that: F (0, T ) F (0, T ) = Ẽ F (0, T ) = [ e R ] T 0 rs ds F (T, T ) B(0, T ) F (0, T ) = Ẽ [F (T, T )]. 1 h nẽ e R i h T 0 r s ds F (T, T ) Ẽ e R i o T 0 r s ds Ẽ [F (T, T )] B(0, T ) 10 / 26

11 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 ] T 0 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 commodity prices, forward prices are identical to futures prices. 11 / 26

12 More General Relationship Between Forwards and Futures Does this matter? The relationship can be written as: F (0, T ) [e R ] T F (0, T ) 1 = cov 0 rsds B(0, T ), F (T, T ). F (0, T ) The following figure shows a scatter of: F (T,T ) - The forward ratio F (T 1,T ) versus - The following proxy for the discount ratio " h1 + r (12)i 12Y 1 + r (1) # 1 m 12 m=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 Dec / 26

13 More General Relationship Between Forwards and Futures The correlation is nontrivial (increasing rates tending to be associated with increasing WTI prices), but the covariance is very small. 2.5 Forward Ratios Versus Discount Ratios ( ) Correlation: 0.32 Covariance: Forward Ratios Rates Increasing Discount Factor Ratios 13 / 26

14 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 and hold the option to expiration: { F (t, T ) F (T, T ), if F (T, T ) < K V (T ) = F (t, T ) K, otherwise which dominates the payoff had the holder exercised at time t. 14 / 26

15 An American feature of an option on a forward contract is trivial 8 Early Exercise of a Call 6 4 Put Payoff Call Payoff Payoff 2 0 Intrinsic Value 2 Swap Payoff Forward Price 15 / 26

16 An American feature of an option on a forward contract is trivial This result is true for any convex payoff f ( ) of the forward price: - By Jensen s inequality we have: f (F (t, T )) = f (ẼT [F (T, T ) F t]) ẼT [f (F (T, T )) F t] where we have used the fact that the forward price is an ẼT -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

17 Options on Futures Commodities options traded on exchanges are options on futures. Example: A call option gives the holder the right to acquire upon exercise: - The futures contract - A cash balance of the difference between the futures price and option strike. Mechanics vary by exchange. 17 / 26

18 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 F t,t K where: - 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

19 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 P t is the time-t price: P t = Ẽ [P T F t ] 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 F t,t K P t 19 / 26

20 Options on Futures: Margined options Since the option payoff is convex, it follows from Jensen s inequality that: ] ] (Ẽ [ FT,T F t K) + [( F Ẽ T,T K) + F t = Ẽ [P T F t ] = P t which means the value at immediate exercise is nonpositive. Consequently these options are never optimal to exercise early. 20 / 26

21 The Basic Option Valuation Framework: Black 76 The classical model for valuation of futures options postulates: - A constant risk-free rate r. - Under the risk-neutral measure the forward price is a geometric Brownian motion (GBM) df t F t which is trivially integrated to give = σdb t F t = F 0 e 1 2 σ2 t+σb t 21 / 26

22 The Basic Option Valuation Framework: Black 76 The usual arbitrage argument proceeds by invoking Itô s formula. A buyer of the option will experience the following variation of its value V t = V (t, F t ): dv t = V t dt + V df t V F t 2 Ft 2 (df t ) 2 22 / 26

23 The Basic Option Valuation Framework: Black 76 To hedge this, the option holder will go short a number of futures contracts, so that portfolio variation is: ( V dv t df t = t σ2 Ft 2 2 ) ( ) V V Ft 2 dt + df t F t Choosing = V / F t, the instantaneous variation of the portfolio becomes deterministic. By no arbitrage, its growth should then equal the carry of the initial cost under the risk-free rate, which result in: V t σ2 F 2 t 2 V F 2 t = rv with boundary data V (T e, F Te,T ) for a European option payoff. 23 / 26

24 The Basic Option Valuation Framework: Black 76 For a standard European option expiring at T e on a forward contract with delivery at time T T e - For example, a call has boundary condition is V (T e, F Te,T ) = d(t e, T ) max ˆF Te,T K, 0. Assuming constant interest rates the result is the Black 76 formulas for the value of a call: C(t, F ) = e r(t t) (F Φ(d 1 ) KΦ(d 2 )) and a put: with P(t, F ) = e r(t t) (KΦ( d 2 ) F Φ( d 1 )) d 1,2 = ln( F K ) ± 1 2 σ2 (T e t) σ T e t where Φ is the c.d.f. of the standard normal distribution. 24 / 26

25 The Basic Option Valuation Framework: Black 76 In the risk neutral world, a futures contract has zero cost of carry. This is equivalent to a stock paying dividends at a rate equal to the risk free rate. Black 76 formula can be obtained from the Black-Scholes formula for a dividend paying stock, when the dividend and risk free rates are equal. 25 / 26

26 Physical versus Risk Neutral Mean-reversion of forward prices is one aspect commodities folklore. Suppose that in the physical measure: df t = µ(t, F t )dt + F t σdb t where for example we could have. µ(t, F t ) = β(l F t ) The same arguments above apply as hedging eliminates the drift. The only exception to this would be if the drift was singular enough to violate absolute continuity. 26 / 26