Derivatives Pricing a Forward / Futures Contract Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles Valuing forward contracts: Key ideas • Two different ways to own a unit of the underlying asset at maturity: – 1.Buy spot (SPOT PRICE: S0) and borrow => Interest and inventory costs – 2. Buy forward (AT FORWARD PRICE F0) • 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 Discount factors and interest rates • Review: Present value of Ct • PV(Ct) = Ct × Discount factor • With annual compounding: • Discount factor = 1 / (1+r)t • With continuous compounding: • Discount factor = 1 / ert = e-rt Derivatives 01 Introduction |3 Forward contract valuation : No income on underlying asset • Example: Gold (provides no income + no storage cost) • Current spot price S0 = $1,340/oz • Interest rate (with continuous compounding) r = 3% • Time until delivery (maturity of forward contract) T = 1 • Forward price F0 ? t = 0 t = 1 Strategy 1: buy forward 0 ST –F0 Strategy 2: buy spot and borrow Should be Buy spot -1,340 + ST equal Borrow +1,340 -1,381 0 ST -1,381 Derivatives 01 Introduction |4 Forward price and value of forward contract rT • Forward price: F0 = S0e • 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 −rT f = S0 − Ke r T • You can check that f = 0 for K = S0 e Derivatives 01 Introduction |5 Arbitrage rT • If F0 ≠ S0 e : arbitrage opportunity rT • Cash and carry arbitrage if: F0 > S0 e • Borrow S0, buy spot and sell forward at forward price F0 rT • Reverse cash and carry arbitrage if S0 e > F0 • Short asset, invest and buy forward at forward price F0 Derivatives 01 Introduction |6 Arbitrage: examples rT • Gold – S0 = 1,750, r = 1.14%, T = 1 S0 e = 1,770 • If forward price =1,800 • Buy spot -1,750 +S1 • Borrow +1,750 -1,770 • Sell forward 0 +1,800 – S1 • Total 0 + 30 • If forward price = 1,700 • Sell spot +1,750 -S1 • Invest -1,750 +1,770 • Buy forward 0 S1 – 1,700 • Total 0 + 70 Derivatives 01 Introduction |7 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 Schematic repo transaction Time t buy bond at Pt deliver bond MARKET TRADER REPO pay Pt DEALER get Pt - haircut Time T sell bond at PT get bond MARKET TRADER REPO get PT DEALER Pay (Pt-Haircut) (1+RepoRate * (T-t)) The repo market Deposit (overnight) Financial Institutional institution investors Non financial firms Collateral Market value (No checking account insured by + haircut FDIC) Huge repo market – exact size unknown 10 February 18, 2013 Spot rates and the yield curve 18/02/13 Derivatives 02 Pricing forwards/futures |11 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 –(0.043)(1) • S0 = 100 e = 95.79 • Step 2: Forward price = future value of current price • I use 6-month spot rate (0.04)(0.50) • F0 = 95.79 e = 97.73 18/02/13 Derivatives 02 Pricing forwards/futures |12 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 A R ≡ forward rate from T to T* • Value of underlying asset: r* -r* T* S0 = A e • Forward price: F = S er T 0 0 * = A e (rT - r* T*) t T T = A e-R(T*-T) r R Ft 18/02/13 Derivatives 02 Pricing forwards/futures |13 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: R(T* - T) A = F0 e • 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 F0 to be paid at time T. • Replacing A and F0 by their values: r *T * − rT R = T * −T In previous example: 4.30%×1− 4%× 0.50 R = = 4.60% 1− 0.50 18/02/13 Derivatives 02 Pricing forwards/futures |14 Forward rate: Example • Current term structure of interest rates (continuous): – 6 months: 4.00% ⇒ d = exp(-0.040 × 0.50) = 0.9802 * – 12 months: 4.30% ⇒ d = exp(-0.043 × 1) = 0.9579 • Consider a 12-month zero coupon with A = 100 • The spot price is S0 = 100 x 0.9579 = 95.79 • The forward price for a 6-month contract would be: – F0 = 97.79 × exp(0.04 × 0.50) = 97.73 • The continuous forward rate is the solution of: • 97.73 e0.50 R = 100 ⇒ R = 4.60% 18/02/13 Derivatives 02 Pricing forwards/futures |15 Forward rate - illustration 18/02/13 Derivatives 02 Pricing forwards/futures |16 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 RS You will repay M(1+RS τ) at maturity • In fact, you sell forward a ZC The face value is M (1+RS τ) The maturity is is T* The delivery price set today is M • The interest will set the value of this contract to zero 18/02/13 Derivatives 02 Pricing forwards/futures |17 Forward borrowing: Gain/loss • At time T* : • Difference between the interest paid RS and the interest on a loan made at the spot interest rate at time T : rs M [ rs- Rs ] τ • At time T: • ΠT = [M ( rs- Rs ) τ ] / (1+rSτ) 18/02/13 Derivatives 02 Pricing forwards/futures |18 FRA (Forward rate agreement) • Example: 3/9 FRA • Buyer pays fixed interest rate Rfra 5% • Seller pays variable interest rate rs 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 18/02/13 Derivatives 02 Pricing forwards/futures |19 FRA: Cash flows 3/9 FRA (buyer) (100 rS 0.50)/(1+rS 0.50) T=0,25 T*=0,75 (100 x 5% x 0.50)/(1+rS 0.50) • General formula:CF = M[(rS - Rfra)τ]/(1+rS τ) • Same as for forward borrowing - long FRA equivalent to cash settlement of result on forward borrowing 18/02/13 Derivatives 02 Pricing forwards/futures |20 Review: Long forward (futures) Synthetic long forward: borrow S0 and buy spot S0 ST Receive underlying asset f0 = 0 0 t T rT Pay forward S0 FSe00= price −−rT rT Value of forward fSFeFFett=−00=() t− contract Derivatives |21 Review Long forward on zero-coupon (1) Face value of zero- coupon given Rτ rT** A = Fe SAe− 0 0 = S T (3) Calculate forward interest rate f0 = 0 * 0 t T τ T rT (2) Calculate forward S0 FSe00= price which set the initial value of the contract equal to 0. −−rT rT fSFeFFett=−00=() t− Derivatives |22 Review Forward investment=buy forward zero- coupon (2) Calculate face value of zero- coupon which set the value of the contract equal to 0 Rτ −rT** AM=+(1 RSτ ) = Fe0 SAe0 = ST f0 = 0 * 0 T τ T (1) Forward price given S0 FM0 = MR(1+SSsττ ) MRr (− ) fSMTT=−=− M= 11++rrssττ Derivatives |23 Review Forward borrowing=sell forward zero- coupon S0 FM0 = f0 = 0 * 0 T τ T ST Rτ ** AM=+(1 Rτ ) = Fe0 −rT S SAe0 = MR(1+SsSττ ) MrR (− ) fMSMTT=−=−= 11++rrssττ Derivatives |24 Review Long Forward Rate Agreement Mrsτ S0 1+ rsτ f0 = 0 * 0 T τ T MR fraτ ** SAe−rT 0 = 1+ rsτ Mr()s− R fra τ fT = 1+ rsτ Derivatives |25 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 Valuing a FRA Mrτ M s +=M 1 r 1+ rsτ + sτ * 0 t T τ T MR fraτ M + ⇔MR(1+fraτ ) 1+ rsτ 1+ rsτ * fMdTtMtfra=×()(1)() −− + Rτ × dTt − Derivatives |27 Basis: definition • DEFINITION : SPOT PRICE - FUTURES PRICE Futures price • bt = St - Ft • Depends on: Spot price • - level of interest rate F = S • - time to maturity (↓ as maturity ↓) T T T time 18/02/13 Derivatives 02 Pricing forwards/futures |28 Extension : Known cash income rT F0 = (S0 – I )e where I is the present value of the income • Ex: forward contract to purchase a coupon-bearing bond 0 0.25 T =0.50 r = 5% S0 = 110.76 C = 6 I = 6 e –(0.05)(0.25) = 5.85 (0.05)(0.50) F0 = (110.76 – 5.85) e = 107.57 18/02/13 Derivatives 02 Pricing forwards/futures |29 Known dividend yield • q : dividend yield p.a.