Option Pricing Chapter 11 Options on Futures Stefan Ankirchner University of Bonn last update: 13/01/2014 at 14:25 Stefan Ankirchner Option Pricing 1
Agenda Forward contracts Definition Determining forward prices Futures contracts Definition The margining mechanism Options on Futures Pricing PDEs Black76 formula Further reading: Chapter 5 & 16 in J. C. Hull: Options, Futures, and other Derivatives, Prentice-Hall, New York. Stefan Ankirchner Option Pricing 2
Forward contracts Definition from Wikipedia: A forward contract, or simply a forward, is a contract between two parties to buy or sell an asset at a specified future time at a price agreed today. In a forward contract one specifies the asset to be delivered, the quantity, the delivery date, the delivery price. The buyer of the contract is said to be long in the asset, and the seller is said to be short. Stefan Ankirchner Option Pricing 3
Forward prices The delivery price is usually chosen such that it does not cost anything to enter the forward contract. This particular price is called forward price. This means that at the time where a forward contract is entered, the forward price is equal to the delivery price. Notice that between the two parties making a forward contract there is only a cash resp. asset flow at the delivery date, but not at the contract date. Stefan Ankirchner Option Pricing 4
Investment and consumption assets There are many different types of assets underlying forward contracts. We distinguish between the following two: Investment assets stocks bonds gold... Consumption assets commodities (e.g. crude oil, copper,...) orange juice... Stefan Ankirchner Option Pricing 5
Forward prices for investment assets The forward price of an investment asset can be determined from its spot price and other observable market factors. Notation S 0 = spot price of the asset T = delivery date r = risk free rate (continuous compounding) F 0 = forward price Theorem Consider an investment asset providing no additional income (e.g. a stock paying no dividends, or a zero-coupon bond). The only arbitrage free forward price for the asset to be delivered at T is F 0 = e rt S 0. (1) Stefan Ankirchner Option Pricing 6
Constructive proof of (1) Proof of (1): Suppose first that F 0 > e rt S 0. Set up the following portfolio: forward contract underl. asset cash position -1 +1 S 0 portfolio value at time 0 is zero: V 0 = 0 + S 0 S 0 = 0 portfolio value at time T is positive with probability one: V T = (S T F 0 ) + S T e rt S 0 = F 0 e rt S 0 > 0 Thus the market admits arbitrage. Stefan Ankirchner Option Pricing 7
Constructive proof of (1) cont d Next suppose that F 0 < e rt S 0. Set up the following portfolio: forward contract underl. asset cash position +1-1 +S 0 portfolio value at time 0 is zero: V 0 = 0 S 0 + S 0 = 0 portfolio value at time T is positive with probability one: V T = (S T F 0 ) S T + e rt S 0 = e rt S 0 F 0 > 0 Again the market admits arbitrage. Therefore F 0 = e rt S 0 is the unique arbitrage free price. Stefan Ankirchner Option Pricing 8
Proof of (1) via risk neutral pricing A forward can be seen as a derivative with payoff S T K, where K is the delivery price. According to the pricing principle, the arbitrage free price is the expected payoff with respect to the risk neutral measure Q. As it costs nothing to enter a forward contract, in an arbitrage free market is must hold true that E Q (S T F 0 ) = 0. (2) Under Q, the discounted asset price is a martingale; hence E Q (e rt S T ) = S 0. Therefore, with (2), which entails (1). F 0 = E Q (S T ) = e rt E Q (e rt S T ) = e rt S 0, Stefan Ankirchner Option Pricing 9
Forward price of dividend paying stocks Theorem Consider a dividend paying stock, and let I be the discounted value of all dividends paid up to T. The only arbitrage free forward price with delivery at T is F 0 = e rt (S 0 I ). (3) Sketch of the proof: If F 0 > e rt (S 0 I ), then consider the portfolio forward contract underl. asset cash position -1 +1 S 0 The portfolio value at time 0 is zero, and at T it satisfies V T = (S T F 0) + S T e rt S 0 + e rt I = F 0 e rt (S 0 I ) > 0, and hence the market admits arbitrage. The case F 0 < e rt (S 0 I ) can be treated similarly. Stefan Ankirchner Option Pricing 10
The margining mechanism Futures prices Futures contracts Like a forward, a futures contract (or simply a futures) is an agreement to buy/sell an asset at a future time at a price specified already today. The main difference to forwards: futures are exchange-traded, whereas forwards are traded OTC (over-the-counter). Stefan Ankirchner Option Pricing 11
The margining mechanism Futures prices Some stylized differences between forwards and futures futures forwards exchange traded OTC traded highly standardized tailor-made margin payments no margining no counterparty risk counterparty risk Stefan Ankirchner Option Pricing 12
The margining mechanism Futures prices The margining mechanism Anyone trading futures has to set up a margin account at the exchange. At the moment a futures contract is entered, an initial margin has to be deposited. At the end of every trading day the futures position is marked-to-market, and the margin account is adjusted accordingly. The precise mechanism is best explained with an Example: Suppose that on July 4, 2011, an investor buys one gold futures at a price of 1500$/oz. Delivery is December 2011, and the contract size is 100 ounces. Suppose that the exchange requires an initial margin of 2000$. Stefan Ankirchner Option Pricing 13
The margining mechanism Futures prices Example cont d Assume that prices fall after the investor has entered the contract, and that the futures closes at 1492$/oz on July 4. The value of the investor s position has declined by (1500 1492) $ 100oz = 800$. oz The margin account is reduced by 800$, and has a new balance of 1200$. Suppose that on July 5 prices soar, and the futures closes at 1520$/oz. Then the exchange transfers (1520 1492) $ 100oz = 2800$ oz to the margin account, having then a new balance of 4000$.... the account is adjusted like this every day up to delivery... The investor can withdraw from the margin account the cash exceeding the initial margin, but has to make sure that a minimum maintenance margin is always set. Stefan Ankirchner Option Pricing 14
The margining mechanism Futures prices Do futures prices coincide with forward prices? Definition. futures price = the delivery price of a traded futures contract (note that it does not cost anything to enter a futures) Under the assumptions that the interest rate r is constant, there is no default risk, the futures price, in theory, is equal to the forward price. In particular, the futures price of an investment asset with no additional income is given by e rt S 0. Caution: Under stochastic interest rate r(t), t 0, the futures price Fut and the forward price For, with delivery T, are given by Fut = E Q [S T ] For = S 0 E Q [ T 0 e r(s) ds] (see e.g. Ch.5 in Shreve: Stoch. Calculus for Finance). Stefan Ankirchner Option Pricing 15
The margining mechanism Futures prices Futures price dynamcis under the risk neutral measure Theorem Consider a futures on an investment asset without additional income. Then the futures price is a martingale wrt the risk neutral measure Q. Proof under the additional assumption that the interest rate is constant. Remark. If interest rates are stochastic, then in general forward prices are not martingales, whereas futures prices are. Stefan Ankirchner Option Pricing 16
Properties Self-financing portfolios Options on Futures Definition. The owner of an option on a futures contract, or simply a futures option, has the right, but not the obligation, to enter a futures contract at a specific future date. A call is the option to buy the futures, and a put is the option to sell it. Option strike price = contract price Examples Treasury bond futures options Crude Oil futures options see CBOT for many further examples... Question: Why options on futures and not on the spot itself? Stefan Ankirchner Option Pricing 17
Properties Self-financing portfolios Futures option value at expiration Notation T = expiration date of the option ( the delivery date of the underlying futures) F t = futures price at t K = strike price Payoff resp. value at expiration of a call: (F T K) + of a put: (K F T ) + Stefan Ankirchner Option Pricing 18
Properties Self-financing portfolios Self-financing futures portfolios Consider a portfolio consisting of ξ futures contracts and η bonds at time t. Let S 0 t = bond price at t V t = portfolio value at t After any margin payment: futures position has value zero. Therefore, V t = ηs 0 t. Let t + δ be the next trading day. By how much does the portfolio value change? Let F = F t+δ F t be the futures price change. Then the margin account is adjusted by ξ F the bond position earns an interest of (e r δ 1)V t rδv t the new portfolio value is V t+δ ξ F + rδv t Stefan Ankirchner Option Pricing 19
Properties Self-financing portfolios Self-financing futures portfolios V t+δ V t ξ F + rδv t Letting δ 0, we get dv t = ξ(t)df t + rv t dt. (4) Equation (4) is the self-financing condition for futures portfolios. Remark. The solution of the SDE (4) is given by t ) V t = e (V rt 0 + ξ(s)e rs df s. 0 Proof. Stefan Ankirchner Option Pricing 20
Properties Self-financing portfolios Put-call parity for futures options Theorem Suppose that a European call on a futures, with strike K and expiration T, is traded at a market price of C. Then the only arbitrage free price for a put with same strike K and expiration T is given by P = C e rt F 0 + e rt K, (5) where F 0 is the current futures price. Proof. Stefan Ankirchner Option Pricing 21
Deriving pricing PDEs Black76 formula Black s model for futures options In a famous model by Black 1 the futures price is assumed to satisfy the dynamics df t = σf t dw t, where W is a BM with respect to the risk-neutral measure Q. On the following slides we derive the pricing PDE for a futures option with payoff h : R R +. The option is a call if h(x) = (x K) +, and a put if h(x) = (K x) +. 1 F. Black. The pricing of Commodity Contracts, Journal of Financial Economics, 3 (1976), 167-179. Stefan Ankirchner Option Pricing 22
Deriving pricing PDEs Black76 formula Letting the 4 steps roll 1) Assume that the futures option is replicable and that its value at time t [0, T ] is equal to v(t, F t ), where v C 1,2. 2) v(t, F t ) is an Ito process with 2 decompositions: From the self-financing condition we have dv(t, F t ) = ξ(t)df t + r v(t, F t )dt and from Ito s formula = ξ(t)σf t dw t + r v(t, F t )dt, dv(t, F t ) = v t (t, F t )dt + v f (t, F t )df t + 1 2 v ff (t, F t )df t df t = v t (t, F t )dt + v f (t, F t )σf t dw t + 1 2 v ff (t, F t )σ 2 F 2 t dt. Stefan Ankirchner Option Pricing 23
4 steps cont d Deriving pricing PDEs Black76 formula 3) Matching the coefficients yields first ξ(t) = v f (t, F t ), and then that v(t, f ) has to solve the PDE v t (t, f ) + 1 2 σ2 f 2 v ff (t, f ) r v(t, f ) = 0, (6) with terminal condition v(t, x) = h(x). Stefan Ankirchner Option Pricing 24
4 steps cont d Deriving pricing PDEs Black76 formula 4) Discounted Feynman-Kac shows that the solution of the PDE (6) is given by v(t, x) = e r(t t) E t,x [h(f T )]. Stefan Ankirchner Option Pricing 25
Deriving pricing PDEs Black76 formula Price of a futures call: the Black76 formula Assume a futures price of F t = f at time t. Then the time t arbitrage free price of a European call futures option, with strike K e and expiration date T, is given by where Black76 call(f, K, T t, σ X, r) = e r(t t) f Φ(d 1 ) e r(t t) KΦ(d 2 ), Proof. d 1 = log ( ) f σ K + X 2 2 (T t) σ, T t d 2 = d 1 σ T t. Stefan Ankirchner Option Pricing 26
Deriving pricing PDEs Black76 formula futures put and link to Black Scholes formula The arbitrage free price of a European put futures option is given by Black76 put(f, K, T t, σ, r) = e r(t t) KΦ( d 2 ) e r(t t) f Φ( d 1 ). Remark. Note that Black76 call(f, K, τ, σ, r) = BS call(fe rτ, K, τ, σ, r). Stefan Ankirchner Option Pricing 27