Interest Rate Course Lecture 9. June

Transcription

1 Interest Rate Course Lecture 9 June

2 Last days Want to find stochastic models consistent with observed i) yield curves and ii) dynamics of yield curve One factor models Two factor models Other approaches

3 Interest Rate Recap One factor models: make stochastic model for short end: dr = a(r,t)dt + b(r,t)dw From that, obtain ZCB bond prices using hedging type argument + market price of risk argument Get Partial Differential Equation which reduces to pair of ODEs if affine type structure is exploited. But market price of risk a bit sketchy.

4 Market Price of Risk Factor models req d market price of risk λ(r,t). From separation of variables argument used to go from Ito s lemma on hedged bond portfolio to partial differential equation in each individual bond. Market price of risk indep. of bond maturity. interpretation as excess return for risky assets. But that s all we know about it.

5 Let s revisit this a bit. Given an interest rate model dr = μ(r,t)dr + σ(r,t)dw, the hedging argument forced us to rephrase things in terms of a risk-neutral drift factor μ Q (r,t), which was obtained from the real measure or true drift thus: μ Q (r,t) = μ(r,t) λ(r,t)σ(r,t).

6 Mathematical Trick only? What was modelling content of this step? It works out that real world dynamics dr = μ(r,t)dr + σ(r,t)dw require: λ(r,t) = *μ(r,t)- μ Q (r,t)+/σ(r,t). Expressions must go to zero in a consistent way.

7 Affine term structure models λ(r,t) left the stage as discretely, and as mysteriously, as it had arrived. Worked with risk neutral drift factor only. This gave us all the bond prices, and so the yield curves, It was all we needed for panel data. The quants are also happy because it s all you need to price options as well.

8 Hints about meaning: Equity options There we have μ(s,t) = (μ)(s), σ(s,t) = (σ)(s), and μ Q (S,t) = (r)(s), so we ended up with λ(s,t) = *μ(s,t)- μ Q (S,t)+/σ(S,t) = *μ*s- r*s+/σ*s. = (μ-r)/σ, This Sharpe Ratio is a constant well freighted with lots of financial meaning.

9 Lucky But we were very lucky here. We had strong financial arguments for all of our choices of μ, μ Q, and σ, and the nice simple form of λ was in some sense a validation of the intuition behind these other choices. But in interest rate modeling, even in the simple one factor interest rate modeling world, we had much less consensus on the right options model to choose.

10 Lots of confusion Pick affine structure to get nice looking yield curves (takes its logical conclusion in Ho-Lee, Hull-White, and their ilk). Other inputs are more P measure (like interest rates can never be negative) It s not inconceivable that spot rates could be CIR and the risk free rates could be Vasicek.

11 A big rug Wilmott: Because we can t observe the function λ(r,t) except possibly via the whole yield curve, I tend to think of it as a great big carpet under which we can brush all kinds of nasty, inconvenient, things.

12 Try to get insights Still, what if both μ and μ Q were Vasicek (and time-independent)? Then μ(r) = κ(ѳ-r), σ(r) = σ, μ Q (r) = κ Q (Ѳ Q - r), So λ(r) = *μ(r)- μ Q (r)+/σ = (κѳ-κ Q Ѳ Q )/σ (κ-κ Q )r/σ It would be nice to have a constant market price of risk here, implying κ=κ Q Suggests that λ = (κ/σ)(ѳ-ѳ Q ),

13 Real vs risk neutral dynamics λ = (κ/σ)(ѳ-ѳ Q ), Both κ and σ are positive Interpret the market price of risk as negative here (because bond prices move opposite to rates). Suggests real rates mean revert to lower levels than risk-neutral rates, which sort of makes sense. (why yield curve normally slopes upwards, even though interest rates can t normally be increasing). Less well motivated but similar CIR assumption yields same result.

14 What does slope of yield curve tell us? Z(r,τ) = exp[a(τ)-rb(τ)], $1 face bond at time τ r(τ) = interest rate for time τ Z(t,T) = exp[- r(t,t)(t-t)] Or r(t,t) = -ln[z(t,t)]/(t-t). So ln[z(r,τ)]/τ = rb(τ)/τ A(τ)/τ. To illustrate, for Vasicek; A(τ) =,σ 2 /(2κ 2 )-,τ-b(τ)- -,σ 2 B 2 (τ)-/(4κ) + Ѳ Q (B(τ)-τ), B(τ) = *1-exp(-κτ)+/κ. (Lecture 7 slide 48)

15 Push some symbols around At the short end (τ 0), B(τ) = [1 (1 κτ + ½ κ 2 τ 2 + )+/κ = τ ½ κτ 2 +, so So τ B(τ) = ½ κτ 2 To leading order, A(τ) = - ½ Ѳ Q κτ 2, so B(τ)/τ = 1 ½ κτ, A(τ)/τ = - ½ Ѳ Q κτ, so So ln[z(r,τ)]/τ = r( 1 - ½ κτ ) + ½ Ѳ Q κτ. So, at the short end, the yield curve is: r(τ) = r short + ½ κ (Ѳ Q r)τ +

16 More generally It can be shown that this is true for any affine model, and so r(τ) = r short + ½ μ Q (r)τ + or, r(τ) = r short + ½ [μ(r) λ(r)σ(r)]τ + Slope of short end is ½ [μ(r) λ(r)σ(r)] Since λ < 0, suggest more risk higher slope in yield curve. (consistent with larger Ѳ Q )

17 Estimating One factor models Many popular models take form dr = μ(r) dt + σr β dw Vasicek(β=0),CIR (β = ½),GBM ( Dothan )β = 1. Chan, Karolyi, Longstaff, & Sanders (1992), working with US data, estimate β = 1.36 (this suggests lognormal ish form for spot rate dynamics, despite theoretical difficulties).

18 Can play with this on spreadsheet Form of volatility doesn t depend on P vs Q aka market price of risk stuff. Built regression spreadsheet, let s go through it a bit. LeastSquaresSheet.xls EstimatingBetaSynth.xls EstimatingBetaReal.xls

19 For more recent Canadian Data Seems hard for me to reject that β = ½. Obviously this is preliminary but it s never a bad idea to revisit empirical studies every so often.

20 Estimating drift term Hard to do from spot data Reasons similar to that for stocks; small relative to diffusion. Also, real vs risk neutral measure issues do play a role here. Could try to back out mean reversion properties from probability density functions

21 Excursion: Fokker Planck If dr = μ(r) dt + σr β dw, r(0) = r0, then probability density function p(r,t) follows a diffusion-like equation This is called the Fokker Planck or the Kolmogorov Forward equation δp/δt = ½σ 2 (δ 2 /δr 2 )(r β p) (δ/δr)[μ(r)p] Check plug β = 0 and μ = 0 to get random walk PDE; plug μ = const > 0 to see mean move right.

22 Steady state PDE If interest rates are strongly mean reverting, we d expect their density function to reach some steady state eventually. Lim t p(r,t) = p*(r). Then (δ/δt) = 0 and (δ/δr) = d/dr 0= ½σ 2 (d 2 /dr 2 )(r β p*) (d/dr)[μ(r)p*] Can integrate once, and then by parts, to obtain: μ(r) = σ 2 βr 2β-1 + ½ σ 2 (r β d[lnp*]/dr

23 Market Price of Risk No info about market price of risk in spot process, need to go to slope of yield curve to decide, as discussed earlier.

24 Back to 1 factor recap Problem insufficient degrees of freedom to turn all knobs. Could go Ho-Lee/Hull-White type route, but then overfits.

25 Two factor models Put in two sources of market risk Similar hedging argument, but with 2 hedging, 1 pricing assets instead of Still get 2+1-d PDE, If structure is affine, get system of (3) ODEs. How do we interpret second risk factor? Wrote down 2DVasicek in Lecture 8.

26 Some named 2Factor Models Fong & Vasicek (1991) dr = κ 1 (θ 1 -r)dt + σ 1 dw 1 dφ = κ 2 (θ 2 - φ)dt + σ 2 dw 2, <dw 1 dw 2 > = ρdt (Affine structure) can interpret φ as square root of the volatility of the spot rate (r as the spot rate).

27 Brennan & Schwartz (1982) dr = (a 1 +b 1 (l-r))dt + σ 1 rdw 1 dl = l(a 2 -b 2 r+c 2 l)dt + σ 2 ldw 2 <dw 1 dw 2 > = ρdt Interpret r as short end, l as long end (see how short end mean is attracted by the long end). Long and short rates must satisfy internal consistency requirements Is more complicated than this, but can see it is essentially lognormal in solution. Lognormal enough to cause blowup of bank account like we discussed in Lecture 8

28 Longstaff and Schwartz This is like 2D CIR: dx = a(x*-x)dt + xdw 1, dy = b(y*-y)dt + ydw 2, <dw 1 dw 2 > = ρdt Spot interest rate is r = cx + dy Simple affine structure explicit formulae for simple interest rate products including ZCBs.

29 Empirical Spot rate Behaviour One factor models Two factor models How many factors is enough?

30 Cochrane and Piazessi Have done a lot of interesting work recently on affine term structure models. Some of their work is on principal component analysis of yield curve time dynamics. Show that can estimate most of this with just 2 factor models. Also have some very interesting work on how to interpret message from forward markets.

31 C&P NBER 9178, Sept 2002 %(varδy) explained by this factor R 2 forecasting rx t+1 from this factor R 2 forecasting rx t+1 from up to this factor Level Slope Curvature Spreads Zigzag 93.7% 4.4% 0.8% 0.6% 0.6% 2.6% 5.2% 7.3% 8.9% 9.1% 2.6% 22.6% 26.5% 26.7% 35.1%

32 Unpacking this Used Principal Component Analysis First 2 factors explain nearly all of the variance in yield curve changes. (2 factor models should be enough). A good yield curve model is not enough to forecast spot rates Used Fama-Bliss 1-5 year ZCB prices from CRSP,

33 Other approaches to interest rate modelling

34 Forward, Zero, and Yield Z(t,T) = value at time t of bond paying $1 at time T (T > t) r(t,t) (= r(t,t,t) ) = time t interest rate for money borrowed/lent between time t and time T. (T>t) Z(t,T) = exp[- r(t,t)(t-t)] Or r(t,t) = -ln[z(t,t)]/(t-t).

35 Forward Rates Forward rates: r(t,s,t) = interest rate, set at time t, for money borrowed/lent between time s and time T, t < s < T. Lim T s+: r(t,s,t) = f(t,s) = forward rate applying between time s and time s + ds No arbitrage argument exp[- tt f(t,u)du] = Z(t,T) (Note this is different from borrowing short/lending long or vice versa as all rates are locked in at time t)

36 Relation with yields r(t,t)(t-t) = tt f(t,u)du, r(t,s+ds)(s+ds-t) = t s+ds f(t,u)du, r(t,s)(s-t) = ts f(t,u)du, so f(t,s) = (d/ds) [r(t,s)*(s-t)]. Since ln[z(t,s)] = -r(t,s)*(s-t), we can also write this as f(t,s) = -(d/ds)ln[z(t,s)]. Take t = 0 to get: f(s) = f(0,s) = (d/ds) [sr(s)], as shown in slide 20 of lecture 3.

37 Relationships A relationship between zero coupon bonds prices the yield curve, and the forward curves quoted at the same time. In everything we ve discussed so far we ve made models for the yield curve

38 Model Zeros instead of yields? In one factor models, dr = a(r,t)dt + b(r,t)dw. (we take a break from our usual notation). Why not make a diffusion-based model for the price of a ZCB? dz =μ(z,t)dt + σ(z,t)dw? Much like the idea of making a random model for the values of the stock price. We take derivatives with respect to calendar time (date) t, not expiry date T.

39 Problems with modelling zeros As we get closer to maturity, the random fluctuations can t be very large. (A day before maturity we re going to have a bond that s worth nearly a dollar no matter what!). The instant before maturity, Z(T-,T) can t fluctuate at all, so we need σ(t,t) = 0. This is bad for two reasons one, we must impose a lot of structure onto σ, and worse, the structure will need to be different for bonds with different maturities

40 So, why not start with Forwards? This might be nice because forward rates are more independent from one another than are yields. Also, forward rates are easy to estimate from very liquid swap & FRA market.

41 Toy model Consider toy model for short rates: dr = Ѳdt + σdw. This is affine, and easy to solve (very similar to Ho-Lee, slides of lecture 8) Z(t,T) = exp[σ 2 /6(T-t) 3 ½Ѳ(T-t) 2 (T-t)r] f(t,t) = -d/dt[ln(z(t,t)] (from a few slides back) So f(t,t) = -d/dt[σ 2 /6(T-t) 3 ½Ѳ(T-t) 2 (T-t)r] So f(t,t) = - ½σ 2 (T-t) 2 + Ѳ(T-t) + r

42 SDE for forward rates We can write down a sde for the evolution of the forward rates: df = (d/dt)[- ½σ 2 (T-t) 2 + Ѳ(T-t) ]dt + dr = [σ 2 (T-t) Ѳ]dt + Ѳdt + σdw Or, df = σ 2 (T-t)dt + σdw Note that the drift in our forward dynamics is simply a transformation of the volatility! This is more generally true.

43 HJM This is the idea behind Heath Jarrow Morton interest rate modeling. Write down initial value problem for the forward rate:

44 Heath Jarrow Morton Major breakthrough in pricing of interest rate products Built framework including all 1- and 2- factor models Start with model for entire forward curve. Since forward curves are known today, yield curve fitting is direct and immediate in this model.

45 ZCB in terms of forwards; SDE for ZCB Write down df(t,t) = α(t,t)dt + σ(t,t)dw Here both σ and dw are vectors, so we need to think of the final product as an inner or dot product. σ(t,t) = Σσ i (t,t)dw i Initial values f(0,t) = f M (0,T) No-arbitrage in the risk- neutral measure, α(t,t) = σ(t,t) tt σ(t,u)du (which fits for our toy model).

46 Evolution of forward curve f(t,t) = f(0,t) + 0t df = f(0,t) + 0t df = f(0,t) + 0t [α(s,t)ds + σ(s,t)dw s ] =f(0,t) + 0t σ(u,t) ut σ(u,s)ds+du + 0t σ(s,t)dw s (recall inner products for σ and dw) r(t) = f(t,t) = f(0,t) + 0t σ(u,t) ut σ(u,s)ds]du + 0t σ(s,t)dw s

47 Spot rate and forward rate r(t)=f(0,t)+ Σ 0t σ i (u,t) tt σ i (u,s)ds]du +Σ 0 t σ i (s,t)dw i s Key insight here is that the whole innovation path W s needs to be recorded; the model is not in general Markov. However if σ i (t,t) = a(t)b(t) it will be markov.

48 Simulating with HJB To price more complicated securities: Simulate evolution of whole forward-rate curve Calculate value of all cashflows under each evolution Calculate PV of these cashflows discounting with realized spot rate r(t)

49 Simulation Step 1 Use df(t;t) = α(t,t)dt + Σσ i (t,t)dw i, α(t,t) = σ(t,t) tt σ(t,s)ds. End of this simulation have realization of whole forward rate path. Hence realized prices of ZCB for all maturities up to time T. Using this forward rate path can calculate value of all cashflows that would have occurred. Then, discount using realized r(t) Repeat until convergence.

50 Libor Market Models Really more for derivatives pricing Designed to be consisent with market quotes for caplets and swaptions (in other words, need to match term structure of volatility in some sense). Two mutually inconsistent versions: lognormal forward LIBOR model (LFM) Lognormal forward swap model (LSM) See Brigo & Mercurio for more details

51 The bottom line It s hard to model interest rates For equities, we could reason, and check with real data, that return was the right thing to model. That almost gives you everything right there even a really nice and intuitive market price of risk. For interest rates, we don t have much to go on. And so we need to use judgement.

52 Judgement What are we using the model for? Are we using it to hedge short dated options? If so, any reasonable model might be OK as a guide to hedging. If, however, we are using models to price long dated bonds with embedded options, the models are harder to justify.

53 Next day We will finish the course tomorrow We will discuss the elephant in the interest rate room credit risk. We ll provide a brief look at the two main modelling frameworks: reduced form models and structural models. We ll relate structural models to a recent proposal for bank capital, reverse convertible bonds.