2023年6月24日发(作者：)

Stochastic Interest Rates:A Crucial Correlationo calculate the value of a stockoption, you need to model sto-chastic share prices. To deter-mine what a callable bond is worth,you must model stochastic interestrates. But to evaluate the options in-herent in a convertible bond—ahydra with one head fixed incomeand the other equity—you must treatboth the share price and the interestrates as stochastic variables function for valuation of con-vertible bonds, OVCV, is based on atwo-factor model that treats bothshare prices and interest rates as sto-chastic variables. And only by meansof a two-factor model can you effec-tively take into account the correla-tion between interest rates and because of developmental costsand the computing power needed torun a two-factor model, many con-vertible bond models attempt to makedo with a single factor, with interestrates held constant. Indeed, mostpractitioners believe that althoughMARKET SOLUTIONSBecause it ignores the difference the correlation effect can make

to net value, a single-factor model doesn’t quite get the job doneTFigure DO OVCV. Tab in, and enter 70 in the OASfield.

Press to see Diamond Offshore Drilling’s convertible bond valueusing stochastic interest rates makes

a calculation slightly more accurate,the net difference is not large and sonot worth the added complexity. Forexample, Michael Brennan and Ed-uardo Schwartz looked at the pricedifference between using constant

interest rates and stochastic ones forthe same convertible at various inter-est rates and concluded that “for areasonable range of interest rate lev-els the errors from the certain inter-est rate model are likely to be slight,and therefore, for practical purposesOVCV: A multitalented functionCORPI•••••n calculating the value of a convertible bond, the OVCV function models not only stochastic interest rates but alsoboth hard and soft calls, puts, and even dilution—à la warrants. In addition, it models dividends and creditspreads, and the advanced model even treats can use OVCV to:break a convertible’s value into its component partscalculate sensitivities on both the equity and interest-rate sides: vega, delta, gamma, duration, and convexitydetermine the implied values of share volatility or credit spreadprice dual-currency bonds, using interest-rate parity to calculate implied forward foreign exchange ratesplot graphs that show how values change under varying scenariosThe data needed for these calculations appear on wake-up, so there’s no need to search for or enter them: the yieldcurve, stock price, historical volatility, hard and soft calls, put schedules, and cash flows are all instantly offers you a powerful two-factor model and all the data needed to run it. No slide rule required.

—E.B. & ERGJuly 1997 91The correlation calculationo calculate the correlation between interest ratesand share prices, we must first understand exactlywhich variables we are trying to calculate the correlationof. To do that, we use the defining equations for the jointtwo-variable stochastic process:dS=

␮Sdt+

␴SdZ1

dr=

␮rdt+

␴rdZ2ᎏᎏSrHere

␴Sand

␴rare the (constant) volatilities of the twoprocesses and

␮Sand

␮rare their drifts. Under a risk-neu-tral measure,

␮S

= rand

␮rmatches the observed yieldcurve. The expressions dZ1and dZ2are increments of aunit Brownian motion, and it is the correlation of

␴SdZ1and

␴rdZ2that is needed. This is exactly the same as thecorrelation between dlog Sand dlog r—by Ito’s lemma.

We thus form the series xi

ϵ

log Si+1– log Siand yi

ϵ

logri+1– log ri

and calculate their correlation. The series xiisjust the log of the return. The sample mean and standarddeviation of xiareTThe correlation between the two series is thengiven by

␳xy

=

xi

–

␮xyi

–

␮y΋(n – 1).␴x␴yi=1Fortunately, you don’t have to calculate these cor-relations yourself; CORR calculates them for CORC 1 (if this slot is taken, use a num-ber other than one), and enter the data as in figure3. We have chosen USG1YL as a proxy for interestrates because, being a weighted average of bondswith maturities out to three years, it is less sus-ceptible to Federal Reserve control. It is also one ofthe indexes used in standard value-at-risk calcula-tions. Note how we use the differences in the logsin the correlation calculation by entering a Din thefirst column and an Ein the sixth type 2 to get the correlation matrix. Figure4 shows that the S&P 500 has a correlation close tominus 0.5, whereas the EMC convertible has a corre-lation of minus 0.294. —E.B. & D.K.͸n

␮x

= xi

΋n and

␴x

=

i=1͸n

i=1͸n

(xi

–

␮x)2΋(n – 1), may be preferable to use this sim-pler model [constant interest rate]for valuing convertible bonds” (“An-alyzing Convertible Bonds,” Journal ofFinancial and Quantitative Analysis15:4 [November 1980]).That claim may be correct for theprimary bond market, but it’s notgenerally valid in the secondary mar-kets. Take, for example, a bustedcallable convertible—a convertiblewhose share price is so far below itsconversion price that its conversionoption is almost worthless—that’spaying a coupon close to par in lightof the current yield curve. Such abond will behave exactly like a nor-mal callable bond, and to ignore sto-chastic interest rates in valuing sucha convertible would be as inaccurateas it would be in the normal callablebond as an investor in convertiblebonds, why should you care aboutsuch a bond? Because it’s busted andbehaves like a normal callable bond,you can leave it to the fixed-incomedepartment. After all, it isn’t reallyaconvertible bond now, anyway.92 July 1997 BLOOMBERGFair enough, except there’s moreto it than that. You need to model thestochastic interest rates because shareprice movements and interest-ratemovements are correlated and thecorrelation strongly affects the valu-ation. That point—that stochasticinterest rates are important because

of the correlation effect—seems tobe continually overlooked by re-searchers and by almost all get a sense of how importantthe correlation effect is, considerFigure 7 from the OVCVscreen. Tab down and enter –0.5

in the VOLAT. & YIELDCORRELATIONfield; press . Type