Submitted for possible presentation at the 2018 FMA Meeting

Transcription

1 Skewed Interest Rate Patterns and a Skewness-Adjusted Black-Derman-Toy Calibration Model R. Stafford Johnson (Contact Author) Department of Finance Xavier University 108 Smith Hall 3800 Victory Parkway Cincinnati, Ohio (Mobile) JohnsonS@xavier.edu Amit Sen Department of Economics Xavier University 38 Smith Hall 3800 Victory Parkway Cincinnati, Ohio Sen@Xavier.edu Submitted for possible presentation at the 018 FMA Meeting 1

2 Highlights Skewed Interest Rate Patterns and a Skewness-Adjusted Black-Derman-Toy Calibration Model In this paper, we present empirical evidence that points to the presence of skewness in interest rates, especially under increasing and decreasing interest rate periods. We next show how the Johnson, Pawlukiewicz, and Mehta skewness-adjusted option pricing model can be used to calibrate a binomial tree to an increasing-rate scenario in which the end-of-the period distribution is characterized by a positive expected logarithmic returns and negative skewness and to decreasing rate case that is characterized by a distribution with a negative expected logarithmic return and positive skewness. We then show how the implied yield curves and implied forward rates generated from the skewness-adjusted binomial interest rate model result in yield curves that are consistent with expected end-of-the-period distributions. Finally, we conclude the paper by showing how skewness can be incorporated into the Black, Derman, and Toy calibration model and showing with a numerical simulation the arbitrage that occurs when the BDT variability conditions are not adjusted to reflect skewness.

3 Abstract The Black, Derman, and Toy (BDT) [1990] calibration model generates a binomial interest rate tree synchronized to the current spot yield curve. As a result, the model generates values for option-free bonds that are equal to their equilibrium prices, thus making the model arbitrage free. Given this feature, the model is readily extendable to valuing bonds with embedded options or bond and interest rate derivatives that are also arbitrage free. This arbitrage-free feature of the calibration model is one of the main reasons that many practitioners favor this model over equilibrium models that generate trees by estimating the up and down parameters based on the mean and variability of the underlying spot rate. The variability condition governing the upper and lower spot rates in the BDT calibration model, however, assumes that the interest rate s logarithmic return is normally distributed. Several empirical studies have provided evidence that the distributions of many securities and indexes exhibit persistent skewness. In this paper, we provide empirical evidence showing that increasing or decreasing interest rate trends are often characterized by end-of-the-period distributions that are skewed. Using a 3-period tree, we next show that when an equilibrium binomial interest rate model is adjusted for skewness such that its end-of-the-period binomial distribution is calibrated to a skewed distribution, the implied spot yield curves generated from the adjusted model will be consistent with interest rate expectation where spot yield curves have a tendency to be positively (negatively) sloped when the market expects interest rate to increase (decrease). Finally, we show that when the BDT model is calibrated to a positively or negatively-sloped yield curve that matches the implied yield curves of a skewness-adjusted equilibrium model, it loses its arbitrage-free feature if the model s variability condition is not adjusted for skewness. The paper contributes to the literature on binomial interest rate modeling by providing empirical evidence of the significance of skewness when interest rates are increasing or decreasing, showing the consistency of an end-of-the period distribution and the implied yield curves for expected increasing and decreasing interest rate scenarios using a skewness-adjusted equilibrium model, and showing how the BDT model can be adjusted to reflect skewness in order to maintain its arbitrage-free feature. 3

4 1. Introduction Skewed Interest Rate Patterns and a Skewness-Adjusted Black-Derman-Toy Calibration Model Two approaches to modeling stochastic interest rate movements using a binomial model are the equilibrium and the calibration models. The equilibrium model solves for the up (u) and down (d) parameter that make the mean and variance of the distribution of the logarithmic return equal to their empirical values (see Rendelman and Bartter [1980], Cox, Ingersoll, and Ross [1981], and Cox, Ross, and Rubinstein [1979]). The bond values obtained using estimated u and d parameter are not likely to be equal to the bond s equilibrium prices (prices obtained by discounting the cash flow by spot rates). As a result, additional assumptions regarding risk premiums must be made to explain market prices. Since the work of Ho and Lee [1986], it has been widely recognized that binomial interest rate models must possess the no-arbitrage property. The second approach to modeling interest rate movements is to calibrate the tree to the current spot yield curve (see Black, Derman, and Toy [1990], Black and Karasinski [1991], and Heath, Jarrow, and Morton [008]). The well-known Black, Derman, and Toy (BDT) calibration method generates a binomial tree that is consistent with an estimated relationship between the volatility of the upper and lower spot rates and yields option-free bond values that reflect the current term structure. Since the resulting binomial tree is synchronized with current spot rates, this model yields values for optionfree bonds that are equal to their equilibrium prices. Given this feature, the model can readily be extended to valuing bonds with embedded options or bond and interest rate derivatives. The BDT calibration model thus has the property that if the assumption regarding the evolution of interest rates is correct, then the model s bond price and derivative values are supported by arbitrage arguments. Accordingly, calibration models are often referred to as arbitrage-free models. This arbitrage-free feature of the calibration model is one of the main reasons that many practitioners favor this model over the equilibrium model. The equilibrium and arbitrage-free calibration models assume that the interest rate s logarithmic return is normally distributed. Several empirical studies have provided evidence that the distributions of stock, stock index, and foreign currency returns exhibit persistent skewness (for example, Koedijk et al. [199], Kon [1990], Aggarwal and Rao [1990], and Turner and Weigel (199)). There have also been a number of studies that have examined the return distributions of stocks, indexes, and currencies and its implications on option pricing models (Jarrow and Rudd [198], Corrado and Tie Su [1996], Stein and Stein [1991], Wiggins [1987], and Hestin [1993], Johnson, Pawlukiewicz, and Mehta [1997], and Câmara and Chung [006]). These studies have identified a number of stylized facts, including option mispricing using option models and the jump-diffusion option model. The existence of skewness poses a modeling challenge for pricing bond 4

5 derivatives and for valuing bonds with embedded options using a binomial interest rate model. In this paper, we address this challenge by first presenting empirical evidence showing the significance of skewness when interest rates are increasing and decreasing. These empirical observations suggest that in cases where interest rates are expected to increase or decrease, pricing biases may result when using either the equilibrium model or the calibration model where the underlying binomial interest rate tree is based on the assumption of normality. We next illustrate using a three-period binomial tree example how the Johnson, Pawlukiewicz, and Mehta (JPM) skewnessadjusted model for a three-period tree can be used to calibrate a binomial interest rate tree for stable, increasing, and decreasing interest rate cases in which the end-of-theperiod distribution is characterized by skewness. Third, we show how the implied spot and forward yield curves generated from the skewness-adjusted model result in flat, normal, and inverted spot yield curves that are consistent with the expected end-of-theperiod distribution. Finally, we conclude the paper by showing how skewness can be incorporated into the Black, Derman, and Toy (BDT) calibration model and by showing with a three-period and 30-period model the possible mispricing of interest rate and bond derivatives that can result when the BDT variability conditions are not adjusted to reflect skewness. The paper contributes to the literature on binomial interest rate modeling by providing empirical evidence of the significance of skewness when interest rates are increasing or decreasing, showing the consistency of an end-of-the period distribution and the implied yield curves for expected increasing and decreasing interest rate scenarios using a skewness-adjusted equilibrium model, and showing how the BDT model can be adjusted to reflect skewness in order to maintain its arbitrage-free feature.. Skewed interest rate patterns.1 Skewed binomial distributions Binomial distributions of spot rates, S, and their corresponding logarithmic return, gn = ln(sn/s0), are shown in Exhibit 1. The exhibit shows three end-of-the-period distributions resulting from a binomial process in which the number of periods to expiration is n = 30 and the initial spot interest rate is S0 = The probability distribution in the top exhibit is generated from a binomial process in which the upward parameter (u) is equal to 1.0, the downward parameter (d) is equal to 1/1.0 ( lnu = lnd ) and the probability of an increase in one period (q) is equal to 0.5. In this case, the distribution approaches a normal distribution with E(gn) = 0, V(gn) = , and skewness, Sk(gn), equal to zero: n E(g n ) = p nj g nj j=0 5

6 E(g n ) = n[q ln u + (1 q) ln d] = n E(g 1 ) (1) E(g 30 ) = 30[0.5 ln(1.0) + (1 0.5) ln (1/1.0) = 0 V(g n ) = E[g n E(g n )] = p nj [g n E(g n )] n j=0 V(g n ) = nq(1 q)[ln u/d)] = n V(g 1 ) () V(g 30 ) = 30 (0.5)(1 0.5) [ln(1.0 1/1.0)] = Sk(g n ) = E[g n E(g n )] 3 = p nj [g n E(g n )] 3 n j=0 Sk(g n ) = n[q(1 q) 3 q 3 (1 q)] [ln u/d)] 3 = n Sk(g 1 ) (3) Sk(g n ) = 30[(0.5)(1 0.5) 3 (0.5) 3 (1 0.5)] [ln (1.0 (1 1.0)) ] 3 = 0 Where: g1 is the logarithmic return for one period, j is the number of increases in n periods, pnj is the probability of j increases in n periods that is defined as: p nj = n! (n j)! j! q j (1 q) n j The middle distribution reflects an increasing interest rate case in which u = 1.0, d = 0.99 ( lnu > lnd ), and q = Here the distribution is negatively skewed with a positive mean: E(gn) = , V(gn) = , and Sk(gn) = The bottom distribution reflects a decreasing interest rate case in which u = 1.01, d = ( lnu < lnd ), and q = 0.5; the distribution is skewed with a negative mean: E(gn) = , V(gn) = , and Sk(gn) = The three distributions shown in Exhibit 1 reflect several properties associated with binomial distributions. First, the expected value, variance, and skewness of the logarithmic return are equal to their moment values for one period times the number of periods defining the total period. Second, the skewness for each period's distribution is zero when there is an equal probability of the rate increasing or decreasing in one period, and skewed when there is not (as illustrated in the cases when q = 0.75 and q = 0.5). Thus, a sufficient condition for symmetry is that there be an equal probability of the underlying security increasing or decreasing in each period. 6

7 Probability Probability Probability Exhibit 1: Constant, increasing, and decreasing interest rate distributions Distribution: n = 30, u = 1.0, d = 0.98, q = 0.50, Mean = 0.00 Var = , Skew = 0.00 Logarithmic Return Distribution: n = 30, u = 1.0, d = 0.99, q = 0.75, Mean = Var = , Skew = Logarithmic Return Distribution: n = 30, u = 1.01, d = 0.99, q = 0.5, Mean = Var = , Skew = Logarithmic Return Finally, the expected value is equal to zero for the case in which q = 0.5. This property is the result of assuming not only that there is an equal probability of an increase or decrease each period, but also that u and d parameters are inversely proportional, or equivalently that the proportional increase in each period (lnu) is equal in absolute value to the proportional decrease (lnd). However, if the distribution of the logarithmic return at the end of n periods had, for example, a positive expected value 7

8 and zero skewness, then the underlying binomial process would have been characterized by the proportional increases in each period exceeding in absolute value the proportional decreases, with the probability of the increase in each period being 0.5; if the distribution also had negative skewness, then the probability of the increase in one period would have exceeded 0.5. On the other hand, if the distribution of the logarithmic return had a negative expected value and zero skewness, then the underlying binomial process would have been characterized by the proportional decreases in each period exceeding in absolute value the proportional increases and with q = 0.5; if the distribution also had a positive skewness, then q would have been less than Increasing and decreasing interest rates empirical evidence of skewness As noted, several empirical studies have provided evidence that the distributions of stock, stock indexes, and currency exhibited skewness. To determine whether interest rates also were characterized by periods of persistent skewess, we conducted a statistical analysis of the logarithmic returns for the 1-year, -year, 5-year and 10-year U.S. Treasury yields using the D Agostino, Belanger, and D Agostino tests of normality [1990]. The statistical analysis we applied for interest rates included: (a) calculating daily logarithmic returns for the period from February 15, 1977 to December 19, 016; (b) Computing the means, standard deviations, and skewness for rolling 50-day periods; (c) Testing the significance of skewness using the D Agostino, Belanger, and D Agostino tests; (d) Identifying the periods of increasing interest rates as (1) those characterized by a positive mean and negative skewness that was significant and () those characterized by negative skewness that was significant and a mean that was becoming less negative, reflecting a change in direction; and (e) Identifying the periods of decreasing interest rates as (1) those characterized by a negative mean and positive skewness that was significant and () those characterized by positive skewness that was significant and a mean that was positive but decreasing, reflecting a change in direction. Exhibit summarizes the results for the interest rate using the 50-day window for the 5-year Treasury yields. The results for the 1-year, -year, and 10-year yields can be viewed from Mendeley Data depository. The figures in the exhibit show two timeseries graphs for each rate: (a) the rate with the 50-day moving means and skewness, and (b) the ρ value for skewness and kurtosis. The gray-shaded areas in the graphs indicate an increasing interest-rate period characterized by skew < 0 and significant and μ > 0 or still negative but increasing, as well as a decreasing interest-rate periods characterized by skew > 0 and significant and μ < 0 or positive but decreasing. The number and length of the gray areas in the graph for the rates indicate a large number of increasing and decreasing cases in which skewness is significant. These findings, in turn, point to the importance of using a skewness-adjusted binomial interest rate model when pricing bond and bond derivatives when the underlying interest rate is expected to increase or decrease. 8

9 DGS5 Mean*1000, Skewness Exhibit : Interest-rate movements Figure 1: Skewness and Mean for 5-Year Treasury Constant Maturity Rate WL=50, February 15, December 19, 016 DGS5 Mean*1000 Skewness Time 9

10 P-value Exhibit : Interest-rate movements (continued) Figure : P-value for Skewness Statistic and Kurtosis Statistic for 5-Year Treasury Constant Maturity Rate, WL=50, February 15, December 19, 016 p-value(skewness) p-value(kurtosis) Time 10

11 3. Skewness-Adjusted Binomial Option Pricing Model A binomial process that converges to an end-of-the-period distribution of logarithmic returns that is normal will have equal probabilities of the underlying security increasing or decreasing each period, whereas one that converges to a distribution that is skewed will not. In the Johnson, Pawlukiewicz, and Mehta (JPM) model, the upward and downward parameters and q values defining a binomial process are found by setting the equations for the binomial distribution s expected value, variance, and skewness equal to their respective empirical values, then solving the resulting equation system simultaneously for u, d, and q: E(g n ) = n[q ln u + (1 q) ln d] = μ e (4) V(g n ) = nq(1 q)[ln u/d)] = V e (5) Sk(g n ) = n[q(1 q) 3 q 3 (1 q)] [ln u/d)] 3 = δ e (6) where: e, Ve, e = the empirical values of the mean, variance, and skewness of the logarithmic return for a period equal in length to n periods. The values of u, d, and q that satisfy this system of equations are: μe u = e μe d = e n n + [(1 q)v e n q ] 1 [ qv e (1 q)n ]1 q = 1 1 [4V e 3 nδ + 1] 1 e (7) (8) if δ e > 0; +if δ e < 0 (9) If e is positive (negative), then q is less (greater) than 0.5; if skewness is zero, then q = 0.5 and Equations (7) and (8) simplify to the Cox, Ross, and Rubinstein (CRR) binomial option pricing model formulas for u and d: u = e μ e n + [V e n ]1 d = e μ e n [V e n ]1 (10) (11) In the skewness-adjusted binomial model, the presence of skewness affects the relative contribution of the mean to the values of u and d. In the case of a positive mean, the mean becomes more important in determining the value of u, the greater the negative skewness. By contrast, in the case of a negative mean, the mean becomes 11

12 more important in determining the value of d, the greater the positive skewness. In addition, the skewness-adjusted model does not have the same asymptotic properties as the normally distributed CRR model. For large n, the CRR model depends only on the variance of the underlying asset. In the skewness model, u and d depend on all three moments for the case of large n. 3.1 Bond and option valuation with Skewness-adjusted binomial tree Under an increasing interest rate scenario, an option-free bond and bond and interest rate options should be valued by a binomial model that reflects a positive expected logarithmic return and negative skewness. In contrast, under a decreasing rate scenario, such bonds and the options on bonds should be valued by a binomial model that reflects a negative expected return and positive skewness. As an example, suppose the spot interest rate is currently at 10% and there is a market expectation of increasing rates over the next three years with the expected distribution of logarithmic returns of spot rates having the following estimated parameters after three years of μe = , Ve = , and δe = The u, d, and q values for a three-period binomial interest rate tree that would calibrate a binomial distribution to this distribution would be u = 1.1, d = 0.95, and q = 0.8. μe u = e μe d = e n + [(1 q)v e ] 1 n q = e (1 0.8) + [ ] 1 = 1.10 (3)(0.8) n + [(1 q)v e ] (0.8) [ ] n q = e = (3)(1 0.8) q = 1 1 [4V e 3 nδ e + 1] 1 1 = [ 4( )3 + (3) ( ) 1] = 0.8 Exhibit 3 shows the binomial valuation of a three-year, option-free 10% annual coupon bond with face value of $100, a call option on the bond with an exercise price (X) of 100 and expiration at the end of period, and a put option on the bond with an exercise price 100 and expiration at the end of period. The bond values are obtained by using a two-period binomial tree of one-period spot rates, with the length of the period being one year and with the estimated u, d, and q values reflecting the decreasing interest rate case. The option-free bond s value of B0 = is obtained by: (1) determining the bond s three possible values in period given the three possible spot rates, () rolling these three values to period 1, where the two possible values in that period are attained by calculating the present values of their expected cash flows, and (3) rolling these two values to the current period, where the current value is found by again determining the present value of the two expected cash flows. 1

13 Exhibit 3: Binomial Bond and Spot Option Values for Increasing Interest Rate Case Binomial Bond and Spot Option Values for Increasing Rate Case S0 10% Coupon 10, F 100, Maturity :n 3 Period 3 cash flow u 1.1, d 0.95, q 0.8 X 100, Expiration nf S 10% ( ) 0.( ) B0 (1.10) e 0.8(0) 0.(0.1633) C (1.10) a 0.8(0) 0.8(0.333) C (1.10) e 0.8(1.435) 0.(0.976) P (1.10) a 0.8(.344) 0.(0.97) P (1.10) S B C IV Max[ ,0] 0 C e u P IV Max[ ,0].344 a u P S C C P P u u e u a u d us 0 11% 0.8( ) 0.( ) (1.11) (0) 0.(0) 0 (1.11) Max[0, 0] 0 0.8(1.8733) 0.(.4074) (1.11) Max[1.435,.344].34 ds 9.5% 0.8( ) 0.( ) Bd (1.095) e d (0) 0.(.8943) (1.095) IV Max[ ,0] a u e u Max[0.1633, 0.333] 0.8(0.4074) 0.(0) (1.095) IV Max[ , 0] 0 a u 0 Max[0.976,0] S B C P S B C P S B C P dd uu uu uu uu uu ud ud ud ud dd dd u S 0 1.1% Max[ ,0] 0 Max[ ,0] uds % Max[ ,0] 0 Max[ ,0] d S % Max[ ,0] Max[ ,0] 0 The price of the European call in Exhibit 3 is and the American call is The European price is obtained by determining the call s intrinsic value at expirations and then rolling the tree to the present where the call price at each node is equal to present value of the expected call values for the next period. The price of the American option is obtained by constraining the price at each node to be the maximum of its intrinsic value or the binomial value. The European put value of and American value of , in turn, are determined similar to the call options. The expectation of the increasing interest rate scenario shown in Exhibit 3 produces a tree in which the binomial bond prices for zero coupon bonds reflect a positively-sloped spot yield curve and higher implied forward rates. The yields on the implied spot yield curve (ym = y1, y, and y3) and the implied forward rates (fmt = M-year 13

14 bond t years from present) for one year (f11 and f1) and for two years from the present (f1, f ) are shown in Exhibit 4a. The positively sloped yield curve and higher implied forward rates shown in the exhibit are, in turn, consistent with an expectation of higher interest rates. In contrast to a increasing rate case, suppose the market expects declining rates over the next three years, with the expected distribution of logarithmic returns of spot rates having the following estimated parameters after three years of μe = , Ve = , and δe = The u, d, and q values for a three-period binomial interest rate tree that would calibrate a binomial distribution to this distribution would be u = , d = , and q = 0.. The binomial pricing with this tree that is calibrated to a decreasing interest rate case results in a greater value of the optionfree bond of B0 = than the value of obtained using a tree calibrated to the decreasing interest rate scenario shown in Exhibit 3. The price of the European call for the decreasing rate case is and the American call is The European put value is and American value is The decreasing interest rate tree also generates binomial bond prices for zero coupon bonds that yield a negatively sloped spot yield curve and decreasing implied forward rates, which are consistent with a market expectation of decreasing rates (Exhibit 4b). Finally, suppose the market expects stable interest rising rates over the next three years, with the expected distribution of logarithmic returns of spot rates having the following estimated parameters after three years of μe = 0, Ve = , and δe = 0. The u, d, and q values for a three-period binomial interest rate tree that would calibrate a binomial distribution to this distribution would be u = , d = 1/u = , and q = 0.5. The binomial valuations of the option-free bond is , which is greater than the increasing interest rate case and smaller than the decreasing case. The price of the European call for the decreasing rate case is and the American call is The European put value is and the American value of The stable interest rate tree shown generates binomial bond prices for zero coupon bonds that yield a flat sloped spot yield curve and constant implied forward rates, which are consistent with a market expectation of stable rates (Exhibit 4c). 3. Binomial futures and futures options prices Exhibit 5 shows the carrying-cost prices on a futures contract on the bond for the increasing rate case, with the futures expiring at the end of period two. The equilibrium futures prices is determined by the carrying-cost model: f 0 = [B 0 PV(C)] (1 + y nf ) n f 14

15 Exhibit 4: Implied Spot and Forward Yield Curves a. Increasing Rate Case: μe = , Ve = , and δe = ; u = 1.1, d = 0.95, and q = 0.8 Binomial prices and yields (y) on zerocoupon bonds with F = $100 and maturities of n = 1,, and 3 periods Maturity Binomial Spot Years Price Yield (y) Implied Forward Rates One Year Two Years y 1 $100 / $ / y $100 / $ /3 $100 / $ y3 b. Decreasing Rate Case: μe = , Ve = , and δe = ; u = , d = , and q = 0. Binomial prices and yields (y) on zero-coupon bonds with F = $100 and maturities of n = 1,, and 3 periods $100 / $ Maturity Binomial Spot Years Price Yield (y) y 1 Implied Forward Rates y $100 / $ / One Year Two Years /3 $100 / $ y3 c. Stable Rate Case: μe = 0, Ve = , and δe = 0; u = , d = 1/u = , and q =

16 Where: PV(C) = present value of coupons paid on the bond during the life of the futures contract. nf = number of periods to the expiration on the futures contract. ynf = n-period risk-free rate for the expiration period on the futures contract. Exhibit 5: Binomial Bond, Futures, and Option Values for Increasing Interest Rate Case Binomial Bond, Futures, and Futures Option Values for Increasing Rate Case Su us0 11%, Bu S0 10% 10 Coupon 10, F 100, Maturity : n 3 fu [ ](1.11) Period 3 cash flow 110 e 0.8(0) 0.(0.887) u 1.1, d 0.95, q 0.8 C u (1.11) n ft [Bt PV(Coupons)](1 y ) f nf IV Max[ ,0] 0 X 98.71,Expiration : nf a C u Max[ , 0] e 0.8(0.5833) 0.(0) Pu S0 10%, B (1.11) f [ ]( ) IV Max[ ,0] ( ) a Pu Max[0.404, ] e 0.8( ) 0.( ) C (1.10) a 0.8( ) 0.(1.149) C (1.10) e a 0.8(0.404) 0.(0) P0 P (1.10) Sd ds0 9.5%, Bd fd [ ](1.095) e 0.8(0.887) 0.(.1843) Cd (1.095) IV Max[ ,0] a Cd Max[ , 1.149] e 0.8(0) 0.(0) Pd 0 (1.095) IV Max[ , 0] 0 a Pd Max[0.0] 0 Suu u S0 1.1%, Buu fuu Buu Cuu Max[ ,0] 0 Puu Max[ ,0] Sud uds %, Bud fud Bud Cud Max[ ,0] Pud Max[ ,0] 0 Sdd d S0 9.05%, Bdd fdd Bdd Cdd Max[ , 0].1843 Puu Max[ , 0] 0 The binomial futures price is generated by assuming the market spot yield curve is equal to the implied spot yields. For example, for the increasing rate case the current futures price of on the bond discounts the $10 coupon in year one by y1 = 10% and the $10 coupon in year by y = %. f 0 = [ ( ) ] ( ) = The equilibrium futures price for the increasing case also has a yield on futures contract that is equal to the implied forward rate generated from the implied spot yield curve for 16

17 the increasing case. That is, the implied yield on the futures given this equilibrium futures price of is equal to %, which matches the implied forward rate on a one-year bond two years forward: y f = F + coupon f 0 1 = = f 1 = (1 + y 3 ) 3 (1 + y 1 )(1 + f 11 ) 1 = (1.1071) 3 (1.10)( ) 1 = For the decreasing rate case (μe = , Ve = , and δe = ; u = , d = , and q = 0.) the current futures price is on the contract expiring in two years and paying 110 one year from the futures expiration. This equilibrium price has an implied yield (yf) that matches the implied forward rate on a one-year bond two years from the present (f1) of 8.786%: f 0 = [ (1.10) ] ( ) = y f = F + coupon f 0 1 = = f 1 = (1 + y 3 ) 3 (1 + y 1 )(1 + f 11 ) 1 = ( ) 3 (1.10)( ) 1 = Finally, for the stable rate case (μe = 0, Ve = , and δe = 0; u = , d = 1/u = , and q = 0.5) its equilibrium futures price of has a futures yield that matches its implied forward rate on a one-year bond two years forward of 10%: f 0 = [ (1.10) ] (1.10) = y f = F + coupon f 0 1 = = 0.10 f 1 = (1 + y 3 ) 3 (1 + y 1 )(1 + f 11 ) 1 = (1.10) 3 (1.10)(1.10) 1 = 0.10 Exhibits 5 also shows the binomial tree values for on-the-money call and put options on the futures contract. The European option prices are obtained by determining 17

18 the option s intrinsic value at expirations and then rolling the tree to the present where the option price at each node is equal to present value of the expected option values for the next period. The price of the American option is obtained by constraining the price at each node to be the maximum of its intrinsic value or the binomial value. 4. Skewness-adjusted BDT calibration model A binomial interest rate tree generated using the u and d estimation approach is constrained to have an end-of-the-period distribution with parameter values that match the analyst s estimated values. The tree is not constrained, however, to yield a bond price that matches its equilibrium value. In contrast, calibration models are constrained to match the current term structure of spot rates and therefore yield prices for optionfree bonds that are equal to their equilibrium values. The first calibration model was the one derived by Black, Derman, and Toy. The BDT calibration model generates a binomial tree by first finding spot rates that satisfy a variability condition between the upper and lower rates. Given the variability relation, the model then solves for the lower spot rates that satisfies a price condition in which the bond values obtained from the tree are consistent with the equilibrium bond price given the current yield curve of spot rates. The BDT model is therefore constrained to have option-free bond prices that match their equilibrium. However, its variability condition is not constrained to have an end-of-the-period distribution with parameter values that match the analyst s estimated values. In fact, unless the BDT variability conditions are adjusted to reflect the end-ofthe-period distribution, calibrating a binomial tree to positively or negatively spot yield curves that reflect increasing or decreasing futures interest rates can lead to mispricing of bond derivatives and opportunities for abnormal returns using the calibration model. 4.1 Skewness-adjusted BDT variability condition In the BDT calibration model, the variability condition governing the upper and lower one-period spot rates for the first period is given a S u = S d u d S u = S d eμe + n V e n e μ e n V e n S u = S d e V e n (1) The variability condition can be modified to incorporate skewness by defining the relationship between Su and Sd in terms of the skewness-adjusted u and d parameters: 18

19 μe S u = S e + V e(1 q) n nq d μe V eq e n n(1 q) S u = S u = S d e V e(1 q) nq q = 1 1 [4V e 3 1 nδ + 1] e + V eq n(1 q) (13) 4. BDT price condition The price condition for a one-period interest rate tree requires that the lower rate along with the variability condition yield a binomial value for a two-period option-free bond that is equal to the current equilibrium price. Assuming zero discount bonds with a face value of $100 and a current two-period spot rate of y, this price condition requires finding Sd where: 100 (1+ y qb = + (1 S u ) 1+ 0 q) B d 100 (1+ y Ve (1 q) Veq + nq n(1 q) q[100 /(1+ Sde = ) 1+ S0 )]+ (1 q)[100 /(1+ S d )] Given the one-period tree, the possible spot rates in period are found by specifying a similar skewness-adjusted variability condition between Sdd and Sud and Sud and Suu, and then solving iteratively for the Sdd value that yields a binomial value for a threeperiod zero discount bond that is equal to the bond s equilibrium price of 1/(1+y3) 3. The tree s spot rates for subsequent periods are found in a similar way. 4.3 Pricing Differences with and without Skewness Adjustment Three-Period Case Suppose the current yield curve has one-, two-, and three- year spot rates that match the implied spot rates for the increasing rate interest rate case: y1 = 10%, y = %, and y3 = 10.71% (see Exhibits 3 and 4a). The corresponding equilibrium bond prices for 1-year, -year, and 3-year option-free bonds with face values of $100 19

20 would be B1 = 100/(1.10) = , B = 100/( ) = 8.145, and B3 = 100/(1.1071) 3 = Suppose that in addition to the current market yield curve equaling the implied spot yield curve for the increasing interest rate case, the annualized logarithmic mean, variance, and skewness also reflect a three-period increasing interest rate scenario in which Ve = , μe = , and δe = It should be noted if expectations are assumed to be incorporated in the spot yield curve, the yield curve positive slope would also be consistent with an expectation of higher interest rates as well. Given the mean, variance, and skewness of the expected distribution values, the resulting skewness-adjusted parameter values are u = 1.1, d = 0.95, and q = 0.8. To generate the two spot rates of Su and Sd for the calibrated tree s first period (length of the step being one year), the skewness-adjusted variability condition is S u = S u = S d e V e(1 q) nq + V eq n(1 q) = S d e (1 0.8) (3)(0.8) + ( )(0.8) (3)(1 0.8) = S d q = 1 1 [4V e 3 1 nδ + 1] 1 = = e + 1 [ 4( ) 3 1 (3)( ) + 1] = 0.8 and the Sd value that equates the binomial price of a two-year zero coupon bond (F = $100) to the equilibrium price (100/ = 8.145) is 9.5%, and the upper rate satisfying the variability condition is Su = (1.1/0.95) 9.5% = 11%. For the tree s second period, q is 0.8, the variability conditions are S ud = S dd e (1 0.8) (3)(0.8) S uu = S ud e (1 0.8) (3)(0.8) + ( )(0.8) (3)(1 0.8) = S dd + ( )(0.8) (3)(1 0.8) = S ud and the Sdd value that equates the binomial price of a three-year zero to the equilibrium price (100/ = ) is 9.05%, and the successive upper rates satisfying the variability conditions are Sud = 10.45%. and Suu = 1.1%. The resulting calibrated binomial tree (S0 = 10%, Sd = 9.5%, Su = 11%, Sdd = 9.05%, Sud = 10.45% and Suu = 1.1%) exactly matches the binomial interest rate tree shown in Exhibit 3. This result is to be expected since the tree being calibrated to a spot yield curve matches the spot yield curve implied by the mean, variance, and skewness of the expected distribution values of the increasing interest rate scenario. Thus, the equilibrium model and the skewness-adjusted calibration model yield the same binomial interest rate tree when the market spot yield curve matches the implied yield curve from the expected distribution. As shown in Exhibits 3 and 5, the model would price a three-period, 10% coupon bond at , a two-period European call on the bond at ( for an American 0

21 call), a European put at (1.745 for American), and a futures contract on the bond expiring in period two at with the same values for the futures option prices that are shown in Exhibits 10. Note that a corollary condition would be that if the market spot yield curve is the best estimate of market expectation of interest rate at the end of the period and the estimated variance is correct, then the implied mean and skewness would be μe = and δe = Suppose that the current spot yield curve is again equal to the implied spot yield curve for the increasing rate scenario, but the variability condition and probability are not adjusted for skewness. This would be a case if the variability condition implies an endof-the period distribution of stable interest rates and a flat yield curve, while the market yield curve in which the binomial tree is being calibrated implies a market expectation of increasing rates. Here q is 0.5, the unadjusted variability conditions are S u = S d e V e n = S d e S ud = S dd e ; S uu = S ud e and the lower spot rates needed to equate the binomial-generated bond prices to their equilibrium prices are Sd = 10.07% and Sdd = 10.15%. Exhibit 6 shows the resulting calibrated binomial interest rate tree with the both the skewness adjustment (top box) and this case in which there is no skewness adjustment (bottom box). The exhibit also shows the binomial tree prices for the three-year, 10% coupon bond, the futures prices, and the binomial values of the spot and futures call and put options on the bond with the spot options each having an exercise price of 100 and expiration of two years and each of the futures option having an exercise price equal to the futures price of Given the calibration model s price constraint, the model s price of the threeperiod 10% bond matches the equilibrium price of The value of the call option for the unadjusted model is zero (given that all of the possible binomial bond prices are less than the exercise price) compared to the skewness-adjusted European call value of and American value of 0.058, and the value of the European put is and the American is compared to the skewness-adjusted values of for the European and for the American. Given the calibration model s price constraint, the futures price for the unadjusted tree of matches that of the skewnessadjusted tree. However, in subsequent periods the futures prices, like the spot prices, differ for the two cases. As a result, the two cases yield different futures option prices. Similar price differences for adjusted and unadjusted skewness variability conditions are also the case for a decreasing rate case. It should be noted that in the 1

22 three-period stable interest case in which the expected mean and skewness were zero and Ve = , the implied yield curve was flat at 10%. If the market yield curve matches the implied yield curve, then the calibrated yield curve will be the same as the binomial trees. In this case, there is no need to adjust the variability conditions for skewness. Exhibit 6: Calibrated Tree with and without Skewness Adjustment: Increasing Rate Case: Spot Call and Puts, X = 100, Futures Call and Put: X = Calibrated Tree with Skewness Adjustment Calibrate yield curve: y1 = 10%, y = %, and y3 = 10.71% V e = ,? e = , and? e =? u = 1.1, d = 0.95, q = 0.8 Implied yieldcurve: y1= 10%, y = %, and y3= 10.71% Ve (1 q) V q e nq n(1 q) S u S d e S u S d e q (.) (3)(.8) (.8) 3(.) F + C = / ( ) )( ) F + C = Spot Rate Bond Price, Futures Price Spot: C e, C a Spot: P e, P a Futures: C e, C a Futures: P e, P a Spot Option: X = 100 Futures Option: Expiration: n = X =98.71, Exp.: n = F + C = nf f0 [B0 PV(C)](1 yn ) f0 [ f 10 ]( ) ( ) F + C = 110 Period Calibrated Tree without Skewness Adjustment Calibrate yield curve: y 1 = 10%, y = %, and y 3 = 10.71% V e = , q = 0.5 Implied yieldcurve: y 1 = 10%, y = %, and y 3 = 10.71% Ve / n //3 Su Sde Sde q F + C = F + C = Spot Rate Bond Price, Futures Price Spot: C e, C a Spot: P e, P a Futures: C e, C a Futures: P e, P a Spot Option: X = 100 Futures Option: Expiration: n = X =98.71, Exp.: n = F + C = F + C = 110 Period 0 1 3

23 period case Exhibit 7 shows the binomial bond prices for zero coupon bonds with maturities from one quarter to 30 quarters, the quarterly yields on the implied spot yield curve (ym: y1, y, y30:), and their implied quarterly forward rates for one year (four quarters), two years (eight quarter), four years (16 quarters), six years (4 quarters), and 7.5 years (30 quarters) from the present. The middle panel shows bond prices for an increasing interest rate scenario in which the distribution at the end of a 7.5 year period is characterized by expected parameter values of μe = , Ve = , and δe = The quarterly u and d values that calibrate the 30-period binomial tree to this end of the period distribution are u = 1.0, d = 0.99 and q = 0.75 using Equations (7), (8), and (9): μe u = e μe d = e n + [(1 q)v e ] (1 0.75) + [ ] n q = e = (30)(0.75) n + [(1 q)v e ] 1 n q q = 1 1 [4V e 3 nδ e + 1] = e (0.75) [ ] 1 = 0.99 (30)(1 0.75) 1 = [ 4( )3 + (30) ( ) 1] = 0.75 The resulting positively sloped yield curve and higher implied forward rates are, in turn, consistent with an expectation of higher interest rates. The right panel in Exhibit 7 shows zero-coupon bond prices, implied spot rates, and implied forward rates for a decreasing rates case in which the market expects lower rates in the future with the expected distribution of logarithmic returns of spot rates having the following estimated parameters after thirty quarters of μe = , Ve = , and δe = The quarterly u and d values that calibrate the 30-period binomial tree to this end of the period distribution are u = 1.01, d = 0.98, and q 0.5. This tree generates binomial bond prices for zero coupon bonds that result in a negatively sloped spot yield curve and decreasing implied forward rates, which are consistent with a market expectation of lower rates. The left panel in Exhibit 7 shows zero-coupon bond prices, implied spot rates, and implied forward rates for stable interest environment in which the distribution at the end of the 7.5 year period is characterized by expected parameter values of μe = 0, Ve = , and δe = 0. The quarterly u and d values that calibrate the 30-period binomial tree to this end-of-the-period distribution are u = 1.0, d = 1/1.0, and q = 0.5. This tree generates binomial bond prices for zero coupon bonds that yield a flat spot yield curve and constant implied forward rates. 3

24 Exhibit 7: Implied Spot Yield Curves from 30-Period Skewness-Adjusted Binomial Model F = 100, initial spot rate = 10%;.5% (quarterly) Stable Increasing Decreasing Maturity Binomial Spot Implied Forward Rates: Quarters (Q) Forward Binomial Spot Implied Forward Rates: Quarters (Q) Forward Binomial Spot Implied Forward Rates: Quarters (Q) Forward Quarters Spot Price Yield 4 Q 8 Q 16 Q 4 Q 30 Q Spot Price Yield 4 Q 8 Q 16 Q 4 Q 30 Q Spot Price Yield 4 Q 8 Q 16 Q 4 Q 30 Q

25 As shown in the three-period case, if the current market spot yield curve were the same as the implied spot yield curve, then the equilibrium model and calibration model would be the same provided the skewness-adjusted variability conditions are used to calibrate the binomial tree to the market yield curve (see Exhibit 9). For example, in the increasing rate case, the BDT skewness-adjusted variability condition for the 30-period case are S u = S u = S d e V e(1 q) nq S u = S d e (1 0.75) (30)(0.75) + V eq n(1 q) + ( )(0.75) (30)(1 0.75) = S d The resulting 30-period calibrated binomial tree matches the binomial interest rate tree generated using the equilibrium model in which μe = , Ve = , and δe = (u = 1.0, d = 0.99, and q = 0.75). Exhibit 9 shows the first eight quarters of the 30-period calibrated binomial interest rate tree with the skewness adjustment variability along with the corresponding bond prices for the.5% quarterly coupon bond, and the prices for the spot options on the bond. As shown, the equilibrium model and the calibrated model with the skewness adjusted variability condition both price a 30-period,.5% quarterly coupon bond at 91.39, a two-year onthe-money European call on the bond at 0.05 (0.40 for an American call), and an onthe-money European put at 1.08 (1.1 for American). If the current spot yield curve is again equal to the implied spot yield for the increasing rate scenario, but the BDT variability conditions and probability are not adjusted for skewness, then the resulting binomial tree will have comparatively lower interest rates at most of the nodes. In this case, q = 0.5 and the unadjusted variability condition is S u = S u = S d e V e(1 0.5) n(0.5) + V e(0.5) n(1 0.5) S u = S d e V e n = S d e = S d As noted, the unadjusted variability condition implies an end-of-the period distribution of stable interest rates and a flat yield curve, which is inconsistent with the market yield curve that reflects a market expectation of increasing rates. Exhibit 10 shows the first eight quarters of the 30-period calibrated binomial interest rate tree without the skewness adjustment variability along with the corresponding bond prices for the.5% quarterly coupon bond, and the prices for the spot options on the bond. Given the calibration model s price constraint, the price of the bond is the same as the 5