Introduction. Practitioner Course: Interest Rate Models. John Dodson. February 18, 2009

Transcription

1 Practitioner Course: Interest Rate Models February 18, 2009

2 syllabus text sessions office hours

3 date subject reading 18 Feb introduction BM 1 25 Feb affine models BM 3 4 Mar Gaussian models BM 4 11 Mar whole-curve models BM 5 SPRING BREAK 25 Mar the market model BM 2, Apr the swap model BM Evaluation There will be two graded assignments for this module due March 11 due April 8 (beginning of next module) Each will be worth half of the total points for the module.

4 There are a number of rich and interesting extensions, some of which are covered in the text non-performance or default foreign currencies pre-payment tax inflation illiquidity, bid/offer spread sovereign vs. interbank exotic derivatives But we will not have time for this. Our goal will be to cover the standard models in use today for the valuation of non-defaultable cash and vanilla derivative securities based on a single yield curve.

5 Note that P risk-free discount factor r spot instantaneous risk-free rate f forward instantaneous risk-free rate R continuous zero-coupon bond yield τ conventional years L simple spot rate F simple forward rate P(T ) = e R(T ) T = L(T ) τ(t ) and P(S) P(T ) = F (T, S) τ(t, S), S > T

6 r(t) stochastic risk-free rate B(t) stochastic bank account balance D(t, T ) stochastic discount factor The bank account is defined by the initial condition B(0) = 1 and the SDE db(t) = B(t) r(t) dt The discount factor is defined by D(t, T ) = B(t) B(T ) = T e t r(t ) dt It represents the amount of money at time t that would grow on deposit to be worth exactly 1 at time T > t.

7 Fitting the Initial We will assume that the graph T P(T ) is observable today, e.g. Bootstrap bond prices CB (c, {T i }) = P (T n ) + c Interpolate bond yields Chain forward rates n τ (T i 1, T i ) P (T i ) i=1 1 P(T ) y(t ) = T 0 P(T ) dτ(t ) 1 P(T ) = 1 + L(T 1 ) τ(t 1 ) F (T 1, T 2 ) τ(t 1, T 2 ) F (T n 1, T n ) τ(t n 1, T )

8 Equivalent Martingal Measure If there are enough securities in the market, then interest rate dynamics can be completely hedged. In this case, the stochastic discount factor is integrable with respect to some measure, and we have [ P(t, T ) = E t [D(t, T )] = E t e T t r(t ) dt ] where the subscript t indicates that the expectation is conditional on the filtration F t. Black-Scholes PDE Evaluating the expectation above may not be practical. The Feynman-Kac result suggests a connection between such expectations and linear second-order PDE s.

9 In fact we will demonstrate that the simplest hedgeability 1 requires P(T, T ) = 1, P t (T, T ) = r and P t (t, T ) + P r (t, T ) b(t, r) P rr (t, T ) σ 2 (t, r) = r P(t, T ) for all t < T, where dr dr = σ 2 dt and b comes from the implicit function theorem. Short-rate Models This PDE has several classes of analytical solutions. We will study single-factor solutions next week and the multi-factor version the following week. 1 All bonds being perfectly correlated.

10 Bonds The net present value at t of a stream of certain cashflows with cumulative value given by c( ) is t P(t, T ) dc(t ) For a prototypical bond, there is a fixed coupon rate c and maturity date T, c(t ) = c τ(t T ) + H(T T ) so the net present value is T V (t, T, c) = c P(t, T ) dτ(t ) + P(t, T ) t

11 Swaps Interest rate swaps have two legs: a floating leg and a fixed leg The fixed leg has c fixed (T ) = s(t, T ) τ(t T ) where s(t, T ) is the fair contractual swap rate at t The floating leg has c float (T ) = H(T t) + H(T T ) Think of depositing $1, collecting and distributing the periodic interest, then withdrawing it. In order for the net present value at t to be zero, we must have 1 P(t, T ) s(t, T ) = T t P(t, T ) dτ(t ) N.B.: There is a connection between swaps and bonds. The swap rate is also the coupon rate (and the internal rate of return) on a bond whose value at t is par.

12 Swaps A party to a swap is either a receiver or payer of the fixed leg cashflows. Since the owner of a bond receives fixed interest, we usually denote a receiver swap position with a positive notional. Conversely a payer swap position has a negative notional. Valuation Say we need to value a swap that was settled at some date t = 0 in the past. We can show that V swap (t, T, s(0, T )) = 1 P(t, T ) s(t, T ) (s(t, T ) s(0, T )) for continuous floating resets. The dynamics of a swap value depends on the joint dynamics of the swap rate and the annuity factor 1 P(t,T ) s(t,t ).

13 Bonds again Duration The annuity factor is the modern version of the classical concept of bond duration. Indeed, we can show that the value of a bond is where so V (t, T, c) = 1 A(t, T ) (s(t, T ) c) A(t, T ) = T t 1 V V s P(t, T ) dτ(t ) = A s=c The annuity factor is a proxy for the first-order exposure of a bond s value to changes in rates. notional A(t, T ) 10 4 is sometimes called the bond s present value of a basis point.

14 The valuation of caps and floors depend explicitly on the nature of interest rate volatility, and so are often the basis for calibrating volatility models. Cap A cap pays off whenever the floating rate is above the strike level. V cap (t, T, K) = T in the continuous reset version. t [ (r(t E t ) K ) ] + D(t, T ) Floor A floor pays out whenever the floating rate is below the strike level. V floor (t, T, K) = T t [ (K E t r(t ) ) ] + D(t, T ) dt dt

15 Cap/Floor Parity Since P(t, T ) = E t [D(t, T )] and P T (t, T ) = E t [ r(t ) D(t, T )] we can derive a parity arbitrage relationship similar to that for European puts and calls. V floor (t, T, K) V cap (t, T, K) = V swap (t, T, K) Moneyness We can unambiguously define K = s(t, T ) as at-the-money, with higher strikes being in-the-money for floors and out-of-the-money for caps and vice versa for lower strikes.

16 & Bond Options The other class of vanilla interest rate derivatives are the swaptions. A swaption is the right, but not the obligation, to enter a swap. They are often denoted as right-to-receive or right-to-pay the fixed leg. Pay/Receive Parity In exact analogy to European put-call parity, if one is long the right to receive and short the right to pay fixed, then by arbitrage this is equivalent to one being long the underlying forward-start swap. Embedded Bond Options We can apply our learnings about swaptions to options on or embedded in (default-free) bonds. E.g., a callable bond is equivalent to a (non-callable) bond and a short receiver swaption.

17 & Bond Options For a (continuous reset) swaption with maturity T 0 and underlying swap maturity T 1, the value at t < T 0 < T 1 is V RTR (t, T 0, T 1, K) = T1 ( E t [D(t, T 0 ) P(T0, T ) K + P T (T 0, T ) ) ] + dt T 0 and similarly for a right-to-pay swaption. Decomposition The difference between this and a cap/floor is that a swaption cannot be decomposed into a sum of options. A cap/floor is exercised continuously, while a swaption is exercised only once.

18 Jamshidian s Decomposition The non-decomposition of swaptions can be an impediment to analysis. Single-Factor Uncertainty If we can write for t < T 0 < T E t [D(T 0, T )] = E t [Π(T 0, T, r(t 0 ))] where the expectation is over the terminal value r(t 0 ) and not the whole path, and if Π(T 0, T, r) is decreasing in r for T > T 0, then there is an r such that T1 K Π(T 0, T, r ) dτ(t ) = 1 Π(T 0, T 1, r ) T 0 N.B.: D, P, and Π are related. P(t, T ) = Π(t, T, r)

19 Jamshidian s Decomposition The holder of a right-to-pay swaption would exercise at T 0 iff r > r, because the swap at the strike rate would have a positive net present value. Prior to exercise the swaption is worth T1 V RTR (t, T 0, T 1, K) = K T 0 E t [D(t, T 0 ) (Π(T 0, T, r(t 0 )) Π(T 0, T, r ) ) + ] dτ(t ) + E t [ D(t, T0 ) (Π(T 0, T 1, r(t 0 )) Π(T 0, T 1, r )) +] and similarly for a right-to-pay fixed swaption. The advantage of this approach becomes apparent if we can determine analytical values for options on single cashflows. But the key assumption that all rates are perfectly correlated may not be reasonable.

20 Emerging Lessons from the Crisis of funding availability: central banks market liquidity: sovereign funds and private equity trading counterparty non-performance: clearing infrastructure hazards of diversification: contagion systemic risk and the coordination problem: market supervision, deposit insurance, and government guarantees