Lecture 2 - Advanced Fixed Income Analytics Backus& Zin...

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Unformatted text preview: Advanced Fixed Income Analytics Backus & Zin April 1, 1999 Options 1: The Term Structure of Volatility 1. Approaches to valuation review and extension 2. The Black-Scholes formula history and meaning 3. Option sensitivities 4. Term structure of volatility 5. Extended Vasicek 6. Options on zeros 7. Options on coupon bonds 8. Options on eurocurrency futures 9. Normals and log-normals 10. Summary and nal thoughts Lecture 2 Advanced Fixed Income Analytics 2-2 1. The Importance of Black-Scholes Elegant solution to option price once you get used to it A purely academic innovation Now the industry standard in most markets Played critical role in growth of option markets Together with related work by Samuelson and Merton, triggered an explosion of work that laid the basis for modern nance in universities and on Wall Street New approach to valuation: replication, a precursor to the kinds of arbitrage-free models we are using Constant interest rate: fortunately not essential or applications to xed income would have died out rapidly Advanced Fixed Income Analytics 2. Two Approaches to Valuation Problem: value random future cash ows ct+n Recursive valuation one period at a time: In period t + n , 1, value is pt+n,1 = Et+n,1 mt+n ct+n In period t + n , 2, value is pt+n,2 = Et+n,2 mt+n,1 pt+n,1 Etc. In period t, value is pt = Et mt+1 pt+1 All-at-once" valuation: Apply multiperiod stochastic discount factor" Mt;t+n: pt = Et Mt;t+nct+n Same answer if Mt;t+n = mt+1mt+2    mt+n Two-period zero: recursive approach pt = Et mt+1pt+1  = Et mt+1Et+1 mt+2  1  = Et mt+1 mt+2  1 : Two-period zero: all-at-once" approach pt = Et Mt;t+2  1 : 2-3 Advanced Fixed Income Analytics 2-4 3. Option Basics A European call option is the right to buy an asset at a xed price K at a xed date n periods in the future S is the price of the underlying K is the strike price n is the maturity of the option a call generates cash ow at t + n of ct+n = St+n , K + where x+ means maxx; 0 the positive part of x The price C of a call satis es h i Ctn = Et Mt;t+n St+n , K + : A put option is a comparable right to sell an asset at price K : ct+n = K , St+n+ A forward contract arranged at date t speci es the purchase at t + n of an asset for price Ftn the forward price, xed at t Prices of puts P  and calls C  are related by put-call parity : Ctn , Ptn = bnFtn , K  t This allows us to concentrate on calls, knowing we can convert any answer we get to puts. Advanced Fixed Income Analytics 2-5 4. The Black-Scholes Formula Assume: log Mt;t+n; log St+n is bivariate normal we say Mt;t+n; St+n is log-normal Then Black-Scholes formula: Ctn = bnFtn d , bnK d , w t t bn t n Ft  w2 d = = = = = n-period discount factor forward price of the underlying cumulative normal distribution function Var t log St+n logFtn =K  + w2=2 w Comments: a little di erent from the original, but more useful for our purposes industry standard in most markets similar formulas were derived by Merton dividend-paying stocks, Black futures, and Biger-Hull and Garman-Kohlhagen currencies one of the best parts: the only component of the formula that's not observed in cash markets is volatility we can turn this around: given call price, nd volatility describes implicitly how changes in F; W; b a ect C Advanced Fixed Income Analytics 2-6 4. The Black-Scholes Formula continued Sample inputs streamlined notation: F = 95:00 K = 97:50 slightly out of the money maturity = 1 year b = exp,:0500 = 0:9512 w = 17:30 Intermediate calculations: d = ,0:0636 d = 0:4746 d , w = ,0:2366 d , w = 0:4065 Call price: C = b F d , K d , w = 0:9512 950:4746 , 97:50:4065 = 5:193 Put price for same strike use put-call parity: P = C , bF , K  = 5:193 , 0:951295 , 97:5 = 7:571 we most often use this in reverse: convert a put to a call and use the Black-Scholes call formula. Advanced Fixed Income Analytics 2-7 5. Black-Scholes: How We Got There Step 1: Integral formulas Setting: let x1 ; x2 = log X1; log X2 be bivariate normal with mean  and variance : 2 3 2 2 3 1 5 ;  = 4 1 12 5  = 4 2 2 12 2 Think of x1 ; x2 = log Mt;t+n; log St+n. Formula 1: Z1 E X1 = 2 1,1 ,1 ex e,x ,  =2 dx1 2 = exp1 + 1 =2 1 1 1 2 2 1 Formula 2: Z Z ,1 jj,1=2 1 1 ex e ,x, E X1jX2  K  = 2 ,1 log K = E X1d; with d = 2 , log K + 12 1 ,1 x,=2 dx2dx1 2 Formula 3: Z Z ,1 jj,1=2 1 1 ex +x e ,x, E X1X2jX2  K  = 2 ,1 log K = E X1X2d; with 2 2 , log K + 12 + 2 d= 1 2 2 ,1 x,=2 dx2dx1 Advanced Fixed Income Analytics 2-8 5. Black-Scholes: How We Got There continued Step 2: Black-Scholes Call prices satisfy Ctn = Et Mt;t+nSt+njSt+n  K  , KEt Mt;t+njSt+n  K  A forward contract arranged at date t speci es the purchase at t + n of an asset for a price Ftn set at t Forward price Ftn satis es 0 = Et Mt;t+n St+n , Ftn  The usual pricing relation with current payment of zero. Solution includes use Formula 1: 2 2 bnFtn = Et Mt;t+nSt+n = exp1 + 2 + 1 + 2 + 2 12 =2 t 2 bn = Et Mt;t+n = exp1 + 1 =2 t 2 Ftn = Et Mt;t+nSt+n =Et Mt;t+n = exp2 + 12 + 2 =2: Term 1 use Formula 3: Et Mt;t+nSt+njSt+n  K  = bnFtn d t with n 2 2 d = 2 , log K + 12 + 2 = logFt =K  + 2 =2 : 2 2 Term 2 use Formula 2: Et Mt;t+njSt+n  K  K = bnK d0 ; t with n 2 0 = 2 , log K + 12 = logFt =K  , 2 =2 = d , d 2 2 2 Advanced Fixed Income Analytics 2-9 5. Black-Scholes: How We Got There continued Comments Clever substitution:  volatility w is the only thing not observable directly from other markets  the features of M matter only via their e ect on b and F Role of log-normality:  assumption is that log of underlying | whatever that might be is normal  prices of many assets exhibit more frequent large changes than this implies kurtosis, skewness Is model appropriate?  it's a good starting point  and the industry standard in most markets  a useful benchmark even when not strictly appropriate mathematically Advanced Fixed Income Analytics 2-10 6. Implied Volatility Annual units it's standard practice to measure maturity in years if periods are h years, then maturity is N = hn volatility v below in annual units is: Nv2 = w2  v = w=N 1=2 v is often multiplied by 100 and reported as a percentage Black-Scholes formula becomes and this will be our standard: CtN = bN FtN d , bN K d , N 1=2v t t N 2  d = logFt =K=2+ Nv =2 N1 v Implied volatility: given a call price, nd the value of v that satis es the formula since the price is increasing in v, we can generally nd a solution numerically limits: the minimum call price is bN FtN , K + v = 0, the t N F N for v = 1 maximum is bt t example continued: if call price is 6.00, v = 19:54 Advanced Fixed Income Analytics 2-11 6. Implied Volatility continued Numerical methods bene t from good rst guess If solution fails, check limits Brenner-Subrahmanyam approximations for options at-the-money forward F = K : 1=2 N 1=2v = 2bF C N 1=2v bF C = 21=2 Example continued Let K = 95 at-the-money forward With v = 17:30 same as before, call price is C = 6:229 Brenner-Subrahmanyam approximation: 2 1=2 v = 0: 6:229 951295 = 17:28! Comment: this is more accuracy than is justi ed by the underlying data, but it illustrates the quality of the approximation. Advanced Fixed Income Analytics 7. Sensitivities 2-12 E ect of small changes in F : @C delta" = @F = bd in-the-money options more sensitive than out-of-the-money E ect of small changes in v: vega" = @C @v = bF 0 dN 1=2 more sensitive at d = 0 | roughly at the money E ect of small changes in r use b = e,rN : rho" = @C @r = ,NC Comments: standard approach to quantifying risk in options based on Black-Scholes: other models imply di erent sensitivities volatility critical: dealers tend to be short options customers buy puts and calls, so they are typically short volatility" even if they are delta hedged" xed income poses basic problem with the approach: hard to disentangle say delta and rho , since both re ect changes in interest rates  need to tie all of the components to underlying variables | a z , so to speak Advanced Fixed Income Analytics 2-13 8. Volatility Term Structures In the theoretical environment that led to Black-Scholes, volatility v is the same for options of di erent maturities and strikes In xed income markets, volatility v varies across several dimensions: maturity of the option N  tenor" of underlying asset strike price K  Volatility term structures for March 19, 1999 interest rate caps: Maturity yrs Volatility  1 12.50 2 15.50 3 16.55 5 17.25 7 17.15 10 16.55 swaption volatilities : Maturity 1 mo 6 mos 1 yr 5 yrs 1 yrs 10.00 13.00 15.65 16.40 2 yrs 11.75 14.00 15.55 15.85 Tenor 5 yrs 10 yrs 13.88 13.88 14.75 14.60 15.20 14.90 15.10 14.20 20 yrs 11.30 11.80 11.90 10.55 Advanced Fixed Income Analytics 2-14 9. Extended Vasicek Model Model generates Black-Scholes option prices for zeros Extended Vasicek model translated from Hull and White: model consists of , log mt+1 = 2=2 + zt + "t+1 zt+1 = 1 , ' t + 'zt + "t+1 bond prices satisfy log bn = ,Ant , Bnzt t Bn = 1 + ' + '2 +    + 'n,1 details:  ts structure of Black-Scholes formula: log Mt;t+n; log bt+n are bivariate normal for all maturities n and tenors  extended" refers to the t" subscripts on and An allows us to reproduce current spot rates exactly Forward prices: a forward contract locks in a price F on an n + -period zero in n periods price now bn times F  must equal current price: bnF = bn+ for completeness, we should probably write F as F n; Key issue: what are the volatilities? Advanced Fixed Income Analytics 2-15 10. Volatility Term Structures in Vasicek How does volatility vary in theory! with maturity and tenor? Volatility of bond prices: log bt+n = ,A , B zt+n Var t log bt+n = B 2Var tzt+n E ect of maturity: Var tzt+n  = = 2 1 + '2 + '4 +    + '2n,1 0 2n 1 A 1 , '2 2 @1 , ' If 0 ' 1, this grows more slowly than n E ect of tenor: how B varies with Annualized per-period volatility is 0 10 1 Var tzt+n = B 2 @ 2 A @ 1 , '2n A vn; = B 2 hn hn 1 , '2 Comments: note that volatility has two dimensions: it's a matrix, like the swaption volatility matrix we'll see shortly e ects of mean reversion apparent through  maturity: v declines with n if 0 ' 1 constant if ' = 1  tenor: v increases with , but the rate of increase is greater if ' is closer to one model required | not optional | for B ; Var tzt+n Advanced Fixed Income Analytics 2-16 10. Volatility Term Structures in Vasicek continued Volatility term structures for 6-month and 5-year zeros 5 4.5 Volatility (Annual Percentage) 4 3.5 5−year zeros 3 2.5 2 1.5 6−month zeros 1 0.5 0 0 20 40 60 80 Maturity of Option in Months Less mean reversion ' closer to one . . . reduces rate of decline with maturity increases volatility of long zeros 100 120 Advanced Fixed Income Analytics 2-17 11. Options on Coupon Bonds A coupon bond has known cash ows on xed dates label set of dates by J = fn1; n2; n3; : : :g eg, with a monthly time interval and semiannual payments, a 5-year bond would have payments at J = f6; 12; 18; 24; 30; 36; 42; 48; 54; 60g label cash ows by cj for each j 2 J Bond price is pt = X j 2J cj bn t j If prices of zeros are log-normal, price of a coupon bond is not log of a sum not equal to sum of logs  Black-Scholes not strictly appropriate Nevertheless: treat Black-Scholes as a convenient reporting tool eg, its implied volatility an approximation to the exact formula Advanced Fixed Income Analytics 2-18 12. Options on Eurodollar Futures Money market futures: similar contracts on eurodollars CME, short sterling LIFFE, euros LIFFE, euroyen TIFFE, etc maturities out 10-15 years price F related to yield" Y by F = 100 , Y settles at Y = 3-m LIBOR Call options on these contracts are claims to future cash ow 90 F , K +  360 equivalent to put options on the yield 90 90  100 , Y , K +  360 =  100 , K , Y +  360 90 = KY , Y +  360 Similarly: put options are equivalent to calls on the yield 90 90 K , F +  360 =  100 , K , 100 , Y +  360 90 = Y , KY +  360 Advanced Fixed Income Analytics 2-19 12. Options on Eurodollar Futures continued Option quotes on Mar 16 for options on the June contract: F = 94:955 N = 0:25 3 months b = 0:9874 roughly 5 for 3 months call price mid of bad ask for K = 95:00: C = 0:0425 put price mid of bad ask for K = 95:00: P = 0:0925 Yield-based implied volatilities streamlined notation: C = b100 , F d , b100 , K d , N 1=2v log 100 , F =100 , K  + Nv2=2 d = N 1=2v This is more work than nding volatility for the futures directly, but connects futures to related OTC markets for rates. put is equivalent to a call on the yield:  volatility solution to formula is v = 0:0687 call is equivalent to a put on the yield:  put-call parity gives comparable call price as CY = PY + b 100 , F  , 100 , K  = 0:0425 + :98740:045 = 0:0869  volatility is v = 0:0629 Comment: do this slowly and carefully! Advanced Fixed Income Analytics 2-20 12. Options on Eurodollar Futures continued Current term structure of yield volatilities: Contract Maturity Futures Price Jun 99 0.25 94.955 Sep 99 0.50 94.860 Dec 99 0.75 94.550 Mar 00 1.00 94.640 Jun 00 1.25 94.550 Sep 00 1.50 94.475 Strike Put Price Volatility 95.00 0.0925 0.0687 94.75 0.1075 0.1087 94.50 0.2500 0.1502 94.75 0.3650 0.1541 94.50 0.3500 0.1628 94.50 0.5700 0.1766 Strikes are closest to at-the-money. Discount factors are bN = e,N r , with r = 0:05. Comments: contracts are for xed tenor but di erent maturities good source of yield volatilities standard model input suggests that markets are more complex than Vasicek standard pattern shows declines in volatility at long maturities, like Vasicek, but increases at short end Advanced Fixed Income Analytics 2-21 13. On Normals and Log-Normals We have made two di erent assumptions about interest rates rates are normal: built into Vasicek logs of prices are normal, interest rates are approx normal rates are log-normal: our analysis of eurodollar futures is based on log-normal yields; caps and swaptions are generally treated the same way 0.25 Probability Density Function 0.2 log−normal 0.15 0.1 normal 0.05 0 0 5 10 3−Month LIBOR in 12 Months 15 This apparently technical issue can have a large e ect on option prices, especially for options far out of the money Advanced Fixed Income Analytics 2-22 Summary 1. The Black-Scholes formula is both a theoretical benchmark for option pricing and an industry standard for quoting prices. 2. Black-Scholes can be viewed as a theoretical framework for thinking about options a model, a convention for quoting prices a formula, or both. 3. In xed income markets, volatility varies across both the maturity of the option and the tenor" of the underlying instrument. 4. In Vasicek, the relation between volatility across maturities and tenors is controlled largely by the mean reversion parameter. 5. Market prices suggest two issues that deserve further attention: the di erence between normal and log-normal interest rates the shape of the volatility term structure ...
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