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>> Chp.19 Term Structure of Interest Rates (II) 报告人:陈焕华 指导老师:郑振龙 厦门大学金融系 教授 >> Continuous time models • Term structure models are usually more convenient in continuous time. • Specifying a discount factor process and then find bond prices. • A wide and popular class models for the discount factor: >> Implications • Different term structure models give different specification of the function for ur , r , • r starts as a state variable for the drift of discount factor process, but it is also the short rate process since Et d / rft dt • Dots(.) means that the terms can be function of state variables.(And so are time-varying) • Some orthogonal components can be added to the discount factor with on effect on bond price. >> Some famous term structure models • 1.Vasicek Model: d rdt dz, dr (r r )dt r dz Vasicek model is similar to AR(1) model. • 2.CIR Model d rdt dz, dr (r r ) r r dz • The square root terms captures the fact that higher interest rate seem to be more volatile, and keeps the interest rate from zero. >> Continuous time models Having specified a discount factor process, it is simple matter to find bond prices t N (N ) Pt Et ( ) t Two way to solve – 1. Solve the discount factor model forward and take the expectation – 2. Construct a PDE for prices, and solve that backward >> Implication • Both methods naturally adapt to pricing term structure derivatives : call options on bonds, interest rate floors or caps, swaptions and so forth, whose payoff is Pt (N ) s C Et x ( s )ds s t t • We can take expectation directly or use PDE with option payoff as boundary conditions. >> expectation approach d d2 d ln 1 / 2 2 (r 1 / 2 2 )dt dz T T T 2 ln (rs 1 / 2 s )ds s dz s 0 s 0 0 (N ) 0 P E0 ( e sT0 ( rs 1 / 2 2 s ) ds sT0 s dz ) • Example: in a riskless economy • With constant interest rate, (N) 0 P P0 e rT T rs ds s 0 e , >> Remark • In more situations, the expectation approach is analytically not easy. • But in numerical way, it is a good way. We can just stimulate the interest rate process thousands of times and take the average. >> Differential Equation Approach • Similar to the basic pricing equation for a security price S with no dividend Et (dS / S ) rdt Et ( dS d ) S • For a bond with fixed maturity, the return is dP( N , t ) 1 P( N , t ) dt P P N • Then we can get the basic pricing equation for the bonds with given maturity: Et ( dP 1 P( N , t ) dP d )( r )dt Et ( ) P P N P >> Differential Equation Solution • Suppose there is only one state variable, r. Apply Ito’s Lemma P 1 P 2 P dP ( ur r )dt r dz 2 r 2 r r 2 • Then we can get: P 1 P 2 P P ur r rP r 2 r 2 r N r 2 Market Price of Risk and >> Risk-neutral Dynamic Approach • The above mentioned PDE is derived with discount factors. • Conventionally the PDE is derived without discount factors. • One approach is write short-rate process and set market price of risk to P 1 2 P 2 P P ur r rP r 2 r 2 r N r >> Implication • If the discount factor and shocks are imperfectly correlated, • Different authors use market price of risk in different ways. • CIR(1985) warned against modeling the right P hand side as r (.) , it will lead to positive expected return when the shock is zero, thus make the Sharpe ration infinite. • The covariance method can avoid this. >> Risk-Neutral Approach A second approach is risk-neutral approach • Define: d / rdt , dr (ur r )dt r dzr • We can then get P 1 2 P 2 P (ur r ) r rP 0 2 r 2 r N • price bonds with risk neutral probability: Pt (N ) T rs ds s 0 E [e ] * t >> Remark • The discount factor model carries two pieces of information. – The drift or conditional mean gives the short rate – The covariance generates market price of risk. • It is useful to keep the term structure model with asset pricing, to remind where the market price of risk comes from. • This beauty is in the eye of the beholder, as the result is the same. >> Solving the bond price PDE numerically • Now we solve the PDE with boundary condition numerically. • • Express the PDE as P P 1 2P 2 (ur r ) r rP 2 N r 2 r • The first step is >> Solving the bond price PDE • At the second step P / r N , P / r 0, 2 2 P P(2N , r ) P(N , r ) N (ur r ) rP (N , r ) N N P(2N , r ) N 2 (ur r ) P(N , r ) 2 >> 5. Three Linear Term Structure Models • Vasicek Model, CIR Model, and Affine Model gives a linear function for log bond prices and yields: ln P( N , r ) A( N ) B( N )r • Term structure models are easy in principle and numerically. Just specify a discount factor process and find its conditional expectation or solve the differential equation. >> Overview • Analytical solution is important since the term structure model can not be reverse-engineered. We can only start from discount factor process to bond price, but don’t know how to start with the bond price to discount factor. Thus, we must try a lot of calculation to evaluate the models. • The ad-hoc time series models of discount factor should be connected with macroeconomics, for example, consumption, inflation, etc. >> Vasicek Model • The discount factor process is: d rdt dz dr (r r )dt r dz • The basic bond differential equation is: P 1 2 P 2 P P (r r ) r rP r 2 r 2 r N r • Method: Guess and substitute >> PDE solution:(1) • Guess P( N , r ) e A( N ) B ( N ) r – Boundary condition: A(0) B(0)r 0 for any r, so A(0) 0, B(0) 0 • The result is >> PDE solution:(1) • To substitute back to PDE ,we first calculate the partial derivatives given P( N , r ) e A( N )B( N ) r >> PDE solution:(1) • Substituting these derivatives into PDE • This equation has to hold for every r, so we get ODEs >> PDE solution(2) • Solve the second ODE with >> PDE solution(3) • Solve the first ODE with >> PDE solution(3) >> PDE solution(3) >> PDE solution(4) • Remark: the log prices and log yields are linear function of interest rates p( N , r ) A( N ) B( N )r , y ( N , r ) • y B( N ) 0 r N A( N ) B( N ) r N N means the term structure is always upward sloping. >> Vasicek Model by Expectation • The Vasicek model is simple enough to use expectation approach. For other models the algebra may get steadily worse. • Bond price >> Vasicek Model by Expectation • First we solve r from • The main idea is to find a function of r, and by applying Ito’s Lemma we get a SDE whose drift is only a function of t. Thus we can just take intergral directly. • Define >> Vasicek Model by Expectation • Take intergral >> Vasicek Model by Expectation • So • We have >> Vasicek Model by Expectation • Next we solve the discount factor process • Plugging r >> Vasicek Model by Expectation >> Vasicek Model by Expectation • The first integral includes a deterministic function, so gives rise to a normally distributed r.v. for • Thus mean is normally distributed with >> Vasicek Model by Expectation • And variance >> Vasicek Model by Expectation • So • Plugging the mean and variance >> Vasicek Model by Expectation • Rearrange into • Which is the same as in the PDE approach >> Vasicek Model by Expectation • In the risk-neutral measure >> CIR Model >> CIR Model • Guess • Take derivatives and substitue • So >> CIR Model • Solve these ODEs • Where >> CIR Model >> Multifactor Affine Models • Vasicek Model and CIR model are special cases of affine models (Duffie and Kan 1996, Dai and Singleton 1999). • Affine Models maintain the convenient form that the log bond prices are linear functions of state variables(The short rate and conditional variance be linear functions of state variables). • More state variables, such as long interest rates, term spread, (volatility),can be added as state variable. >> Multifactor Affine Model >> Multifactor Affine Model Where , K K , y , y, b K 1, Con : i i ' y 0 i 0 CIR.M . i 0 V .M . >> PDE solution • Guess • Basically, recall that • Use Ito’s Lemma dP 1 P 11 2 P ' dy dy ' dy P P y 2P yy ' >> PDE solution >> Multifactor Affine Model >> Multifactor Affine Model >> Multifactor Affine Model >> Multifactor Affine Model Rearrange we get the ODEs for Affine Model >> Bibliography and Comments • The choice between discrete and continuous time is just for convenience. Campbell, Lo and MacKinlay(1997) give a discrete time treatment, showing that the bond prices are also linear in discrete time two parameters square root model. • In addition to affine, there are many other kinds of term structure models, such as Jump, regime shift model, nonlinear stochastic volatility model, etc. For the details, refer to Lin(2002). >> Bibliography and Comments Constantinides(1992) • Nonlinear Model based on CIR Model, • Analytical solution. • Allows for both signs of term premium. >> Risk-neutral method The risk-neutral probability method rarely make reference to the separation between drifts and market price of risk. This was not a serious problem for the option pricing, since volatility is more important. However, it is not suitable for the portfolio analysis and other uses. Many models imply high and time-varying market price of risk and conditional Sharpe ratio. Duffee(1999) and Duarte(2000) started to fit the model to the empirical facts about the expected returns in term structure models. >> Term Structure and Macroeconomics • In finance, term structure models are often based on AR process. • In macroeconomics, the interest rates are regressed on a wide variety of variables, including lagged interest rate, lagged inflation, output, unemployment, etc. • This equation is interpreted as the decisionmaking rule for the short rate. • Taylor rule(Taylor,1999), monetary VAR literature (Eichenbaum and Evans(1999). >> The criticism of finance model • The criticism of term structure model in finance is hard when we only use one factor model. • Multifactor models are more subtle. • But if any variable forecasts future interest rate, it becomes a state variable, and should be revealed by bond yields. • Bond yields should completely drive out other macroeconomic state variables as interest rate forecasters. • But in fact, it is not. >> High-frequency research • Balduzzi,Bertola and Foresi (1996), Piazzesi(2000) are based on diffusions with rather slow-moving state variable. The one-day ahead densities are almost exactly normal. • Johannes(2000) points out the one day ahead densities have much fatter tails than normal distribution. This can be modeled by fast-moving state variables. Or, it is more natural to think of a jump process. >> Other Development • All the above mentioned models describe the bond yields as a function of state variables. • Knez, Litterman and Scheinkman(1994) make a main factor analysis on the term structure and find that most of the variance of yields can be explained by three main factors, level, slope, hump. It is done by a simple eigenvalue decomposition method. >> Remark • Remark: This method is mainly used in portfolio management, for example, to realize the asset immunation of insurance fund. • It is a good approximation, but just an approximation. The remaining eigenvalues are not zero. Then the maximum likelihood method is not suitable, maybe GMM is better. • The importance of approximation depends on how you use the model, if you want to find some arbitrage opportunity, it has risk. The deviation from the model is at best a good Sharpe ratio but K factor model can not tell you how good. >> Possible Solution • Different parameters at each point in time (Ho and Lee 1986). It is useful, but not satisfactory. • The whole yield curve as a state variable, Kennedy(1994), SantaClara and Sornette(1999) may be the potential way. >> Market Price of Risk • The market price of interest rate risk reflects bond the market price of real interest rate change and the market price of inflation. • The relative contribution is very important for the nature of risk. • If the real interest rate is constant and nominal rates change with inflation, the short term bonds are safest long term investment. >> Market Price of Risk • If the inflation is constant and nominal rates change with the real rate, the long term bonds are safest long term investment. • Little work is done on the separation of interest rate premia between real and inflation premium components. Buraschi and Jiltsov(1999) is one recent effort. >> • Thanks!