1、14-15. Calibration in Black Scholes Modeland Binomial TreesMA6622, Ernesto Mordecki, CityU, HK, 2006.References for this Lecture:John C. Hull, Options, Futures K;T;r;) = S(0)(d1)KerT(d2),whereS(0) is the spot price of the stock, measured in local currency.K is the strike price of the option, in the
2、same currency.T is the excercise date of the option, measured in years,r is the annual percent of riskfree interest rate, is the volatility (also annualized).(x) = 12pi integraltextxet2/2dt is the distribution function of a nor-mal standar random variable,d1 = logS(0)/K+r+2/2TTd2 = d1 T.7The value o
3、f a Put Option has a similar formula:P = KerT(d1)S(0)(d2),Remark The option price does not depend on . This is dueto the fact that the price is computed under the risk neutral prob-abiliy Q.More precisely, the evolution of the stock under Q isS(t) = S(0)expparenleftbig(r 2/2)t + W(t)parenrightbig, (
4、1)where now W(t) is a Wiener process under the risk-neutralmeasure Q. The value can be computed asC(S(0);K;T;r;) = EQerTparenleftbigS(T)Kparenrightbig+where the expectation is taken with respect to Q (i.e. for a stockmodelled by (1)8Example Let us compute the price of a call option written onthe Han
5、g Seng Index2, with (that days) price S(0) = 15247.92,struck at K = 15000 expirying in July, with a volatility of 22%.In order to compute the other parameters we take into account:the underlying of the option is 50 HSI, but this is not relevantfor the option price (why?)The option was written on Jun
6、e 14, and it expires in the bussi-ness day inmediatly preceeding the last bussiness of the contractmonth (July): the expiration date is July 29.We then have days = 32 trading days. As the year 2006 hasyear = 247 trading days, we obtain T = days/year = 32/247.We compute the risk-free interest rate fr
7、om the Futures prices,written on the same stock over the same period. We get a futuresquotation of F(T) = 15298 for July 2006. Then, as F(T) =2From the “South China Morning Post”, June 15, 20069S(0)exp(rT), we obtainr = yeardays logparenleftBigF(T)S(0)parenrightBig= 24732 logparenleftBig1529815248pa
8、renrightBig= 0.025.With this information, we computeC(15248;15000;32/247;0.025;0.22) = 639.72.(Newspaper quotation is 640.)Just we are here, we compute by put-call parity the price of theput pption with the same characteristics. Put-Call partity statesthatC + KerT = P + S(0),that, in numbers, isP =
9、640 + 15000e0.025(32/247) 15248 = 343.5.(Quoted price is 342.)1014c. Implied VolatilityIn the previous example, everything is clear with one relevant ex-ception: Why did we used = 0.22?In fact, the real computation process, in what respects the volatil-ity is the contrary: we know from the market th
10、at the option priceis 640, and from this quotation we compute the volatility. Thenumber obtained is what is called implied volatility, and shouldbe distinguished from the volatility in (1).It must be noticed that there is no direct formula to obtain fromthe Black Scholes formula, knowing the price C
11、.In other words, the equationC(15248;15000;32/247;0.025;) = 639.72.can not be inverted to yield . We then use the Newton-Raphsonmethod to find the root .11Suppose that you want to find the root x of an equation f(x) = y,wheref isanincreasing(ordecreasing)differentiablefunction, andwe have an initial
12、 guess x0. By Taylor developementf(x) f(x0) + fprime(x0)(xx0).If we want x to satisfy f(x) = y, then it is natural to assume thatf(x0) + fprime(x0)(xx0) = y,and from this wex = x0 + y f(x0)fprime(x0) .The obtained value of x is nearer to the root than x0. The NewtonRaphson method consists in computi
13、ng a sequencexk+1 = xk + y f(xk)fprime(xk)that converges to the root.12Let us then find the implied volatility of given the quoted optionprice QP. The derivative of the function C with respect to iscalled vega, and is computed asvega() = S(0)T(d1), with d1 as above.So, given an initial value for 0,
14、we compute C(0), and obtainour first approximation:1 = 0 + C(0)QPS(0)T(d1).If this is close to 0 we stop. Otherwise, we compute 2 and stopwhen the sequence stabilizes.Example Let us compute the implied volatility in the previousexample. Suppose we take an initial volatility of 0 = 0.15.13Step 1.We c
15、ompute C(0.15) = 495.Step 2.We compute vega(0.15) = 2029Step 3.We correct1 = 0.15 + 4954602029 = 0.2216Step 4.We compute C(0.2216) = 643.116. We are really near to theimplied volatitliy.Step 5.We compute again vega(0.2216) = 2101.75,Step 6.Finally we obtain2 = 0.2216 + 640643.162101.75 = 0.220134,an
16、d we are done.1414d. Time-dependent volatiliyBlack Scholes theory assumes that volatility is constant over time.We have seen that, from a statistical point of view, that volatiliyvaries over time.What happens with respect to the risk-neutral point of view? Inother terms, is implied volatility consta
17、nt over time?Month Strike Price Volit % Futures rJune 14400 905 27 15241 0.010July 14400 1079 24 15298 0.025August 14400 1117 23 - -September - - - 15296 0.010In this table3 we see that the implied volatiliy (Volit) also variesover time. This series of implied volatilities for of at-the-money3Taken
18、from “South China Morning Post”, June 15, 2006. The spot price is S(0) = 15247.9215options with different maturities is called the term structure ofvolatilites. We included the Futures prices and the correspondingrisk free interest rates4 computed from this price, through theformula F(T) = S(0)exp(r
19、T)This has a remedy in the frame-work of BS theory. Assume thatr = r(t), i.e. the risk-free interest rate is deterministic, butdepends on time = (t), i.e. the same happens with the volatility.We define the forward interest rate, and the forward volatility, asr(t,T) = 1T tintegraldisplay Ttr(s)ds, (t
20、,T)2 = 1T tintegraldisplay Tt(s)ds.(2)here t is today, and T is the expiry of the option.In this model we have a Black-Scholes pricing formula for a Call4The interest rate is negative due to the fact that futures price is smaller than the spot price16option:C(S(t);K;T;r;) = S(t)(d1)Ker(t)t(d2)where
21、t = T t, andd1 = logS(t)/K + r(t) + (t)2/2t(t)t , d2 = d1 (t)radicalbigt.In practice, we do not know the complete curves r(t,T) and(t,T), where T is the parameter. In order to use the time-dependent BS formula we assume that r(t,T) are (t,T) are con-stant between the different expirations.17Example
22、We want to price a call option, written today, July14, (time t) nearly at the money (with strike, say 14400), expir-ing on July, 21 (T). As we do not know the implied forwardvolatiliy (t,T), we interpolate between (t,T1) = 27, where T1corresponds to maturity June, 29, and (t,T2) = 24 where T2corresp
23、onds to maturity August, 30. We have(t,T)2 = (T T1)(t,T1)2 + (T2 T)(t,T2)T2 T1 .We have T T1 = 16, and T2 T = 5, so(t,T)2 = 16272 + 523221 = 692.6and (t,T) = 26.32. We perform the same computation for therisk-free interest rate:r(t,T) = 16(0.01) + 50.02521 = 0.001718The price of the Call option isC(
24、15248;14400;0.0017;27/247;0.2632) = 1043.73.Observe that the raw linear interpolation of the option prices is16905 + 5107921 = 946.429This is due to the fact that the option price depends highly non-linearly on 19Plan of Lecture 15(15a)Volatility Smile(15b)Volatility Matrices(15c)Review of Binomial
25、Trees(15d)Several Steps Binomial Trees(15e)Pricing Options in the Binomial Model(15f)Pricing American Options in the Binomial Model2015a. Volatility SmileLet us see more in details the quotations of option prices5,Month Strike Price Volit %June 13000 2246 35June 13200 2048 34June 13400 1851 33June 1
26、3600 1656 32June 13800 1462 30June 14000 1272 29June 14200 1086 28June 14400 905 27June 14600 733 26June 14800 571 24June 15000 424 23Month Strike Price Volit %June 15200 296 22June 15400 199 21June 15600 124 21June 15800 72 20June 16000 38 20June 16200 18 19June 16400 7 19June 16600 3 19June 16800
27、1 18June 17000 1 20June 17200 1 225“South China Morning Post”, June 15, 20062113000 14000 15000 16000 170002022.52527.53032.535STRIKEIMPLIEDVOLATILITY SMILEWe see that the volatility, far from constant, varies on the strikeprices, forming a smile, or, more precisely, a smirk.This is clear fact showi
28、ng that real markets do not follow Black-Scholes theory.2215b. Volatility MatricesVolatility matrices combine volatility smiles with volatility termstructures, and are used to compute options prices.14400 14600 14800 15000 15200 15400 15600Jun 27 26 24 23 22 21 21Jul 24 24 23 22 21 21 21Aug 23 - - -
29、 21 20 2015800 16000 16200 16400 16600 16800 17000Jun 20 20 19 19 19 18 20Jul 21 21 20 20 20 20 19Aug 20 20 19 19 19 18 18With this matrix in view, we can compute implied volatilites witha reported strike and arbitrary expiration, and with a reportedexpiration and arbitrary strike.23In order to comp
30、ute the implied volatiliy for a non reported expi-ration and a non reported strike (for instance, expiration on July21 and strike 16500) we can computeFirst, by linear interpolation in time, obtain both values of im-plied volatility at strike 16600 and 16400 for the given expiry.Second, use this val
31、ues, interpolating in strike, to obtain thedesired implied volatility.The problem here is that the process computing first the volatilitesinterpolating in strike, and second in time, can produce a differentvalue.Infact, more complexmodels areneeded, as the procedureofstrikeinterpolation has only an
32、empirical basis.2415c. Review of Binomial TreesOur interest is then to consider more flexible models, with moreparameters. Let us first consider the one step binomial tree.Consider then a risky asset with value S(0) at time t = 0, and, attime t = 1, valueS(1) =braceleftBiggS(0)u with probability pS(
33、0)d with probability 1pHere u and d stand for up and down. We are then assuming thatthe returns X defined byS(1)S(0) 1 = Xsatisfy1 + X =braceleftBiggu with probability pd with probability 1p25Let us calibrate this model, i.e. determine the values of the para-meters u,d,p under the risk neutral measu
34、re.Denoting by r the continuous risk free interest rate, the first con-dition is that ertS(t) is a martingale.In this simple case, this amounts to ES(1) = S(0)er, that gives:up + d(1p) = er.Given a value of the implied volatiliy (computed from sometraded derivative), we impose varX = 2. Let us compu
35、tevarX = var(1 + X) = Ebracketleftbig(1 + X)2bracketrightbigbracketleftbigE(1 + X)bracketrightbig2.26We haveE(1 + X) = up + (1p)dE(1 + X) = u2p + (1p)d2giving the conditionvarX = u2p+d2(1p)(up+d(1p)2 = (u+d)2p(1p) = 2.We have two equations for three parameters u,d,p. In order todetermine the paramet
36、ers, it is usual to impose u = 1/d6These three conditions implyp = er dud, u = e, d = e.6Cox, J., Ross,S. and Rubinstein, M. Option Pricing: A simplified approach. Journal of Financial Eco-nomics, 7 (1979).27Remark In practice we take a time increment instead of one,and use annualized values of r an
37、d . The corresponding formulasarep = er dud , u = e, d = e.Example Let us calibrate the Binomial Tree using the valuesof our first example on option pricing. We have = 32/247,r = 0.025, = 0.22. This givesu = 1.082, d = 0.924, p = 0.500677.2815d. Several Steps Binomial TreesIn practice one assumes th
38、at t = 0,2,.,T, with T = N,and construct the several step binomial tree under the assumptionof time-space homogeneity.This assumption is equivalent to the Black-Scholes assumptionthat the risk-free rate and the volatiliy are constant over time andspace.The result of this assumption is that at each o
39、f the two nodes,resulting from the first step, the future evolution of the asset pricereproduces as in the first step.In order to model the stock prices we label each node by (n,i),where n is the step, and i the number of upwards movements. Atstep n we have i = 0,.,n, and we denote by j = n i thenum
40、ber of downwards movements. We obtain, given i, that the29stock price takes the valueS(n) = S(0)uidni = sn(i),where sn(i) is a notation. Let us now compute the probability ofreaching the value sn(i). We neeed exactly i ups (and j = nidowns), but they can come in different orders. There areCni = n!i!(n1)!,different ways of obtaining i ups, each has a probability p, theyare independent, soPS(n) = S(0)uidj = Cni pi(1p)ni = Pn(i).(where Pn(i) is a notation). The conclussion is that the stock priceevolves according to the formulaS(n) = S(0)uidni, with probability Pn(i), for i = 0,.,n.30
