PowerPoint Version of Binomial Tree Pricing Notes
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Bonds with embedded options UNIVERSITE D’AUVERGNE Drake DRAKE UNIVERSITY Callable Bonds UNIVERSITE D’AUVERGNE Drake Drake University A problem with traditional pricing is that it ignores options imbedded in the bond such as a call option. The call feature increases reinvestment rate risk since the bond will be called if rates are low, especially if lower than the coupon rate. As the yield decreases the price increase lessens because there is a higher probability of the bond being called (price compression) Valuing a Callable Bond UNIVERSITE D’AUVERGNE Drake Drake University You can think of a callable bond as having two components: A noncallable bond and a call option. The price of the callable bond would then be equal to the noncallable bond price minus the call option price. Valuation Model Review UNIVERSITE D’AUVERGNE Drake Drake University Remember from before that the appropriate rate to use is not a single rate, but the zero spot rate or the forward rates (example on next slide) The value of the callable bond will be tied directly to the volatility of interest rates. To price the bond we will use a binomial tree model. 5.25% coupon bond, 3 years to maturity, yearly payments UNIVERSITE D’AUVERGNE Drake Drake University Assume you have the following observed yield curve, spot rates, and forward rates. Maturity Rate 1 yr 2 yr 3 yr YTM Market Value Spot Rate 3.5% 4.0% 4.5% 5.25 100 3.5% 100 4.01% 4.531% 5.25 100 105.25 2 3 102.075 Forward 3.5% 4.5225% 5.5792% (1.035) (1.0401) (1.04531) 5.25 5.25 105.25 102.075 (1.035) (1.035)(1.04523) (1.035)(1.04523)(1.055792) Binomial Tree Model UNIVERSITE D’AUVERGNE Drake Drake University We are going to represent the two possible paths of interest rates in a tree structure. Let each time be denoted as a decision Node N with a subscript denoting whether it represent the higher or lower interest rate environment. The current level of interest rates is r*, it may increase to state H or decrease to state L If the level of interest rates increases to H the bond will have a value of VH in the next period. Likewise if rates decrease to L the value will be Binomial Tree Model UNIVERSITE D’AUVERGNE Drake Drake University l VH NH V l r* N l VL NL Time 0 Time 1 Value at Time 0 UNIVERSITE D’AUVERGNE Drake Drake University The value of the bond at time 0 can be calculated assuming that there is an equal chance of obtaining either state. The expected value at time 0 is then equal to the PV of total amount you would receive in each state multiplied by the probability of the state (in this case 1/2) The total amount you would receive is the value or the bond plus any other cash flows received (For example the coupon payment) Binomial Tree Model UNIVERSITE D’AUVERGNE Drake Drake University VH l C NH H V0 l r* N 1 VH CH VL CL V0 2 1 r * 1 r * Time 0 VL l NL CL Time 1 Simple Example UNIVERSITE D’AUVERGNE Drake Drake University Assume that the current level of interest rates is 3.5% as in our previous example. If the higher interest rate price is $98 with a coupon of $5 and the lower interest rate price is $102 with the same $5 of coupon the value at time 0 would be 1 98 5 102 5 101.449 2 1.035 1.035 Binomial Tree Model continued UNIVERSITE D’AUVERGNE Drake Drake University Starting today we want to think about the future path of interest rates. Start with the time 0 (today) and think about the future path of interest rates. We will assume that over the next year there are two possible outcomes for the one year rate at time 1. Let the lower rate be r1L and the higher rate be r1H Binomial Tree Method continued UNIVERSITE D’AUVERGNE Drake Drake University Assuming the lower rate in the next period the higher rate can be found from the equation. r1H=e2sr1L where: e = natural logarithm 2.71828 (x = lny y=ex) For example let r1L=4.5% and s=10% then r1H = .045e2(.10)=.054963 Binomial Tree Model UNIVERSITE D’AUVERGNE Drake Drake University VH l CH NH r1H=.05496 V0 l r* N Time 0 VL l CL NL r =.045 1L Time 1 Extending the model UNIVERSITE D’AUVERGNE Drake Drake University It is easy to extend the model to add a second year. From each of the decision nodes NH and NL you can just repeat the same tree. UNIVERSITE D’AUVERGNE Binomial Tree Model Drake Drake University V0 l N r=.035 Time 0 VH C l H r1H=.05496 NH VL l NL CL r1L=.045 Time 1 VHH l CHH NHH r2HH VHL l CHL NHL r2HL VLL l CLL NLL r2LL Time 2 The Two Year Bond UNIVERSITE D’AUVERGNE Drake Drake University Using the observed yield curve from before, the two year bond would have a 4% coupon rate implying $4 coupon payments each year. At maturity the bond will have a value of $100 Substitute the value in for VHH, VHL,and VLL. Let the coupon be $4 in each period. Binomial Tree Model UNIVERSITE D’AUVERGNE Drake Drake University V0 l N r=.035 Time 0 VH C =4 l H r1H=.05496 NH VL l NL CL = 4 r1L=.045 Time 1 VHH = 100 l CHH = 4 NHH r2HH VHL = 100 l CHL = 4 NHL r2HL VLL= 100 l CLL= 4 NLL r2LL Time 2 Finding VH and VL UNIVERSITE D’AUVERGNE Drake Drake University The values of the bond can be found at time 1 by applying the earlier formula 1 VH CH VL CL V0 2 1 r * 1 r * 1 VHH CHH VHL CHL 1 100 4 100 4 VH 98.582 2 1 r1H 1 r1H 2 1.05496 1.05496 1 VLL CLL VHL CHL 1 100 4 100 4 VL 99.522 2 1 r1L 1 r1L 2 1.045 1.045 Binomial Tree Model UNIVERSITE D’AUVERGNE Drake Drake University V0 l N r=.035 Time 0 VH=98.852 C =4 l H r1H=.05496 NH VL=99.522 l NL CL = 4 r1L=.045 Time 1 VHH = 100 l CHH = 4 NHH r2HH VHL = 100 l CHL = 4 NHL r2HL VLL= 100 l CLL= 4 NLL r2LL Time 2 Iterative Procedure UNIVERSITE D’AUVERGNE Drake Drake University We assumed an interest rate of 4.5% for r1L this is the correct rate IF V0 can be found given the current values it the tree and V0 is equal to the market price of 100 1 V CH VL CL 1 98.582 4 99.522 4 V0 H 99.567 2 1 r * 1 r * 2 1.035 1.035 Since the price from the tree is too low, the rate r1L must be lower to increase the price. Try a new price and repeat until the correct price 4.074% is found Iterative procedure UNIVERSITE D’AUVERGNE Drake Drake University Given the rate of 4.074 the expected value of the two possible changes in interest rates is equal to the current value, in other words it is “fairly priced.” The change in rates requires finding new values for r1H and for VH and VL Binomial Tree Model UNIVERSITE D’AUVERGNE Drake Drake University V0 l N r=.035 Time 0 VH=99.071 C =4 l H r1H=.04976 NH VL=99.929 l NL CL = 4 r1L=.04074 Time 1 VHH = 100 l CHH = 4 NHH r2HH VHL = 100 l CHL = 4 NHL r2HL VLL= 100 l CLL= 4 NLL r2LL Time 2 Interpretations UNIVERSITE D’AUVERGNE Drake Drake University r1H and r1Lare a set of forward rates from time 1 to time 2 or 1f1 as it was previously called. Notice that since the change in rates makes a difference in the value of the bond, for each forward rate we will have a different value of the bond. The next step, time 3 UNIVERSITE D’AUVERGNE Drake Drake University The model could be extended again to include the next year. The YTM for the three year treasury was 4.5% so the coupons at every time period become 4.50. The goal is to find a value for r2LL that will allow us to move from right to left through the tree to produce a value of 100 again at V0 r2HL=r2LLe2s as before and r2HH=r2HLe2sr2LLe4s the correct rate is then 4.53% Binomial Tree Model UNIVERSITE D’AUVERGNE Drake Drake University V0 l N r=.035 Time 0 VH =98.074 C = 4.50 l H r1H=.04976 NH VL=99.926 l NL CL = 4.50 r1L=.04074 Time 1 VHH = 97.886 l CHH = 4.50 NHH r2HH=.06757 VHL = 99.022 l CHL = 4.50 NHL r2HL =.05532 VLL= 100 l CLL= 4.50 NLL r2LL=.0453 Time 2 Valuing an Option Free Bond UNIVERSITE D’AUVERGNE Drake Drake University The rates in the binomial tree now represent the correct rates for the on the run treasury yield curve that we started with. It is now possible to use it to value the three year 5.25% coupon bond. Starting with year three the values VHH, VHL, and VLL can be found then we can work right to left through the tree Binomial Tree Model UNIVERSITE D’AUVERGNE Drake Drake University V0 l N r=.035 Time 0 VH =99.461 C = 5.25 l H r1H=.04976 NH VL=101.333 l NL CL = 5.25 r1L=.04074 Time 1 VHH = 98.588 l CHH = 5.25 NHH r2HH=.06757 VHL = 99.732 l CHL = 5.25 NHL r2HL =.05532 VLL= 100.689 l CLL= 5.25 NLL r2LL=.0453 Time 2 Value of the bond UNIVERSITE D’AUVERGNE Drake Drake University 1 VH C H VL CL V0 2 1 r * 1 r * 1 99.461 5.25 101.333 5.25 102.075 2 1.035 1.035 The same value as we calculated before!!! Valuing a Call Option UNIVERSITE D’AUVERGNE Drake Drake University Assume that the the bond can be called at the end of the first year or later for a call price of $100. If the value at a node is greater than $100 then the bond will be called (the yield is less than the coupon) and the firm can refinance at a lower rate. Starting on the right, if the value exceeds 100 it needs to be replaced, then the tree is worked right to left again. Binomial Tree Model UNIVERSITE D’AUVERGNE Drake Drake University V0 l N r=.035 Time 0 VH =99.461 C = 5.25 l H r1H=.04976 NH VL=101.001 l NL VL=100 CL = 5.25 r1L=.04074 Time 1 VHH = 98.588 l CHH = 5.25 NHH r2HH=.06757 VHL = 99.732 l CHL = 5.25 NHL r2HL =.05532 VLL= 100.689 VLL =100 l C = 5.25 LL NLL r2LL=.0453 Time 2 Value of callable bond UNIVERSITE D’AUVERGNE Drake Drake University 1 VH C H VL C L V0 2 1 r * 1 r * 1 99.461 5.25 100 5.25 101.4302 2 1.035 1.035 The value has decreased because of the call option The value of the call option is then 102.075-101.4302=0.6448 Put option UNIVERSITE D’AUVERGNE Drake Drake University The same model could be used to value a put option. Now, you look at increases in rates that lower the price. If the value of the bond at the node is less than the puttable value then the option would be exercised and the value of the bond becomes the put value. Assume that our bond has a put option after year one with the puttable value being $100 Binomial Tree Model (Put) UNIVERSITE D’AUVERGNE Drake Drake University V0 l N r=.035 Time 0 VH =100.261 C = 5.25 l H r1H=.04976 NH VL=101.461 l NL CL = 5.25 r1L=.04074 Time 1 VHH = 98.588 VHH=100 l C = 5.25 NHH HH r2HH=.06757 VHL = 99.732 l VHL=100 NHL CHL = 5.25 r2HL =.05532 VLL= 100.689 l CLL= 5.25 NLL r =.0453 2LL Time 2 Value of puttable bond UNIVERSITE D’AUVERGNE Drake Drake University 1 VH C H VL C L V0 2 1 r * 1 r * 1 100.261 5.25 101.461 5.25 102.523 2 1.035 1.035 The value has increased because of the put option. The value of the put option is 102.075 – 102.523= -0.448 Modeling Risk UNIVERSITE D’AUVERGNE Drake Drake University Modeling risk is the risk that the valuation model has produced an incorrect result due to assumptions used in the model. Higher volatility lowers the value of the call option (raises value of put) Lower volatility raises the value of the a call option (lower the value of a put) Option Adjusted Spread UNIVERSITE D’AUVERGNE Drake Drake University The constant spread that when added to all the forward rates in the binomial tree will make the theoretical value equal to the market price. OAS Intuition UNIVERSITE D’AUVERGNE Drake Drake University Converts the difference between the valuation and the market price into a spread measure. The key is the inputs in the model If the tree uses the treasury spot curve, the OAS represents the richness or cheapness of the security plus a credit spread If the tree uses issuer’s spot rate curve, then the credit risk is already incorporated. OAS and total yield spreads UNIVERSITE D’AUVERGNE Drake Drake University The OAS is attempting to separate the amount of the nominal spread that is the result of option risk. Therefore it reports a spread that is adjusted for the option. Example: Assume you have calculated the OAS of a BBB callable corporate bond compared to non callable treasuries to be 120 Bp. would imply that the BBB pays 120 Bp more because of the liquidity and credit risk etc. the spread has removed the portion of the spread attributable to the option. OAS and benchmarks UNIVERSITE D’AUVERGNE Drake Drake University In the previous example the OAS was a representation of credit and liquidity risks. If instead of using the on the run treasury as a benchmark we used the on the run issues for the same issuer (the issuer of the BBB). Then credit risk is also not part of the spread, only liquidity and other factors. OAS is the spread after adjusting for options, what it actually represents however depends upon the benchmark being used… OAS in our example UNIVERSITE D’AUVERGNE Drake Drake University Assume that the 5.25% callable three year coupon bond is currently selling for $101.17 Previously we found the price to be $101.4302 The OAS would be the additional yield added to the binomial interest rate tree at every yield that produced a value for the bond of $101.17, in this case the OAS is 45 Bp Binomial Tree Model UNIVERSITE D’AUVERGNE Drake Drake University V0 l N r=.035 Time 0 VH =99.833 C = 5.25 l H r1H=.005426 NH VL=100.6946 l NL VL=100 CL = 5.25 r1L=.04524 Time 1 VHH = 102.6309 VHH = 100 l CHH = 5.25 NHH r =.07208 2HH VHL = 103.817 VHL = 100 l C = 5.25 HL NHL r2HL =.0598 VLL= 104.809 VLL =100 l C = 5.25 LL NLL r2LL=.0498 Time 2 UNIVERSITE D’AUVERGNE Value of callable bond (with OAS) Drake Drake University 1 VH C H VL C L V0 2 1 r * 1 r * 1 99.83306 5.25 100 5.25 101.1703 2 1.035 1.035 Funding Cost as a Benchmark UNIVERSITE D’AUVERGNE Drake Drake University Often the on the run issues of the LIBOR is used as the benchmark. LIBOR is used as a benchmark borrowing rate that the institution pays to obtain funds. It can then compare its cost of funding to LIBOR by looking at the spread above LIBOR it pays to obtain funds. As long as the assets spread relative to LIBOR is greater than the spread the institution must pay to obtain funding, it is covering the funding cost. Effective Duration and Convexity UNIVERSITE D’AUVERGNE Drake Drake University Effective Duration (option adjusted duration) allows a yield change to change the expected future cash flows. Quick approximation of duration and convexity UNIVERSITE D’AUVERGNE Drake Drake University P- = the price if yield is decreased by x Bp P+ = the price if yield is increased by x Bp P0 = the initial price Dy=change in rate (x Bp in dec form) P P duration - 2(P0 )( Dy) P P 2( P0 ) convexity 2 (P0 )( Dy) Calculating P+ and P- UNIVERSITE D’AUVERGNE Drake Drake University Calculate the OAS 2) Shift the on the run yield curve by a small basis points 3) Construct a binomial interest rate tree based on the new yield curve 4) To each of the short rates add the OAS to adjust the tree 5) Use the adjusted tree to determine the value of P. 1) Valuing a Step Up Callable Note UNIVERSITE D’AUVERGNE Drake Drake University The Binomial model can be expanded to cover other types of options. One possibility is a note whose coupon rte changes over the life of the note. In this case the coupon rate may increase in the future. Initially, the procedure is the same as before. After developing the interest rate tree, the bond is valued using the coupon rates that correspond to what the bond will pay. Valuing a Floating Rate Note UNIVERSITE D’AUVERGNE Drake Drake University On a floating rate bond the payment at the end of the year is determined by the rate at the beginning of the year. Therefore the coupon payment for each node will be based off of the interest rate for that node (the rate at the beginning of the period determines the coupon at the end of the period). The valuation therefore uses the coupon for that node as the payment at the next point in A Capped Floater UNIVERSITE D’AUVERGNE Drake Drake University If the floating rate bond has a cap, then whenever the coupon is above the cap the value of the coupon will be based off of the cap. Analysis of Convertible Bonds UNIVERSITE D’AUVERGNE Drake Drake University A convertible bond can allows the holder to convert the bond into a predetermined number of shares of common stock of the issuer. It may also be callable and puttable. Exchangeable securities allow conversion to the stock of another firm. The conversion privilege may extend over the entire life of the of the issue or a portion of the issue Conversion Ratio UNIVERSITE D’AUVERGNE Drake Drake University The conversion ratio is the number of shares the holder will receive if the conversion option is exercised. Assume that the conversion ratio is 25, this would imply that you would receive 25 shares for each $1,000 of par value. The conversion price is the price per share implied by the conversion ratio. In the example above this would be $1000/25 = $40 If not issued at par, the conversion price is found by dividing the issue price per 1000 by the conversion Other embedded options UNIVERSITE D’AUVERGNE Drake Drake University Often the convertible is also callable, usually with a non callable period at the beginning of the period of the issue The issue may also be puttable. Hard Put – the issuer must redeem with cash Soft Put – the issuer may use common stock, cash, or subordinated notes, or a combination of the three Minimum Value UNIVERSITE D’AUVERGNE Drake Drake University The conversion (or parity) value is the value of the security if it is converted immediately. The minimum value is then the greater of 1) the conversion price and 2) its value as a security without the conversion option (also called the straight value or investment value). If the security does not sell a the greater of the two, then there are arbitrage possibilities. Market Conversion Price UNIVERSITE D’AUVERGNE Drake Drake University If the convertible bond is bought then converted immediately into stock, the buyer is effectively paying a share price based on the value of the security. The market conversion price is marketprice of market convertible security conversion conversionratio price Market Conversion Price UNIVERSITE D’AUVERGNE Drake Drake University If the actual market price increases above the market conversion price the value of the convertible bond should increase by the same percentage. Buying the convertible bond rather than the underlying stock results in basically paying a premium for the stock an can be expressed as a ratio based on the market price. Why pay a premium? UNIVERSITE D’AUVERGNE Drake Drake University The investor may be willing to pay the premium if there is an expectation of receiving a higher current income from the coupon payments than from possible dividends on the stock. One way to address this is looking at the amount of time it takes to recover the premium paid (ignoring the time value of money) Premium Payback Period UNIVERSITE D’AUVERGNE Drake Drake University The amount of time to recover the premium is marketconversion premuin premiumper share payback favorableincome period differential per share marketconversionprice - current marketprice coponinterest- (conversion ratio x commonstock dividend per share) conversionratio Downside Risk UNIVERSITE D’AUVERGNE Drake Drake University It is often assumed that the price of the convertible security cannot fall below the straight value, therefore some participants look at the ratio of the market price to the straight value as a measure of downside risk. However the straight value changes as interest rates change so this does not truly provide a good measure of downside risk . Up side potential UNIVERSITE D’AUVERGNE Drake Drake University The up side depends upon the valuation of the common stock and the potential for gain the stock price. If the straight value is significantly higher than the value implied by the common stock price then it is referred to as a fixed income equivalent or busted convertible. If the conversion value from the stock price is higher than the straight value it is a common stock equivalent. Option based value UNIVERSITE D’AUVERGNE Drake Drake University We have ignored the true value of the security The convertible security value should equal the straight value plus the value of a call option of the stock of the firm. IF the bond has a call feature then the value becomes: the straight value plus the value of the cal option on the stock minus the value of the call option on the bond.
