悉尼大学固定收益证券课件Lecture 5.ppt

models of the term structure,lecture 5,dr. andrew ainsworth,finc3019 fixed income securities,last week.,the yield curve factors affecting the yield curve economic news and bond prices the term structure of interest rates spot rates forward rates theories of the term structure strips market,introduction,binomial trees multiplicative random walk mean reverting selected models of the term structure what constitutes a good model properties of different models effective duration revisited valuing a callable bond using a binomial tree sundaresan ch. 9 (excl 9.4) martellini et al. ch.12.1, ch. 14.1,introduction,the aim of the lecture is to show how we can use an interest rate model to estimate a dynamic model of the term structure in pricing bond, we need to estimate discount rates to value cash flows last week we extracted zero prices to achieve this pricing and hedging fixed income securities that pay uncertain future cash flows depends on how the term structure of interest rates evolves over time we will use an example of a callable bond this takes account of the correlation between the discount rate and the future cash flow,binomial trees,binomial trees,ruud,ruuu,rudd,rddd,period 0,period 1,period 2,period 3,binomial trees,period 0,period 1,period 2,period 3,1,3,3,1,0.125,0.125,0.375,0.375,number of paths,probability,binomial trees,the model is presented in discrete time (rather than continuous time) there must not be arbitrage opportunities we can specify a process of how interest rates of a certain maturity evolve over time (we will use 1-year rates) what are we trying to do? use the specified process estimate expected future one year rates work back through the binomial tree to calculate zero prices from these one year rates then we are able to back out the term structure from the zero prices,multiplicative random walk,in our first example we use a multiplicative random walk equal probability of moving up (q) or down (1-q) through the interest rate tree: q = 0.5 we need to specify the initial short rate this rate can increase by a factor u, or decrease by a factor d the tree is recombining so u = (1/d) an up move followed by a down move is the same as a down move followed by an up move non-recombining trees become large very quickly! for n periods a non-recombining tree has 2n nodes versus n+1 for a recombining tree e.g. 5 year bond: 32 vs 6 possible values in year 5 for a non-recombining and recombining tree, respectively,evolution of one-year interest rates,t = 0,t = 1,t = 2,t = 3,q = 0.5,1-q = 0.5,evolution of one-year interest rates,t = 0,t = 1,t = 2,t = 3,current rate = 10% up factor (u) = 1.25 down factor (d) = 0.8 q = 0.5,1-year discount bond price,lets work out the price of the 1-year discount bond,t = 0,t = 1,2-year discount bond,the tree for the 2-year zero looks as follows we need to calculate the 2-year zero price at date t=0 so we can infer the 2-year ytm,t = 0,t = 1,t = 2,2-year zero coupon bond,first we work out the zero prices in period 1 then we work back to the period 0 price, discounting and using the probabilities (q):,evolution of 2-year zero prices,t = 0,t = 1,t = 2,evolution of three-year zero prices,t = 0,t = 1,t = 2,t = 3,3-year zero coupon bond,lets work back from period 3 to get the period 2 prices of a 3-year zero,3-year zero coupon bond,lets work back to the period 1 prices,3-year spot rate,now lets discount this back to period zero to work out the 3-year zero price: from the 3-year zero price at period 0 we can calculate its ytm and we can add this top our estimate of the term structure,the term structure,ill leave period 4 for you as an exercise after running through the calculations, we get the spot curve:,mean-reverting interest rate process,mean-reverting interest rate process,we can also make the probability of an increase and decrease in interest rates depend on the current level of interest rates this is a more realistic approach as there is likely to be mean reversion in interest rates the process is additive rather than multiplicative for example, if interest rates are equal to 1.5%, the probability of an increase should be greater than the probability of a decrease in this case the probability of an up move and a down move are not necessarily equal so we need to work out a probability tree as well as interest rate tree,mean-reverting interest rate process,upper limit: lower limit: probability of an up move depends on the interest rate at time t: as does the probability of a down move: if rt = 0 there is a 100% probability that rates will increase and if rt = 2 (the maximum), there is a zero probability of an increase in rates when does q = 0.5? what is ?,mean-reverting interest rate process,we include a speed of adjustment measure (d) if this measure is large we will approach the long-term mean quickly if this measure is small we will approach the long-term mean slowly as the number of nodes in the tree approach infinity the mean approaches m the variance approaches dm/2 we can estimate these features from current market data and use them to construct the interest rate tree and the term structure,mean-reverting interest rate process,t = 0,t = 1,t = 2,an example,first we need to estimate the parameters for our model i used the rba cash rate since 1996 as an example to estimate m and d then we can calculate the short rate tree determine probability tree we know this varies at each node then proceed as we did previously to estimate the term structure,evolution of one-year interest rates,t = 0,t = 1,t = 2,current rate = 4% d = 0.37% m = 5.35%,probability of up moves,t = 0,t = 1,t = 2,the probability of an up move from t = 0 is:,calculating zero coupon prices,lets work out the 2 year zero price at t = 1:,selected models of the term structure,selected models of the term structure,the binomial tree processes can also be described: or with a long-term mean not equal to zero: is the change in the short-term interest rate is the change in time is the expected change in st interest rate per time period is the std dev of absolute change in st interest rate is an independent bernoulli distributed variables that has 50% chance of being either -1 or +1,selected models of the term structure,can use the natural logarithm of interest rates this captures relative changes in interest rates (i.e. percentage change) one benefit of this approach is that interest rates cannot become negative,selected models of the term structure,there are a wide variety of interest rate models available to researchers and practitioners we will briefly cover what constitutes a good model there are a few models that i would like you to have an understanding of their basic features: vasicek cox, ingersoll and ross ho and lee,attributes of a good term structure model,rogers (1995) provides criteria to select a suitable term structure model: flexible enough to cover most situations arising in practice capture various shapes of the yield curve realistic, in that the model wont do silly things no negative interest rates simple enough that one can compute answers in reasonable time well-specified, in that required inputs can be observed or estimated a good fit of the model to the data an equilibrium derivation of the model no arbitrage opportunities,a general description of interest rate models,these are continuous-time models limit of previous discrete time equation when t goes to zero stochastic differential equation: the first term is the expected value of the instantaneous change in the interest rate the second term is its standard deviation dw has zero mean and variance of dt - wiener process,the vasicek (1977) model,vasicek (1977) mean-reverting interest rates deterministic component what happens if m = r? random component can be negative or positive m is the long-run mean of the short-term interest rate s2 is the variance k is the speed of adjustment to the long-term mean volatility of short rates is greater than longer-term rates short-term rates can become negative,cox, ingersoll and ross (1985) model,cir: similar to vasicek if st rate above mean it gets pulled down random walk if m = r variance of changes in interest rates are proportional to the level of interest rates in the random component, square root of r reduces volatility when interest rates are high interest rates are always positive what happens if interest rates approach zero? difficult to implement model,arbitrage models,these type of models are calibrated to market data ho and lee (1986) is calibrate to market prices in the ho and lee model of the term structure, interest rates are not mean reverting can become negative sundaresan describes the black, derman and toy (1990) in some detail for those that are interested this model reflects both market prices and volatilities,other term structure models,there are many other more complicated term structure models that are more frequently employed in practice these include: multi-factor models we have focused only on one factor models (the short rate is that factor) other models control for the slope of the yield curve and volatility non-recombining trees trinomial lattices ill leave the study of these models to those that are interested hull is a good text to start with.,other uses for binomial trees and effective duration - revisited,binomial trees. again,we can use the binomial tree to calculate the price of an option-free coupon-paying bond importantly, now we know how to construct a binomial tree we can price a bond with an embedded option first well can calculate the price of a callable bond then well measure the effective duration of the callable bond the lecture on duration assumed that cash flows do not change when interest rates change we can calculate effective duration to examine how the price of bonds with embedded options change in response to a change in yield this involves setting up a binomial interest rate tree to determine whether the bond issuer should call the bond,if a 3-year callable bond has a coupon rate of 6% p.a. it will be called when interest rates fall below 6% as the company can re-issue debt at a cheaper rate,pricing a callable bond,t = 0,t = 1,t = 2,pricing a callable bond,if the bond has a par value of $100 and is callable at $100 we can use the binomial tree to determine the price of the callable bond to do this we discount the face value and coupon back through the tree checking to see if it is optimal to call the bond at any point (i.e. exercise the option) if the discounted price at given node plus the coupon payed at that node is greater than the call price plus the coupon then the bond is called as it is cheaper to refinance we assume q=0.5 the interest rate lattice is given in the previous slide,if the bond exists at maturity it will pay face value plus coupon we work back from t=3 to t=2 discounting at the relevant rate in the tree,pricing a callable bond,t = 0,t = 2,t = 3,t = 1,pricing a callable bond,to determine the price at the upper node at t=2: therefore it is not optimal to exercise at this node at the lower node in period 2, the bond price plus the coupon (106.952) is greater than 106 so the bond is called and the value at this node is 106 continuing this process we get a price of 99.572 at t=0,pricing a callable bond,given the callable bonds market price we can use the simulated price from the tree to estimate the option adjusted spread (oas) note: the market price does not usually equal the theoretical price the oas represents the value of the embedded option and is added to the discount rate to force the present value from the tree to equal the market price if the market price in our example is $99 then we need to a percentage to each rate in the interest rate tree to force our theoretical price to equal 99 we can use solver in excel to do this it turns out the oas is 0.16% so