Derivatives Pricing a Forward / Futures Contract

Transcription

1 Derivatives Pricing a Forward / Futures Contract Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles

2 Valuing forward contracts: Key ideas Two different ways to own a unit of the underlying asset at maturity: 1.Buy spot (SPOT PRICE: S 0 ) and borrow => Interest and inventory costs 2. Buy forward (AT FORWARD PRICE F 0 ) VALUATION PRINCIPLE: NO ARBITRAGE In perfect markets, no free lunch: the 2 methods should cost the same. You can think of a derivative as a mixture of its constituent underliers, much as a cake is a mixture of eggs, flour and milk in carefully specified proportions. The derivative s model provide a recipe for the mixture, one whose ingredients quantity vary with time. Emanuel Derman, Market and models, Risk July 2001 Derivatives 01 Introduction 2

3 Discount factors and interest rates Review: Present value of C t PV(C t ) = C t Discount factor With annual compounding: Discount factor = 1 / (1+r) t With continuous compounding: Discount factor = 1 / e rt = e -rt Derivatives 01 Introduction 3

4 Forward contract valuation : No income on underlying asset Example: Gold (provides no income + no storage cost) Current spot price S 0 = $1,340/oz Interest rate (with continuous compounding) r = 3% Time until delivery (maturity of forward contract) T = 1 Forward price F 0? Strategy 1: buy forward t = 0 t = 1 0 S T F 0 Strategy 2: buy spot and borrow Buy spot -1,340 + S T Borrow +1,340-1,381 Should be equal 0 S T -1,381 Derivatives 01 Introduction 4

5 Forward price and value of forward contract Forward price: F S 0 = 0 e rt Remember: the forward price is the delivery price which sets the value of a forward contract equal to zero. Value of forward contract with delivery price K f = S 0 Ke rt You can check that f = 0 for K = S 0 e r T Derivatives 01 Introduction 5

6 Arbitrage If F 0 S 0 e rt : arbitrage opportunity Cash and carry arbitrage if: F 0 > S 0 e rt Borrow S 0, buy spot and sell forward at forward price F 0 Reverse cash and carry arbitrage if S 0 e rt > F 0 Short asset, invest and buy forward at forward price F 0 Derivatives 01 Introduction 6

7 Arbitrage: examples Gold S 0 = 1,750, r = 1.14%, T = 1 S 0 e rt = 1,770 If forward price =1,800 Buy spot -1,750 +S 1 Borrow +1,750-1,770 Sell forward 0 +1,800 S 1 Total If forward price = 1,700 Sell spot +1,750 -S 1 Invest -1,750 +1,770 Buy forward 0 S 1 1,700 Total Derivatives 01 Introduction 7

8 Repurchase agreement (repo) An important kind of forward contract. Sale of a security + agreement to buy it back at a fixed price. (=reverse cash-and-carry, sale coupled with long forward) Underlying security held as a collateral Most repo are overnight Counterparty risk => haircut = Value of collateral - Loan

9 Schematic repo transaction Time t buy bond at Pt deliver bond MARKET pay Pt TRADER get Pt - haircut REPO DEALER Time T sell bond at PT get bond MARKET get PT TRADER REPO DEALER Pay (Pt-Haircut) (1+RepoRate * (T-t))

10 The repo market Financial institution Deposit (overnight) Collateral Market value + haircut Institutional investors Non financial firms (No checking account insured by FDIC) Huge repo market exact size unknown 10 February 4, 2013

11 Spot rates and the yield curve 4/02/13 Derivatives 02 Pricing forwards/futures 11

12 Forward on a zero-coupon : example Consider a: 6- month forward contract on a 1- year zero-coupon with face value A = 100 Current interest rates (with continuous compounding) 6-month spot rate: 4.00% 1-year spot rate: 4.30% Step 1: Calculate current price of the 1-year zero-coupon I use 1-year spot rate S 0 = 100 e (0.043)(1) = Step 2: Forward price = future value of current price I use 6-month spot rate F 0 = e (0.04)(0.50) = /02/13 Derivatives 02 Pricing forwards/futures 12

13 Forward on zero coupon Notations (continuous rates): A face value of ZC with maturity T * r * interest rate from 0 to T * r interest rate from 0 to T R forward rate from T to T * Value of underlying asset: S 0 = A e-r* T* Forward price: F 0 = S 0 e r T = A e = A e -R(T*-T) (rt - r* T*) t T T * r r * R A F t 4/02/13 Derivatives 02 Pricing forwards/futures 13

14 Forward interest rate Rate R set at time 0 for a transaction (borrowing or lending) from T to T * With continuous compounding, R is the solution of: A = F 0 er(t* - T) The forward interest rate R is the interest rate that you earn from T to T* if you buy forward the zero-coupon with face value A for a forward price F 0 to be paid at time T. Replacing A and F 0 by their values: R r * T = * T * rt T In previous example: R = 4.30% 1 4% = 4.60% 4/02/13 Derivatives 02 Pricing forwards/futures 14

15 Forward rate: Example Current term structure of interest rates (continuous): 6 months: 4.00% d = exp( ) = months: 4.30% d * = exp( ) = Consider a 12-month zero coupon with A = 100 The spot price is S 0 = 100 x = The forward price for a 6-month contract would be: F 0 = exp( ) = The continuous forward rate is the solution of: e 0.50 R = 100 R = 4.60% 4/02/13 Derivatives 02 Pricing forwards/futures 15

16 Forward rate - illustration 4/02/13 Derivatives 02 Pricing forwards/futures 16

17 Forward borrowing View forward borrowing as a forward contract on a ZC You plan to borrow M for τ years from T to T * Define: τ = T * - T The simple interest rate set today is R S You will repay M(1+R S τ) at maturity In fact, you sell forward a ZC The face value is M (1+R S τ) The maturity is is T * The delivery price set today is M The interest will set the value of this contract to zero 4/02/13 Derivatives 02 Pricing forwards/futures 17

18 Forward borrowing: Gain/loss At time T * : Difference between the interest paid R S and the interest on a loan made at the spot interest rate at time T : r s M [ r s - R s ] τ At time T: Π T = [M ( r s - R s ) τ ] / (1+r S τ) 4/02/13 Derivatives 02 Pricing forwards/futures 18

19 FRA (Forward rate agreement) Example: 3/9 FRA Buyer pays fixed interest rate R fra 5% Seller pays variable interest rate r s 6-m LIBOR on notional amount M $ 100 m for a given time period (contract period) τ 6 months at a future date (settlement date or reference date) T ( in 3 months, end of accrual period) Cash flow for buyer (long) at time T: Inflow (100 x LIBOR x 6/12)/(1 + LIBOR x 6/12) Outflow (100 x 5% x 6/12)/(1 + LIBOR x 6/12) Cash settlement of the difference between present values 4/02/13 Derivatives 02 Pricing forwards/futures 19

20 FRA: Cash flows 3/9 FRA (buyer) (100 r S 0.50)/(1+r S 0.50) T=0,25 T*=0,75 General formula:cf = M[(r S - R fra )τ]/(1+r S τ) (100 x 5% x 0.50)/(1+r S 0.50) Same as for forward borrowing - long FRA equivalent to cash settlement of result on forward borrowing 4/02/13 Derivatives 02 Pricing forwards/futures 20

21 Review: Long forward (futures) Synthetic long forward: borrow S 0 and buy spot S 0 S T Receive underlying asset f 0 = 0 0 t T S 0 F = S e 0 0 rt Pay forward price f = S Fe = ( F F ) e rt t t 0 t 0 rt Value of forward contract Derivatives 21

22 Review Long forward on zero-coupon (1) Face value of zerocoupon given f 0 = 0 * * S = Ae rt 0 S T 0 t T τ A = T R Fe τ 0 (3) Calculate forward interest rate * S 0 F = S e 0 0 rt t t 0 t 0 rt f = S Fe = ( F F ) e rt (2) Calculate forward price which set the initial value of the contract equal to 0. Derivatives 22

23 * * S = Ae rt 0 Review Forward investment=buy forward zerocoupon S T (2) Calculate face value of zerocoupon which set the value of the contract equal to 0 A= M(1 + Rτ ) S = R Fe τ 0 f 0 = 0 S 0 0 T F0 = M τ T (1) Forward price given * M(1 + RSτ) M( RS rs) τ ft = ST M = M = 1+ rτ 1+ rτ s s Derivatives 23

24 Review Forward borrowing=sell forward zerocoupon S 0 F0 = M f 0 = 0 0 T τ T * 0 * * S = Ae rt S T A= M(1 + Rτ ) M(1 + RSτ) M( rs RS) τ ft = M ST = M = 1+ rτ 1+ rτ s S Derivatives 24 s = R Fe τ 0

25 Review Long Forward Rate Agreement S 0 Mrτ s 1+ rτ s f 0 = 0 0 T τ T * 0 * * S = Ae rt MR fra 1+ rτ s τ f T = M( r R ) τ s 1+ rτ s fra Derivatives 25

26 To lock in future interest rate Borrow short & invest long Buy forward zero coupon Invest forward at current forward interest rate Sell FRA and invest at future (unknown) spot interest rate Derivatives 26

27 Valuing a FRA Mrτ s 1+ rτ s M + = rτ 1+ s M 0 t T τ T * MR fraτ M + 1+ rτ 1 + s rτ s M(1 + R τ ) fra f = M d T t M + R d T t t * ( ) (1 fraτ ) ( ) Derivatives 27

28 Basis: definition DEFINITION : SPOT PRICE - FUTURES PRICE Futures price b t = S t - F t Depends on: - level of interest rate - time to maturity ( as maturity ) Spot price F T = S T T time 4/02/13 Derivatives 02 Pricing forwards/futures 28

29 Extension : Known cash income F 0 = (S 0 I )e rt where I is the present value of the income Ex: forward contract to purchase a coupon-bearing bond T =0.50 r = 5% S 0 = C = 6 I = 6 e (0.05)(0.25) = 5.85 F 0 = ( ) e (0.05)(0.50) = /02/13 Derivatives 02 Pricing forwards/futures 29

30 Known dividend yield q : dividend yield p.a. paid continuously F 0 = [ e -qt S 0 ] e rt = S 0 e (r-q)t Examples: Forward contract on a Stock Index r = interest rate q = dividend yield Foreign exchange forward contract: r = domestic interest rate (continuously compounded) q = foreign interest rate (continuously compounded) 4/02/13 Derivatives 02 Pricing forwards/futures 30

31 Interest rate parity Underlying asset: one unit of foreign currency $1 r $ : foreign interest rate (2%) $ $ e rt 2% 0.5 $ e = F 0 forward exchange rate /$ Fe 0 rt $ Fe = Se rt 0 0 rt $ Spot exchange rate /$ S Time 0 Se r : domestic interest rate (4%) Time T 0 rt 4% e = F = S e 0 0 ( r r ) T $ Derivatives 31

32 Commodities I = - PV of storage cost (negative income) q = - storage cost per annum as a proportion of commodity price The cost of carry: Interest costs + Storage cost income earned c=r-q For consumption assets, short sales problematic. So: F ( r+ u) T The convenience yield on a consumption asset y defined so that: F 0 0 S = S 0 0 e e ( c y) T 4/02/13 Derivatives 02 Pricing forwards/futures 32

33 Summary Value of forward contract Forward price No income Known income I =PV(Income) f f = S Ke = ( S I) rt rt F = Se F=(S I)e rt Known yield q f =S e -qt K e -rt F = S e (r-q)t Commodities f=se (u-y)t - Ke -rt F=Se (r+u-y)t Ke rt 4/02/13 Derivatives 02 Pricing forwards/futures 33

34 Valuation of futures contracts If the interest rate is non stochastic, futures prices and forward prices are identical NOT INTUITIVELY OBVIOUS: è Total gain or loss equal for forward and futures è but timing is different Forward : at maturity Futures : daily 4/02/13 Derivatives 02 Pricing forwards/futures 34

35 Forward price & expected future price Is F 0 an unbiased estimate of E(S T )? F < E(S T ) Normal backwardation F > E(S T ) Contango F = E(S T ) e (r-k) T If k = r F = E(S T ) If k > r F < E(S T ) If k < r F > E(S T ) r = risk-free rate k = expected return = risk-free + risk premium CAPM: k = r + β(k Market r) 4/02/13 Derivatives 02 Pricing forwards/futures 35

36 Does the forward price predict the future price? Consider a non dividend paying stock. S 0 = E 0 (S 1 ) 1+ E(R) E 0(S 1 ) = S 0 (1+ E(R)) Forward price: F 0 = S 0 (1+ r F ) E 0 (S 1 ) F 0 = S 0 Forward price bias = [ E(R) r F ] Risk premium on underlying asset This is also the expected cash flow on the derivative 4/02/13 Derivatives 03 Hedging with Futures 36

37 E(S 1 F 0 ) = E(S 1 ) F 0 = S 0 (1+ k) S 0 (1+ r) = S 0 (k r) 4/02/13 Derivatives 02 Pricing forwards/futures 37

38 4/02/13 Derivatives 02 Pricing forwards/futures 38