Lecture 3 - Advanced Fixed Income Analytics Lecture 3...

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Unformatted text preview: Advanced Fixed Income Analytics Lecture 3 Backus & Zin April 1, 1999 Binomial Models 1 1. Flow chart 2. Rate trees 3. Contingent claims and state prices 4. Valuation 1: one period at a time 5. Valuation 2: all at once 6. Models: Ho and Lee, Black-Derman-Toy 7. Calibration of parameters 8. Examples of asset valuation 9. Eurodollar options term structure of volatility revisited 10. Summary and nal thoughts Advanced Fixed Income Analytics 3-2 1. Flow Chart Data 6 - Parameters ? Short Rate Tree ? State Prices ? Derivative Valuation ? Cash Flows We'll start in the middle how a rate tree turns into state prices and work out from there Advanced Fixed Income Analytics 3-3 2. A Short Rate Tree Consider this tree for one-period interest rates  short rates": 8.0000 9.0000 PPP 6.0000 6.0000 PPP 1.0000 PPP 5.0000 PPP 4.0000 PPP 3.0000 PP 6.0000 P 2.0000 Key feature: down,up and up,down get you to the same place the math term is lattice" Convention arbitrary, but you have to choose something: With continuous compounding, the one-period discount factor satis es b1 = exp,rh=100 r = ,100=h logb1 If data di er, convert to this basis note that b1 is convention free One-period discount factor tree h = 1=4: 0.9802 PP 0.9778 P PP 0.9851 P P 0.9851 PP 0.9876 0.9975 PPP 0.9925 0.9900 PPP PP 0.9851 P 0.9950 ie, this is the tree for b1 Advanced Fixed Income Analytics 3-4 3. States Terminology: state" means a scenario or situation state-contingent claims, or derivatives, are assets whose cash ows depend on the situation at a future date eg, option payo s depend on the future value of the underlying In binomial models, the state is the location in the tree Label the location in the tree by i; n: i = number of up moves since start n = number of periods since start Examples: i; n = 0; 0 is the initial node i; n = 3; 3 is the upper right node on the preceeding page Note the extra dimension: discount factors value cash ows at di erent dates here we distinguish by state" as well as date ie, we introduce uncertainty For valuation we need a list of states, the ordered pair i; n the cash ows associated with each state, ci; n state prices: the value of one dollar in each state, Qi; n Advanced Fixed Income Analytics 3-5 4. Recursive Valuation: Theory Apply these equations at each node i; n, starting at the end:  qu = ub1   qd = d b1 = 1 , ub1 p = c + qupu + qdpd where i + 1; n + 1 is the up" state and i; n + 1 the down" qu qd is the value of one dollar in the up down state pu pd is the price of the asset in the up down state   u d  is the risk-neutral probability of the up down state c is the asset's cash ow in i; n p is the asset's price in i; n State prices discount future cash ows: that's the role of b1 adjust for risk: the risk-neutral probabilities might be called risk-adjusted" probabilities divide the discount factor: qu + qd = b1   The 50-50 rule: set u = d = 0:5 completely arbitrary, but absolutely standard Advanced Fixed Income Analytics 5. Recursive Valuation: Examples Example 1: 3-period zero Cash ows are 100.00 PP 0.0000 P P 0.0000 PP 0.0000 100.00 P PPP 0.0000 PP PP 0.0000 P 100.00 PP 0.0000 P 100.00 Prices at the end are 100.00 PP na P PP 100.00 na P PP PP na P na P PP na P 100.00 PP na P 100.00 Find prices one period from the end: 100.00 PP 97.775 P PP na P P 100.00 PP na PP 99.750 PPP 100.00 P na PP 98.511 P 100.00 Details for boxed" node 0,2:  state prices are qu = qd = 0:9851=2 = 0:4926  zero's price is p = 0 + 0:4926100 + 100 = 98:511 3-6 Advanced Fixed Income Analytics 5. Recursive Valuation: Examples continued Example 1: 3-period zero continued Find prices two periods from the end: 100.00 PP 97.775 P P 97.292 PP 99.750 100.00 P PPP na PP PP 98.144 P 100.00 PP 98.511 P 100.00 Details for boxed" node 0,1:  state prices are qu = qd = 0:9900=2 = 0:4950  zero's price is p = 0 + 0:495099:750 + 98:511 = 98:144 Find price for initial node: 100.00 PP 97.775 P P 97.292 PP 99.750 100.00 P PPP 96.504 PP PP 98.144 P 100.00 PP 98.511 P 100.00 Details for boxed" node 0,0:  state prices are qu = qd = 0:9876=2 = 0:4938  zero's price is p = 0 + 0:493897:292 + 98:144 = 96:504 3-7 Advanced Fixed Income Analytics 5. Recursive Valuation: Examples continued Example 2: 3-period 8 bond quarterly payments Cash ows are 102.00 PP 2.000 P PP 2.000 P P 102.00 PP 2.000 2.000 PPP 102.00 2.000 PPP P 2.000 P P 102.00 Price path for bond: 102.00 101.73 PPP 102.00 103.21PPP 103.75 PPP 104.36 PPP PP 104.09 P 102.00 PP 102.48 P 102.00 Details for boxed" node 1,1:  state prices are qu = qd = 0:9851=2 = 0:4926  zero's price is p = 2 + 0:4926101:73 + 103:75 = 103:21 note the cash ow of 2 here Same approach: we can value anything! 3-8 Advanced Fixed Income Analytics 3-9 5. Recursive Valuation: Examples continued Example 3: one dollar in state 2,2 pure state-contingent claim Cash ows are 0.0000 PP 1.0000 P P 0.0000 PP 0.0000 0.0000 P PPP 0.0000 PP PP 0.0000 P 0.0000 PP 0.0000 P 0.0000 Price path is: 0.0000 PP 1.0000 P P 0.4926 PP 0.0000 0.0000 P PPP 0.2432 PP PP 0.0000 P 0.0000 0.0000 PPP 0.0000 Details for initial node 0,0:  state prices are qu = qd = 0:9876=2 = 0:4938  zero's price is p = 0 + 0:49380:4926 + 0 = 0:2432 Comment: this example is a little abstract, but it turns out to be useful Advanced Fixed Income Analytics 3-10 6. All-at-Once Valuation A second approach: multiply state prices by cash ows and add State prices Qi; n for our environment are: 0.1189 PP 0.2432 P P 0.4938 PP 0.4877 0.3621 P PPP 1.0000 PP PP 0.4938 P 0.3636 PP 0.2444 P 0.1204 Comments: these are prices now for one dollar payable in the relevant state node think about this: it's not a price path initial node: a dollar now is worth a dollar the node with the box is example 3 we'll see shortly where these come from Example 1: p = 100  0:1189 + 0:3621 + 0:3636 + 0:1204 = 96:50 same answer by di erent route Discount factors and spot rates: X bn = Qi; n i X n+1 b = Qi; nb1i; n i n y = ,100=nh log bn Advanced Fixed Income Analytics 3-11 7. Computing State Prices Du e's formula: 8  d b1i; nQi; n if i = 0  d b1i; nQi; n+  ub1i , 1; nQi , 1; n if 0 qi; n + 1 = : i n+1  ub1i , 1; nQi , 1; n if i = n + 1 Comments: the idea is to compute all the state prices at once, starting at the beginning saves a lot of work on the edges the rst and third lines: price is qu qd times the current state price in the middle second line: since you can reach the node from two previous nodes, it has two components try a few steps to see how it works Chriss Black-Scholes and Beyond , ch 6 has a nice summary he calls them Arrow-Debreu prices Discount factors and spot rates: Maturity n 1 2 3 4 Discount factor bn 0.9876 0.9753 0.9650 0.9540 Spot rate yn 5.000 4.9994 4.7441 4.7111 Advanced Fixed Income Analytics 3-12 8. Models A model is a rule for generating a short rate tree once we have the tree, we know how to do the rest The Ho and Lee model: Short rate rule: rt+1 = rt + t+1 + h1=2 "t+1; with 8 +1 with probability 1 2 : ,1 with probability 1 2 h converts to annual units "t+1 = Properties:  The mean change in r is Etrt+1 , rt = t+1 + h1=2 1=21 + 1=2,1 = t+1  The variance of the change in r is 2 h1=212 + 1=2,12 i Var t rt+1 , rt  = h = h 2  A discrete approximation to Vasicek without mean reversion ' = 1 Advanced Fixed Income Analytics 3-13 8. Models continued The logarithmic model Ho and Lee in logs Tuckman calls this the original Salomon model" Let z = log r so that r = expz  : zt+1 = zt + t+1 + h1=2 "t+1 with 8 +1 with probability 1 2 "t+1 = : ,1 with probability 1 2 Comments:  log r keeps r positive  the volatility parameter applies to the rate, hence consistent with industry practice for quoting implied volatilities Advanced Fixed Income Analytics 3-14 8. Models continued The Black-Derman-Toy model logarithmic model with time-dated volatility Black-Derman-Toy model z = log r: z i; n = z 0; 0 + n X j =1 t+j + 2i , nh1=2 n Find short rates from ri; n = exp z i; n What?  Ho and Lee might be expressed as ri; n = r0; 0 + n X j =1 t+j + 2i , nh1=2  The last term: in state i; n, we have experienced i up moves over n periods. Apparently we experienced n , i down moves, too, so the total e ect of up and down moves is i , n , i = 2i , n  BDT: change to logs and make depend on time  Why is this clever? If we did this through the usual route, up,down and down,up wouldn't end up in the same place if isn't the same each period. BDT nesse this by de ning up and down relative to the mean rate in that period ie, by the horizontal di erence between rates in the tree.  Why is this useful? Because the term structure of volatility isn't at. Advanced Fixed Income Analytics 3-15 9. Choosing Parameters Choosing volatilities : estimate from recent data eg, standard deviation of changes in spot rates infer from option prices interest rate caps, eurodollar futures, swaptions Choosing drift" parameters : reproduce current spot rates exactly! remark: absolutely essential how can you value options if the spot rates are wrong? Du e's formula is extremely helpful: quick way to compute spot rates for a model, so they can be compared to the data algorithm: 1. guess 's 2. compute rate tree 3. use Du e's formula to compute spot rates 4. compare spot rates to data 5. Choose:  if spot rates in the model are the same as the data, you're done  if they're di erent, return to 1 with a new guess this is clearer if you run through it on a spreadsheet Advanced Fixed Income Analytics 3-16 9. Choosing Parameters continued Calibrating the Ho and Lee model Set h = 0:25 3-month eurodollar contracts coming up Choose = 0:5 ballpark number, more later Current spot rates are 4:969; 4:991; 5:030; 5:126; 5:166; 5:207 these match eurodollar futures prices, an issue we can explore in more depth later if you like The resulting interest rate tree is 6.665 PP 6.326 P 6.164 PPP 5.826 6.165 PPP 5.608 PPP PP PP 5.665 5.264 P 5.664 P PP PP PP 4.969 P 5.108 P 5.326 P PP P 4.764 P 5.164 PP 4.826 5.165 P PPP 4.608 PP PP 4.664 P 4.665 PP 4.326 P 4.165 State prices courtesy of Du e's formula: .0291 PP .0592 P .1201 PPP .2371 .1459 PPP .2437 PPP PP PP .4938 P .3609 P .2926 PP PP PP 1.0000 P .4877 P .3563 P PP P .4938 P .3613 PP .2380 .2933 P PPP .2440 PP PP .1206 P .1477 PP .0596 P .0295 Advanced Fixed Income Analytics 3-17 9. Choosing Parameters continued Calibrating the Ho and Lee model continued Verifying spot rates: b1 = 0:4938 + 0:4938 = 0:9877  y1 = ,100=h log b1 = 4:969 b4 = 0:0592 + 0:2371 + 0:3563 + 0:2380 + 0:0596 = 0:9375  y4 = ,100=4h log b4 = 5:126 you need more digits to reproduce this exactly Complete table of discount factors and spot rates Maturity Discount Factor Spot Rate 0.25 0.9877 4.969 0.50 0.9754 4.991 0.75 0.9630 5.030 1.00 0.9500 5.126 1.25 0.9375 5.166 1.50 0.9249 5.207 ie, the spot rates are exactly what we want Advanced Fixed Income Analytics 3-18 10. Options on Eurodollar Futures Recall: options on eurodollar futures are like options on 3-month LIBOR we saw this earlier when we examined the yields" implicit in futures prices there are subtle di erences between forward rates and futures that we'll ignore for now 3-month LIBOR  Y " is related to the 3-month discount factor  b" by Y = 400  1=b , 1 Since our tree has a 3-month time interval, the tree for Y is easily computed from the one-period discount factors: 6.721 PP 6.376 P 6.212 PPP 5.868 6.213 PPP 5.647 PPP PP PP 5.705 5.704 P 5.298 P PP PP PP 5.000 P 5.140 P 5.361 P PP P 4.792 P 5.197 PP 4.855 5.199 P PPP 4.634 PP PP 4.691 P 4.692 PP 4.349 P 4.187  node 1,1 box: b" = exp,5:264=400 = 0:98693 Y " = 400  1=b , 1 = 5:298  not much di erent from the continuously-compounded short rate, but it reminds us that interest rate conventions are important Advanced Fixed Income Analytics 3-19 10. Options on Eurodollar Futures continued Consider an option with strike K on Y in 3 months the option has cash ows of Y , K + : with K = 5 the cash ows are PP na P 0.298 PPP na PPP na PPP PP 0.000 P PP na P PP na P PP na P PP na P PP na P PP na P PP na P PP na P PP na P PP na P na na na na na na Value of option:  all-at-once method multiply cash ows by state prices and add: p = 0:49380:298 = 0:147 Advanced Fixed Income Analytics 10. Options on Eurodollar Futures continued 3-20 Term structure of volatility revisited Objective: compute volatilities for at-the-money options We need forward rates:  with a 3-month time interval h = 0:25, 3-month forward rates satisfy 1 + F n=400 = bn=bn+1  F n = 400 bn=bn+1 , 1  from the discount factors computed above, we get Maturity Discount Factor Spot Rate Forward Rate 0.25 0.9877 4.969 5.045 0.50 0.9754 4.999 5.140 0.75 0.9630 5.030 5.450 1.00 0.9500 5.126 5.360 1.25 0.9375 5.166 5.450 1.50 0.9249 5.207 5.525 the last one is based on b7, which we haven't reported  Comment: by construction, forward rates are 100 futures prices" same prices we reported last time Compute prices of at-the-money options K = F : Maturity Strike Call Price Volatility 0.25 5.045 0.1251 0.1478 0.50 5.140 0.1237 0.1185 0.75 5.450 0.1833 0.1762 1.50 5.360 0.1810 0.0654 1.25 5.450 0.2231 0.1165 1.50 5.525 0.2207 0.1025 Advanced Fixed Income Analytics 3-21 10. Options on Eurodollar Futures continued Term structure of volatility revisited continued How we got these numbers:  option prices: same approach as above  nd cash ows, multiply by state prices, and add; eg, 0:1251 = 0:49385:298 , 5:045  volatility: inputs are price above, strike forward rate, and n-period discount factor use n-period spot rate shortcut: Brenner-Subrahmanyam approximation  good learning experience: pick a speci c maturity and work through all the steps Result: 0.18 * = data, o = model Implied Volatility (Annual Percentage) 0.16 0.14 0.12 0.1 0.08 0.06 0.2 0.4 0.6 0.8 1 Maturity in Years 1.2 1.4 1.6 Advanced Fixed Income Analytics 3-22 10. Options on Eurodollar Futures continued Term structure of volatility revisited continued Comments:  bumpy!  the inevitable result of a discrete model  can be mitigated with smaller time interval  no obvious pattern to term structure of volatility if there is one, it's lost in the noise  unlike BDT, we can't choose volatilities to t current term structure of volatility Volatility smile maturity 9 months: 0.1 Implied Volatility (Annual Percentage) 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 4.5 5 5.5 Strike Price 6 6.5 Advanced Fixed Income Analytics 3-23 Summary 1. Many of the most popular xed income models are based on binomial trees: each period, you go up or down, and up,down and down,up get you to the same place 2. A model consists of a rule for generating interest rates 3. Using such a model to value a derivative asset involves the following steps: choose a model choose parameter values compute the asset's cash ows in each node of the tree this often takes some e ort value the cash ows by either i multiplying them by state prices and summing or ii computing the value recursively" one period at a time, starting at the end 4. Given a model, we can value almost anything with the same technology 5. Models di er in their functional form logs or levels? and in the exibility of their parameters BDT allows input of a volatility term structure, Ho and Lee does not | although it could! 6. For options, the discrete set of possibilities of binomial models can lead to bumpy" prices ...
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