VIEWS: 27 PAGES: 14 POSTED ON: 4/27/2010
Black, Derman & Toy Given: Maturity Yield Volatility 1 0.1 0.2 2 0.11 0.19 We take the market parameters as sacrosanct--we'll build a tree to match t 3 0.12 0.18 Volatilities are for the annualized discount rates to this node. 4 0.125 0.17 It's important to know what these volatilities mean. They refer to the possib 5 0.13 0.16 the security of that term. So only the 2-year case corresponds to the sh Assume: The short rate (I.e., the yield on a one-year T-Bill) follows a binomial process. In the spirit of the binomial model, the difference between the nodes in a time period is determined by the standa Use discrete-time compounding/discounting in the tree. So, the price of a 1-Year T-Bill is: 90.909091 Implying the tree: (What are the 2 possible short rates in 1 year?) Price of the 2-Yr Bill: 81.162243 What we know: From sigma(2), the ratio (of the logs) of the rates. The rate at which we discount the time-1 values is 10%. We therefore have 2 equations in 2 unknow Let's use solver to solve for these 2 values: ru 0.143181 Equation 1: 0.19 rd 0.097916 Equation 2: 81.16224 Price Tree for 2 - Year Zero This solves for the second time period of the short-rate tree, using informat 100 The one-year short rate is 10%, and in 1 year there is a 50% probability tha 87.47525 This allows us to solve for the price of the note one year from now in the up 81.16224332 100 91.08168 This allows us to solve for the price of the note one year from now in the do 100 Note: All of these trees are in the "Risk-Neutral World." There is not enoug nodes and the probabilities, so we set the up and down probabi To solve for the 3rd time in the tree, we also need to use the restriction that the tree recombines. (This is consistent with the as Price of the 3-Yr Bill: 71.178025 ru 0.194187 Equation 1: 0.18 rd 0.0976 Equation 2: 71.17802 rud = rdu 0.137669 Price Tree for 3 - Year Zero Short rate tree (out 3 years) 100 83.73897 0.194187 75.07039 0.143181 100 71.17802478 87.899054 0.1 0.137669 81.52126 100 0.097916 91.107872 0.0976 100 Present Value Tree for 3 - Year Coupon Bond (cum- Interest). 110 102.11287 101.325 110 95.50296049 106.68896 108.7816 110 110.21866 110 Price Tree for 3 - Year Coupon Bond (less accrued Interest). 100 92.112867 91.32496 100 95.50296049 96.68896 98.78155 100 100.21866 100 Price Tree for 3 - Year Coupon Bond Call Option. 0 0.738711 1.765681248 1.6889597 3.145788 5.2186595 Hedge Ratios Call: 0.322812 Put: -0.169348 Price of the 4-Yr Bill: 62.429508 ru 0.217888 Equation 1: 0.17 rd 0.087172 Equation 2: 62.42951 Price Tree for 4 - Year Zero Short rate tree (out 4 years) 100 82.109384 70.456006 100 0.194187 64.56798 86.165925 0.143181 62.42950795 77.169704 100 0.1 0.137669 72.77694 89.421184 0.097916 82.63617 100 0.0976 91.981736 100 Price of the 5-Yr Bill: 54.275994 ru 0.255246 Equation 1: 0.16 rd 0.086534 Equation 2: 54.27599 Price Tree for 5 - Year Zero Short rate tree (out 100 79.66567 67.068577 100 58.883897 83.69831 55.0976 73.568207 100 0.143181 54.27599488 67.089119 87.06109 0.1 64.30958 79.08217 100 0.097916 74.123873 89.8146 83.634555 100 92.03575 100 we'll build a tree to match them. to this node. n. They refer to the possible change over the next year in the ytm on ase corresponds to the short-rate. s determined by the standard deviation: . 5 × ln( rU / rD ) st = Dt ve 2 equations in 2 unknowns ru and rd Targets Criterion 0.19 3.78076E-12 Minimize the criterion cell by changing e26 and e27 81.16224 subject to the constraint that e26 >= e27 rt-rate tree, using information in today's yield curve. re is a 50% probability that the 1-year rate will be 14.32% and 50% probability that it will be 9.79%. ne year from now in the up-state ne year from now in the down-state World." There is not enough information to separately identify the t the up and down probabilities equal to 50%. his is consistent with the assumption that the volatility at this point does not depend on which node we're in.) Targets Criterion 0.18 3.59311E-19 Minimize the criterion cell by changing e45 and e46 71.17802 subject to the constraint that e45 >= e46 out 3 years) 0.750703943 The volatility of the 3-year zero 1.332083053 refers to the range between the up and down 1.154159024 0.154159 states in the next period. Demonstration: 75.07039 Thus, the L54 cell contains the 2-period discount factor for a three-year Zero in the up-state after 1 period. 0.815212602 1.226673873 By analogy: 81.52126 1.107553102 0.107553 Price Tree for 3 - Year Coupon Bond Put Option. 2.887133 1.262763 0.573983306 0 0 0 Targets Criterion 0.17 7.37718E-14 62.42951 out 4 years) 0.217888 0.64568 1.548755 1.156985 0.156985 0.160552 The distance here is dictated by the implied volatility of the 4-year Zero. 0.727769 0.118303 1.374062 1.111737 0.111737 0.087172 Targets Criterion 0.16 3.80097E-12 54.27599 ort rate tree (out 5 years) 0.255246 0.550976 1.814961 0.217887587 1.160692 0.160692 0.194187 0.194767 0.160551578 0.137669 0.148619 0.643096 1.554978 1.116686 0.116686 0.118303247 0.0976 0.113405 0.08717235 0.086534 ctor for a three-year Zero The BDT model assumes that the short-rate process follows a mean-reverting geometric Brownian motion. The ability to fit the yield curve exactly by allowing the long-run mean to vary as we move through time. The volatility fits the zero-coupon bond volatility at each maturity. (These may be obtained from options markets.) The volatility equation is derived as follows: r0 × es Dt This interest rate in the up-state is rU r0 r0 × e - s Dt This interest rate in the up-state is rD In BDT, our volatility equation solves for sigma, as a function of Dt, r0, rU, and rD. Here's the derivation: rU es Dt = rD e -s Dt Now recall that exponents are additive when the arguments are multiplied (subtractive when divided): es Dt = e 2s Dt e -s Dt Thus (taking the log of both sides:) ær ö ln ç U çr ÷ = 2s ÷ Dt è D ø Isolating sigma: 1 æ rU ö Note that everything in this derivation is true, independent of the prob ln ç ÷ This is an important point that suggests that the volatility and nodes 2 ç rD è ÷ ø or the equivalent risk-neutral world. s = Another way of noting this is by seeing that we do not have to know th with the nodes / volatilities. Dt This is the same reason that under the Black-Scholes model, the imp the real world. wnian motion. e move through time. obtained from options markets.) n is true, independent of the probability of going up or down. gests that the volatility and nodes are the same whether we are in the real world ng that we do not have to know the expected interest rate next period to work he Black-Scholes model, the implied volatility is the volatility of the stock in