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. paid continuously -qT rT (r-q)T F0 = [ e S0 ] e = S0 e
• 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)
18/02/13 Derivatives 02 Pricing forwards/futures |30 Interest rate parity
r$: foreign interest rate (2%) Underlying rT asset: one unit $1 $e $ of foreign currency $1.0101e2%× 0.5 =
F0 forward exchange rate €/$
rT$ rT$ rT€ €Fe0 Fe00= Se
FSe= ()rrT€− $ rT 00 Spot exchange € €S €Se0 rate €/$ 0 €0.70 €0.70×e4%× 0.50 = 0.7141 Time T Time 0 r€: domestic interest rate (4%)
Derivatives |31 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: (r+u)T F0 ≤ S0e
• The convenience yield on a consumption asset y defined so that: (c−y)T F0 = S0e
18/02/13 Derivatives 02 Pricing forwards/futures |32 Summary
Value of forward contract Forward price
No income −rT f = S − Ke F = Se rT Known income f (S I) Ke−rT F=(S – I)erT I =PV(Income) = − − 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
18/02/13 Derivatives 02 Pricing forwards/futures |33 Valuation of futures contracts
• If the interest rate is non stochastic, futures prices and forward prices are identical • NOT INTUITIVELY OBVIOUS: – èTotal gain or loss equal for forward and futures – èbut timing is different • Forward : at maturity • Futures : daily
18/02/13 Derivatives 02 Pricing forwards/futures |34 Forward price & expected future price
• Is F0 an unbiased estimate of E(ST) ?
• F < E(ST) Normal backwardation
• F > E(ST) Contango r = risk-free rate (r-k) T k = expected return F = E(ST) e = risk-free + risk premium
• If k = r F = E(ST) CAPM:
• If k > r F < E(ST) k = r + β(kMarket − r) • If k < r F > E(ST)
18/02/13 Derivatives 02 Pricing forwards/futures |35 Does the forward price predict the future price?
Consider a non dividend paying stock.
E (S ) S = 0 1 ⇔ E (S ) = S (1+ E(R)) 0 1+ E(R) 0 1 0
Forward price: F0 = S0 (1+ rF )
E0 (S1)− F0 = S0 [E(R)− rF ]
Risk premium on Forward price bias = underlying asset This is also the expected cash flow on the derivative
18/02/13 Derivatives 03 Hedging with Futures |36 E(S1 − F0 ) = E(S1)− F0
= S0 (1+ k)− S0 (1+ r)
= S0 (k − r)
18/02/13 Derivatives 02 Pricing forwards/futures |37 18/02/13 Derivatives 02 Pricing forwards/futures |38