derivatives options on bonds and interest rates (ii) 11... · 2012-05-03 · 3 may 2012 derivatives...
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Derivatives Options on Bonds and Interest Rates (II)
Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles
3 May 2012
Review
Valuing a (long) forward contract: f = S – PV(K) Forward on zero-coupon: expressed in term of price or of interest rate Payoff at maturity: fT = ST – K <--> fT* = M (R – rT )Δt (short FRA) Payoff >0 or <0 From forward to options (European) Put Call Parity: Call – Put = Forward Options of bonds and interest rates: Call on ZC fT= Max(0, ST – K) <--> Put on IR (floor) fT* = Max(0, M (R – rT ) Δt) Put on ZC fT= Max(0, K – ST) <--> Call on IR (cap) fT* = Max(0, M (rT - R) Δt) Put Call Parity again: Floor – Cap = - FRA <--> FRA + Floor = Cap
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What did we learn so far?
Forward / Futures No need to model price evolution (no arbitrage assumption) Required variables to value forward/futures: Spot price, Yield, Delivery price, Maturity, Interest rate
Options Model for price evolution required In risk neutral world (no arbitrage condition) Additional variable: volatility Black-Merton-Scholes model: dS=rSdt+σSdz Lognormal property: ln ST normally distributed Ito =>Partial Differential Equation (PDE) Solution:
- BS formulas for European options - Numerical procedures (binomial, Monte Carlo, …) for
American or exotic options
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Bond option: where are we?
• To value options on bonds and IR: – Start from model of term structure (not prices)
• Basic requirement: no arbitrage • Fit initial term structure:
– dr Normal: Ho and Lee / Hull and White – dr LogNormal: Black Derman Toy / Black Karinski
• Equilibrium model of term structure – dr Normal: Vasicek
– Use Black
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Ho & Lee
One source of uncertainty: short rate evolution Interest rates normally distributed Many analytical results (not covered in class)
dr =!(t)dt +"dz
Short Rate
r
r
r
rTime
Extension: Hull-White (one factor) dr = !(t)! ar[ ]dt +!dz
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Black-Derman-Toy
d ln r =!(t)dt +"dz
Lognormal version of the Ho & Lee model Advantage: interest rate cannot become negative Disadvantage: no analytical solution
Extension: Black-Karasinski d ln r = !(t)! a ln r[ ]dt +!dz
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Using Derivagem
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Black’s Model
TTXFd σ
σ5.0)/ln(
1 +=
[ ])()( 21 dKNdFNeC rT −= −
[ ])()( 21 dKNdNeSeeC rTqTrT −= −−
But S e-qT erT is the forward price F
This is Black’s Model for pricing options
[ ])()( 21 dKNdFNeP rT −+−−= −
Tdd σ−= 12
The B&S formula for a European call on a stock providing a continuous dividend yield can be written as:
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Example (8th ed. 28.3)
• 1-year cap on 3 month LIBOR • Cap rate = 8% (quarterly compounding) • Principal amount = $10,000 • Maturity 1 1.25 • Spot rate 6.39% 6.50% • Discount factors 0.9381 0.9220 • Yield volatility = 20%
• Payoff at maturity (in 1 year) = • Max{0, [10,000 × (r – 8%)×0.25]/(1+r × 0.25)}
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Example (cont.)
• Step 1 : Calculate 3-month forward in 1 year : • F = [(0.9381/0.9220)-1] × 4 = 7% (with simple compounding)
• Step 2 : Use Black
2851.0)(5677.0120.05.0120.0
)%8%7ln(
11 =⇒−=××+×
= dNd
2213.0)(7677.120.05.05677.02 2 =⇒−=××−−= dNd
Value of cap = 10,000 × 0.9220× [7% × 0.2851 – 8% × 0.2213] × 0.25 = 5.19
cash flow takes place in 1.25 year
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Using DerivaGem
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For a floor :
• N(-d1) = N(0.5677) = 0.7149 N(-d2) = N(0.7677) = 0.7787 • Value of floor = • 10,000 × 0.9220× [ -7% × 0.7149 + 8% × 0.7787] × 0.25 = 28.24 • Put-call parity : FRA + floor = Cap • -23.05 + 28.24 = 5.19 • Reminder : • Short position on a 1-year forward contract • Underlying asset : 1.25 y zero-coupon, face value = 10,200 • Delivery price : 10,000 • FRA = - 10,000 × (1+8% × 0.25) × 0.9220 + 10,000 × 0.9381 • = -23.05 • - Spot price 1.25y zero-coupon + PV(Delivery price)
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Using DerivaGem
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1-year cap on 3-month LIBOR
Cap Principal 100 CapRate 4.50%TimeStep 0.25
Maturity (days) 90 180 270 360Maturity (years) 0.25 0.5 0.75 1Discount function (data) 0.9887 0.9773 0.965759 0.954164IntRate (cont.comp.) 4.55% 4.60% 4.65% 4.69%Forward rate(simp.comp) 4.67% 4.77% 4.86%
Cap = call on interest rateMaturity 0.25 0.50 0.75Volatility dr/r (data) 0.215 0.211 0.206d1 0.4063 0.4630 0.5215N(d1) 0.6577 0.6783 0.6990d2 0.2988 0.3138 0.3431N(d2) 0.6175 0.6232 0.6342Value of caplet 0.3058 0.0722 0.1039 0.1297Delta 49.1211 16.0699 16.3773 16.6739
Floor = put on interest rateN(-d1) 0.3423 0.3217 0.3010N(-d2) 0.3825 0.3768 0.3658Value of floor 0.1124 0.0298 0.0391 0.0436Delta 23.3087 8.3619 7.7667 7.1802
Put-call parity for caps and floorsFRA 0.1934 0.0425 0.0648 0.0861+floor 0.1124 0.0298 0.0391 0.0436=cap 0.3058 0.0722 0.1039 0.1297
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Using DerivaGem
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Using bond prices
• In previous development, bond yield is lognormal. • Volatility is a yield volatility. • σy = Standard deviation (Δy/y) • We now want to value an IR option as an option on a zero-coupon:
• For a cap: a put option on a zero-coupon • For a floor: a call option on a zero-coupon
• We will use Black’s model. • Underlying assumption: bond forward price is lognormal • To use the model, we need to have:
• The bond forward price • The volatility of the forward price
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From yield volatility to price volatility
• Remember the relationship between changes in bond’s price and yield:
yyDyyD
SS Δ
−=Δ−=Δ
D is modified duration
This leads to an approximation for the price volatility:
yDyσσ =
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Back to previous example (Hull 8th ed. 28.3)
1-year cap on 3 month LIBOR Cap rate = 8% Principal amount = 10,000 Maturity 1 1.25 Spot rate 6.39% 6.50% Discount factors 0.9381 0.9220 Yield volatility = 20%
1-year put on a 1.25 year zero-coupon
Face value = 10,200 [10,000 (1+8% * 0.25)]
Striking price = 10,000
Spot price of zero-coupon = 10,200 * .9220 = 9,404
1-year forward price = 9,404 / 0.9381 = 10,025
3-month forward rate in 1 year = 6.94%
Price volatility = (20%) * (6.94%) * (0.25) = 0.35%
Using Black’s model with:
F = 10,025 K = 10,000 r = 6.39% T = 1 σ = 0.35%
Call (floor) = 27.631 Delta = 0.761
Put (cap) = 4.607 Delta = - 0.239
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Using DerivaGem
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Valuing IR Derivatives: beyond Black
• Black’s model is concerned with describing the probability distribution of a single variable at a single point in time
• A term structure model describes the evolution of the whole yield curve • 2 approaches (cf Hull 7th ed. Chap 30):
– Equilibrium models: Vasicek 1977 • Term structure = f(Factors) • In equilibrium models, today’s term structure is an output
– No-arbitrage: Ho-Lee 1986 • Binomial evolution of whole term structure • In a no-arbitrage models, today’s term structure is an input
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Vasicek (1977)
• Derives the first equilibrium term structure model. • 1 state variable: short term spot rate r • Changes of the whole term structure driven by one single interest rate • Assumptions:
1. Perfect capital market 2. Price of riskless discount bond maturing in t years is a function of
the spot rate r and time to maturity t: P(r,t) 3. Short rate r(t) follows diffusion process in continuous time:
dr = a (b-r) dt + σ dz
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The stochastic process for the short rate
• Vasicek uses an Ornstein-Uhlenbeck process dr = a (b – r) dt + σ dz
• a: speed of adjustment • b: long term mean • σ : standard deviation of short rate
• Change in rate dr is a normal random variable • The drift is a(b-r): the short rate tends to revert to its long term mean
• r>b ⇒ b – r < 0 interest rate r tends to decrease • r<b ⇒ b – r > 0 interest rate r tends to increase
• Variance of spot rate changes is constant
• Example: Chan, Karolyi, Longstaff, Sanders The Journal of Finance, July 1992
• Estimates of a, b and σ based on following regression: rt+1 – rt = α + β rt +εt+1
a = 0.18, b = 8.6%, σ = 2%
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Pricing a zero-coupon
• Using Ito’s lemna, the price of a zero-coupon should satisfy a stochastic differential equation:
dP = m P dt + s P dz • This means that the future price of a zero-coupon is lognormal. • Using a no arbitrage argument “à la Black Scholes” (the expected return of
a riskless portfolio is equal to the risk free rate), Vasicek obtain a closed form solution for the price of a t-year unit zero-coupon:
• P(r,t) = e-y(r,t) * t • with y(r,t) = A(t)/t + [B(t)/t] r0
• For formulas: see Hull 4th ed. Chap 21.
• Once a, b and σ are known, the entire term structure can be determined.
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Vasicek: example
• Suppose r = 3% and dr = 0.20 (6% - r) dt + 1% dz • Consider a 5-year zero coupon with face value = 100 • Using Vasicek:
• A(5) = 0.1093, B(5) = 3.1606 • y(5) = (0.1093 + 3.1606 * 0.03)/5 = 4.08% • P(5) = e- 0.0408 * 5 = 81.53
• The whole term structure can be derived: • Maturity Yield Discount factor • 1 3.28% 0.9677 • 2 3.52% 0.9320 • 3 3.73% 0.8940 • 4 3.92% 0.8549 • 5 4.08% 0.8153 • 6 4.23% 0.7760 • 7 4.35% 0.7373 0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0
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Jamshidian (1989)
• Based on Vasicek, Jamshidian derives closed form solution for European calls and puts on a zero-coupon.
• The formulas are the Black’s formula except that the time adjusted volatility σ√T is replaced by a more complicate expression for the time adjusted volatility of the forward price at time T of a T*-year zero-coupon
[ ]aee
a
aTTTa
P 211 )*(
−−− −
−=σ
σ