Derivatives Options on Bonds and Interest Rates. Professor André Farber Solvay Business School Université Libre de Bruxelles

Transcription

1 Derivatives Options on Bonds and Interest Rates Professor André Farber Solvay Business School Université Libre de Bruxelles

2 Caps Floors Swaption Options on IR futures Options on Government bond futures Derivatives 0 Options on bonds and IR 2

3 Introduction A difficult but important topic: Black-Scholes collapses:. Volatility of underlying asset constant 2. Interest rate constant For bonds:. Volatility decreases with time 2. Uncertainty due to changes in interest rates 3. Source of uncertainty: term structure of interest rates 3 approaches:. Stick of Black-Scholes 2. Model term structure : interest rate models 3. Start from current term structure: arbitrage-free models Derivatives 0 Options on bonds and IR 3

4 Review: forward on zero-coupons +M 0 T T* τ -M(+Rτ) Borrowing forward Selling forward a zero-coupon Long FRA: ( r R) τ M ) + rτ Derivatives 0 Options on bonds and IR 4

5 Options on zero-coupons Consider a 6-month call option on a 9-month zero-coupon with face value 00 Current spot price of zero-coupon = 98.4 Exercise price of call option = 99 Payoff at maturity: Max(0, S T 99) The spot price of zero-coupon at the maturity of the option depend on the 3-month interest rate prevailing at that date. S T = 00 / ( + r T 0.25) Exercise option if: S T > 99 r T < 4.04% Derivatives 0 Options on bonds and IR 5

6 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: 99(4.04% rt )0.25 Max(0, ) + r 0.25 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 T Derivatives 0 Options on bonds and IR 6

7 European options on interest rates Options on zero-coupons Face value: M(+Rτ) Exercise price K Option on interest rate Exercise rate R A call option Payoff: Max(0, S T K) A put option Payoff: Max[0, M (R-r T )τ / (+r T τ)] A put option Payoff: Max(0, K S T ) A call option Payoff: Max[0, M (r T -R)τ / (+r T τ)] Derivatives 0 Options on bonds and IR 7

8 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 (r t R) τ] M is the principal amount of the cap R is the cap rate r t 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 r t < R : CF from borrowing = M r t τ If rate r t > R: CF from borrowing = M r T τ + M (r t R) τ = M R τ Derivatives 0 Options on bonds and IR 8

9 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 r t ) τ] M is the principal amount of the cap R is the cap rate r t 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 r t < R : CF from borrowing = M r t τ If rate r T > R: CF from borrowing = M r T τ + M (r t R) τ = M R τ Derivatives 0 Options on bonds and IR 9

10 Black s Model The B&S formula for a European call on a stock providing a continuous dividend yield can be written as: [ Se e N d ) KN( )] qt rt rt C = e ( d2 But S e -qt e rt is the forward price F C This is Black s Model for pricing options = e rt [ FN d ) KN( )] ( d2 P = e rt [ FN d ) + KN( )] ( d2 d ln( F σ / X ) = + T 0.5σ T d 2 = d σ T Derivatives 0 Options on bonds and IR 0

11 Example (Hull 5th ed th ed th ed. 28.3) -year cap on 3 month LIBOR Cap rate = 8% (quarterly compounding) Principal amount = $0,000 Maturity.25 Spot rate 6.39% 6.50% Discount factors Yield volatility = 20% Payoff at maturity (in year) = Max{0, [0,000 (r 8%) 0.25]/(+r 0.25)} Derivatives 0 Options on bonds and IR

12 Example (cont.) Step : Calculate 3-month forward in year : F = [(0.938/0.9220)-] 4 = 7% (with simple compounding) Step 2 : Use Black 7% ln( ) d 8% = = N( d) = d 2 = =.7677 N ( d 2 ) = Value of cap = 0, [7% % 0.223] 0.25 = 5.9 cash flow takes place in.25 year Derivatives 0 Options on bonds and IR 2

13 Using DerivaGem Derivatives 0 Options on bonds and IR 3

14 For a floor : N(-d ) = N(0.5677) = N(-d 2 ) = N(0.7677) = Value of floor = 0, [ -7% % ] 0.25 = Put-call parity : FRA + floor = Cap = 5.9 Reminder : Short position on a -year forward contract Underlying asset :.25 y zero-coupon, face value = 0,200 Delivery price : 0,000 FRA = - 0,000 (+8% 0.25) , = Spot price.25y zero-coupon + PV(Delivery price) Derivatives 0 Options on bonds and IR 4

15 Using DerivaGem Derivatives 0 Options on bonds and IR 5

16 -year cap on 3-month LIBOR Cap Principal 00 CapRate 4.50% TimeStep 0.25 Maturity (days) Maturity (years) Discount function (data) IntRate (cont.comp.) 4.55% 4.60% 4.65% 4.69% Forward rate(simp.comp) 4.67% 4.77% 4.86% Cap = call on interest rate Maturity Volatility dr/r (data) d N(d) d N(d2) Value of caplet Delta Floor = put on interest rate N(-d) N(-d2) Value of floor Delta Put-call parity for caps and floors FRA floor =cap Derivatives 0 Options on bonds and IR 6

17 Using DerivaGem Derivatives 0 Options on bonds and IR 7

18 Using bond prices In previous development, bond yield is lognormal. Volatility is a yield volatility. σ y = 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 Derivatives 0 Options on bonds and IR 8

19 From yield volatility to price volatility Remember the relationship between changes in bond s price and yield: ΔS Δy = DΔy = Dy S y This leads to an approximation for the price volatility: D is modified duration σ = Dyσ y Derivatives 0 Options on bonds and IR 9

20 Back to previous example (Hull 7h ed. 28.3) -year cap on 3 month LIBOR Cap rate = 8% Principal amount = 0,000 Maturity.25 Spot rate 6.39% 6.50% Discount factors Yield volatility = 20% -year put on a.25 year zero-coupon Face value = 0,200 [0,000 (+8% * 0.25)] Striking price = 0,000 Spot price of zero-coupon = 0,200 *.9220 = 9,404 -year forward price = 9,404 / = 0,025 3-month forward rate in year = 6.94% Price volatility = (20%) * (6.94%) * (0.25) = 0.35% Using Black s model with: F = 0,025 K = 0,000 r = 6.39% T = σ = 0.35% Call (floor) = Delta = 0.76 Put (cap) = Delta = Derivatives 0 Options on bonds and IR 20

21 Using DerivaGem Derivatives 0 Options on bonds and IR 2

22 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. Derivatives 0 Options on bonds and IR 22

23 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 977 Term structure = f(factors) In equilibrium models, today s term structure is an output No-arbitrage: Ho-Lee 986 Binomial evolution of whole term structure In a no-arbitrage models, today s term structure is an input Derivatives 0 Options on bonds and IR 23

24 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: P T ( r, t) = 00 e r( T t) where r is the interest rate (the same of all zero-coupons)

25 Zero duration portfolio Suppose r=3% Create a portfolio with zero-duration: Long n t -year zero-coupon bond (Example: Long n 0-year zc) Short t 2 -year zero-coupon (Example: Short 2-year zc) In order to have zero-duration: n = t2p2 t P Interest rate 3% Bonds 2 Portfolio with zero- duration Maturity 0 2 # units Value Price Bond Duration 0 2 Bond Convexity

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

27 No-arbitrage in continuous time The no-arbitrage condition can be demonstrated in a continuous time setting Stochastic process for short rate: dr = a( r) dt + σ ( r) dz P( r, t) = Pricing equation for zero-coupon: r( T t) Long-short portfolio: V e = np Cash P2 + = 0 Zero duration: V r = 0 n = ( T ( T 2 t) P2 t) P This portfolio has no risk and no investment. It should earn zero dv = 0 By Ito: dv = T2 P2 ( T T2 ) σ ² dt = 2 0 This equation holds only if T = T 2. Conclusion: the bond valuation model implies that arbitrage is possible.

28 dt T T P T dt P t T P t T P t T P t T dt rcash P t T P t T n rp nrp dv ² ) ( 2 ²) )² ( 2 ² )² ( ) ( ) ( 2 ( ² )² ( 2 ² )² ( σ σ σ σ σ = = + + = dz r V dt r V a r V t V dv σ σ = ²) ² ² 2 ( 2 2 ) ( ) ( 0 P t T P t T n r V = = As: Details of calculation

29 Monte Carlo experiment # period/yea 250 dt Drift a.00% sigma 3.00% Short rate r 3% Maturity Yield Price Quantity Value Bond % Bond % Cash Total V 0.00 dr - 0.7% =a*dt+normsinv(rand())*sigma*sqrt(dt) r+dr 2.83% Maturity Yield Price Quantity Value Delta Bond t- dt % Bond 2 t2- dt % Cash Total V+dV

30

31 Vasicek (977) Derives the first equilibrium term structure model. state variable: short term spot rate r Changes of the whole term structure driven by one single interest rate Assumptions:. 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 Derivatives 0 Options on bonds and IR 3

32 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 992 Estimates of a, b and σ based on following regression: r t+ r t = α + β r t +ε t+ a = 0.8, b = 8.6%, σ = 2% Derivatives 0 Options on bonds and IR 32

33 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: with y(r,t) = A(t)/t + [B(t)/t] r 0 P(r,t) = e -y(r,t) * t For formulas: see Hull 4th ed. Chap 2. Once a, b and σ are known, the entire term structure can be determined. Derivatives 0 Options on bonds and IR 33

34 Vasicek: example Suppose r = 3% and dr = 0.20 (6% - r) dt + % dz Consider a 5-year zero coupon with face value = 00 Using Vasicek: A(5) = 0.093, B(5) = y(5) = ( * 0.03)/5 = 4.08% P(5) = e * 5 = 8.53 The whole term structure can be derived: Maturity Yield Discount factor 3.28% % % % % % % % % % % % 6.00% 5.00% Derivatives 0 Options on bonds and IR 34

35 Jamshidian (989) 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 σ P = σ a [ e ] a( T* T ) e 2a at Derivatives 0 Options on bonds and IR 35

36 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) = % Ru = 2% + % * sqrt(0.5) Rd=2% - % * sqrt(0.5) More on this later Derivatives 0 Options on bonds and IR 36

37 Option valuation How to value a 6-month call option on a -year ZC with a strike price K=99? Build a binomial tree for the bond price: This is the market price! Prices calculated using the 6-month rate prevailing at time t = 0.5 Derivatives 0 Options on bonds and IR 37

38 Market price of interest rate risk = ln( ) / 0.5 = 2.00%!.50% = 99.0! e ".50%!0.50 " Shape of term structure determined by 2 forces: - Expected future spot rate - Risk aversion (required risk premium) Derivatives 0 Options on bonds and IR 38

39 Different possible views of current term structure Small IR increase High risk premium High IR increase Smal risk premium Derivatives 0 Options on bonds and IR 39

40 Back to -period binomial pricing Derivatives 0 Options on bonds and IR 40

41 Multiperiod interes rate tree r i, j! r = r i+, j i, j + "r +! "t r i+, j+ = r i, j + "r #! "t i is the number of period j is the number of downward movement in interest rate Derivatives 0 Options on bonds and IR 4

42 Pricing a 3-period zero-coupon Suppose risk-neutral probability constant Problem: Market price different from price calculated by model Derivatives 0 Options on bonds and IR 42

43 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 that yield the market price. Derivatives 0 Options on bonds and IR 43

44 Valuing a -year cap on 6-month rate Protection against an upward movement of 6-month rate. 6-month rate with simple compounding Payoff at maturity = (e 3.9%!0.5 ")! 2 = 3.95% =00! Max(0,3.95%" 2%)! 0.5! e "3.9%!0.5 Derivatives 0 Options on bonds and IR 44

45 Swaption A 6-month swaption on a -year swap Option maturity: 6 month Swap maturity: year (at option maturity) Swap rate:.25% per period ( period = 6 month) Remember: Swap = Floating rate note - Fix rate note Swaption = put option on a coupon bond Bond maturity:.5 year Coupon: 2.5% (semi-annual) Option maturity: 6-month Strike price = 00 Derivatives 0 Options on bonds and IR 45

46 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. f ij = p i f i+, j + ( pi ) fi+, j+ + r Δt j e i, coupon i+ i is the number of period j is the number of downs Δt is the time step Derivatives 0 Options on bonds and IR 46

47 Valuing the swaption Derivatives 0 Options on bonds and IR 47

48 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 Derivatives 0 Options on bonds and IR 48

49 Ho-Lee model r i+, j = r i, j +! i! "t +"! "t r i+, j+ = r i, j +! i! "t #"! "t The thetas are parameters used to fit the term structure. Derivatives 0 Options on bonds and IR 49

50 Ho-Lee illustration Derivatives 0 Options on bonds and IR 50

51 Ho-Lee illustration (2) Derivatives 0 Options on bonds and IR 5

52 Black Derman Toy z i+, j = z i, j +! i! "t +"! "t z i+, j+ = z i, j +! i! "t #"! "t r i, j = e z i, j Derivatives 0 Options on bonds and IR 52

53 BDT illustration Derivatives 0 Options on bonds and IR 53