Derivatives Options on Bonds and Interest Rates · 1 May 2011 Derivatives 10 Options on bonds and IR |6 Payoff of a call option on a zero-coupon • The exercise rate of the call

Derivatives Options on Bonds and Interest Rates

Professor André Farber Solvay Business School Université Libre de Bruxelles

1 May 2011 Derivatives 10 Options on bonds and IR |2

•  Caps •  Floors •  Swaption •  Options on IR futures •  Options on Government bond futures


Introduction

•  A difficult but important topic: •  Black-Scholes collapses:

1. Volatility of underlying asset constant 2. Interest rate constant

•  For bonds: –  1. Volatility decreases with time –  2. Uncertainty due to changes in interest rates –  3. Source of uncertainty: term structure of interest rates

•  3 approaches: 1. Stick of Black-Scholes 2. Model term structure : interest rate models 3. Start from current term structure: arbitrage-free models


Review: forward on zero-coupons

•  Borrowing forward ↔ Selling forward a zero-coupon

•  Long FRA:

0 T T* τ

+M

-M(1+Rτ)

)1

)(ττ

rRrM

+

−


Options on zero-coupons

•  Consider a 6-month call option on a 9-month zero-coupon with face value 100

•  Current spot price of zero-coupon = 98.14 •  Exercise price of call option = 99 •  Payoff at maturity: Max(0, ST – 99) •  The spot price of zero-coupon at the maturity of the option depend on the

3-month interest rate prevailing at that date. •  ST = 100 / (1 + rT 0.25) •  Exercise option if:

•  ST > 99 •  rT < 4.04%


Payoff of a call option on a zero-coupon

•  The exercise rate of the call option is R = 4.04% •  With a little bit of algebra, the payoff of the option can be written as:

•  Interpretation: the payoff of an interest rate put option •  The owner of an IR put option:

•  Receives the difference (if positive) between a fixed rate and a variable rate

•  Calculated on a notional amount •  For an fixed length of time •  At the beginning of the IR period

)25.01

25.0)%04.4(99,0(T

T

rrMax

+−


European options on interest rates

•  Options on zero-coupons •  Face value: M(1+Rτ) •  Exercise price K A call option •  Payoff:

Max(0, ST – K) A put option •  Payoff:

Max(0, K – ST )

•  Option on interest rate

•  Exercise rate R A put option •  Payoff:

Max[0, M (R-rT)τ / (1+rTτ)]

A call option •  Payoff:

Max[0, M (rT -R)τ / (1+rTτ)]


Cap

•  A cap is a collection of call options on interest rates (caplets). •  The cash flow for each caplet at time t is:

Max[0, M (rt – R) τ] •  M is the principal amount of the cap •  R is the cap rate •  rt is the reference variable interest rate •  τ is the tenor of the cap (the time period between payments)

•  Used for hedging purpose by companies borrowing at variable rate •  If rate rt < R : CF from borrowing = – M rt τ •  If rate rt > R: CF from borrowing = – M rT τ + M (rt – R) τ = – M R τ


Floor

•  A floor is a collection of put options on interest rates (floorlets). •  The cash flow for each floorlet at time t is:

Max[0, M (R –rt) τ] •  M is the principal amount of the cap •  R is the cap rate •  rt is the reference variable interest rate •  τ is the tenor of the cap (the time period between payments)

•  Used for hedging purpose buy companies borrowing at variable rate •  If rate rt < R : CF from borrowing = – M rt τ •  If rate rT > R: CF from borrowing = – M rT τ + M (rt – R) τ = – M R τ


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:


Example (Hull 5th ed. 22.3 – 6th ed. 26.3 – 7th 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)}


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

1 May 2011

Using DerivaGem

Derivatives 10 Options on bonds and IR |13


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)

1 May 2011

Using DerivaGem



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

1 May 2011

Using DerivaGem



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


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 =


Back to previous example (Hull 7h 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

1 May 2011

Using DerivaGem



Interest rate model

•  The source of risk for all bonds is the same: the evolution of interest rates. Why not start from a model of the stochastic evolution of the term structure?

•  Excellent idea •  ……. difficult to implement •  Need to model the evolution of the whole term structure! •  But change in interest of various maturities are highly correlated. •  This suggest that their evolution is driven by a small number of underlying

factors.

1 May 2011

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


1 May 2011

Modelling the term structure evolution

•  Modelling the evolution of the term structure is complex. •  Assuming that all IR might change by the same amount

(parallel shift) would suppose there exist arbitrage opporunities.

•  To see this, suppose that the term structure is flat. •  The value of a (T-t)-year zero-coupon is:

•  where r is the interest rate (the same of all zero-coupons)

)(100),( tTrT etrP −−×=

1 May 2011

Zero duration portfolio

•  Suppose r=3% •  Create a portfolio with zero-duration:

–  Long n t1 -year zero-coupon bond (Example: Long n 10-year zc) –  Short 1 t2-year zero-coupon (Example: Short 1 2-year zc)

•  In order to have zero-duration: 11

22

PtPtn =

Interest rate 3%Bonds 1 2 Portfolio with zero-‐durationMaturity 10 2 # units ValuePrice 74.08 94.18 Bond 1 0.25 18.84Duration 10 2 Bond 2 -‐1.00 -‐94.18Convexity 100 4 -‐75.34

1 May 2011

Impact of convexity

Conclusion: No arbitrage implies more complex evolution of the term structure

1 May 2011

No-arbitrage in continuous time The no-arbitrage condition can be demonstrated in a continuous time setting

021 =+−= CashPnPV

dzrdtradr )()( σ+=Stochastic process for short rate:

Pricing equation for zero-coupon: )(),( tTretrP −−=

Long-short portfolio:

Zero duration: 11

22

)()(0PtTPtTn

rV

−−

=→=∂∂

0²)(21

2122 =−= dtTTPTdV σBy Ito:

This portfolio has no risk and no investment. It should earn zero dV = 0

This equation holds only if T1 = T2. Conclusion: the bond valuation model implies that arbitrage is possible.

1 May 2011

dtTTPT

dtPtTPtTPtTPtT

dtrCashPtTPtTnrPnrPdV

²)(21

²))²(21²)²(

)()(

21(

²)²(21²)²(

21

2122

221111

22

221121

σ

σσ

σσ

−=

−−−−−

=

⎥⎦

⎤⎢⎣

⎡ +−−−+−=

dzrVdt

rVa

rV

tVdV σσ

∂∂

+∂∂

+∂∂

+∂∂

= ²)²²

21(

11

22

)()(0PtTPtTn

rV

−−

=→=∂∂As:

Details of calculation

1 May 2011

Monte Carlo experiment # period/year 250dt 0.004Drift a 1.00%sigma 3.00%Short rate r 3%

Maturity Yield Price Quantity ValueBond 1 10.00 3.00% 74.08 0.25 18.84Bond 2 2.00 3.00% 94.18 -‐1.00 -‐94.18Cash 75.34Total V 0.00

dr -‐0.17% =a*dt+NORMSINV(RAND())*sigma*SQRT(dt)r+dr 2.83%

Maturity Yield Price Quantity Value DeltaBond 1 t1-‐dt 10.00 2.83% 75.38 0.25 19.16 0.33Bond 2 t2-‐dt 2.00 2.83% 94.51 -‐1.00 -‐94.51 -‐0.34Cash 75.35 0.01Total V+dV 0.0028 0.0028

1 May 2011


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


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%


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.


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


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

1 May 2011

Binomial no-arbitrage model


Suppose that we observe the following term structure:

Expected binomial evolution of the 6-month rate:

Underlying assumptions : Expected 6-month rate = 2% Standard deviation (per annum) = 1% Ru = 2% + 1% * sqrt(0.5) Rd=2% - 1% * sqrt(0.5) More on this later

1 May 2011

Option valuation

•  How to value a 6-month call option on a 1-year ZC with a strike price K=99?

•  Build a binomial tree for the bond price:


Prices calculated using the 6-month rate prevailing at time t = 0.5

This is the market price!

1 May 2011

Market price of interest rate risk


= ln( 99.0198.02

) / 0.5

= 2.00%!1.50%

= 99.01! e"1.50%!0.50 " 98.02

Shape of term structure determined by 2 forces: -  Expected future spot rate -  Risk aversion (required risk premium)

1 May 2011

Different possible views of current term structure


Small IR increase High risk premium

High IR increase Smal risk premium

1 May 2011

Back to 1-period binomial pricing


1 May 2011

Multiperiod interes rate tree


ri, j !ri+1, j = ri, j +"r +! "t

ri+1, j+1 = ri, j +"r #! "t

i is the number of period j is the number of downward movement in interest rate

1 May 2011

Pricing a 3-period zero-coupon


Suppose risk-neutral probability constant

Problem: Market price different from price calculated by model

1 May 2011

Fitting the term structure


One solution is to let the risk neutral probability vary. Using Goalseek, let Excel determine the RNProba to use in period 1 that yield the market price.

1 May 2011

Valuing a 1-year cap on 6-month rate


= (e3.91%!0.5 "1)!2 = 3.95%

=100!Max(0,3.95%" 2%)!0.5! e"3.91%!0.5

6-month rate with simple compounding

Payoff at maturity

Protection against an upward movement of 6-month rate.

1 May 2011

Swaption


A 6-month swaption on a 1-year swap Option maturity: 6 month Swap maturity: 1 year (at option maturity) Swap rate: 1.25% per period (1 period = 6 month)

Remember: Swap = Floating rate note - Fix rate note Swaption = put option on a coupon bond

Bond maturity: 1.5 year Coupon: 2.5% (semi-annual) Option maturity: 6-month Strike price = 100


Valuation formula

•  The value of any bond or derivative in this model is obtained by discounting the expected future value (in a risk neutral world). The discount rate is the current short rate.

trijiijii

ij jiecouponfpfp

f Δ++++ +−+

=,

11,1,1 )1(

i is the number of period j is the number of “downs” Δt is the time step

1 May 2011

Valuing the swaption


1 May 2011

Models with constant risk-neutral probability


Adjusting RN probabilities as we did before to fit the term structure does not guarantee that all RN probabilities are between zero and one. To avoid this problem, the industry uses risk neutral tree without reference to the true interest tree. Two popular models in which RN proba = ½:

- Ho-Lee model - Simple Black, Derman and Toy (BDT) model

1 May 2011

Ho-Lee model


ri+1, j = ri, j +!i !"t +" ! "t

ri+1, j+1 = ri, j +!i !"t #" ! "t

The thetas are parameters used to fit the term structure.

1 May 2011

Ho-Lee illustration


1 May 2011

Ho-Lee illustration (2)


1 May 2011

Black Derman Toy


zi+1, j = zi, j +!i !"t +" ! "t

zi+1, j+1 = zi, j +!i !"t #" ! "t

ri, j = ezi, j

1 May 2011

BDT illustration


Documents

Derivatives Options on Bonds and Interest Rates · 1 May 2011 Derivatives 10 Options on bonds and IR |6 Payoff of a call option on a zero-coupon • The exercise rate of the call