AMS Portfolio Theory and Capital Markets

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1 AMS Portfolio Theory ad Capital Markets I Class 6 - Asset yamics Robert J. Frey Research Professor Stoy Brook iversity, Applied Mathematics ad Statistics frey@ams.suysb.edu This material itroduces stochastic calculus. March 06, 005. Biomial Lattice Redux Biomial Lattices - A useful special case of a lattice is the biomial lattice. At each step oe of the braches of a ode combies with oe of the braches of aother. Thus, istead of leaf odes, there are oly + leaves. Example - Cosider the followig model for the price dyamics of a fiacial asset. Let p t deote the price of the asset at time t. At time t + with probability the asset will be u p t ad with probability = the asset will be d p t. If we set d = /u, the after up steps ad m dow steps the price will be u -m p t regardless of the specific sequece of up ad dow steps it took to get there.. Geometric Biomial Lattice The model, called the geometric biomial lattice, is a useful model for describig the behavior of asset prices. Cosider the price model above i which p 0 =, = 0., = 0.5, ad H + r L =.0, the poits represetig three steps of the model are

2 ams-q0-lec-06-p.b We ca plot a full te steps o graph with time alog the x-axis ad price alog the y-axis Limitig istributio Cosider a geometric biomial lattice as described above with steps ad each step beig of duratio = /. If we umber the odes from top to bottom i step > 0 from 0 to, the the probability that ode j is reached is determied by the biomial distributio j = K j O j H - L - j, j e 80,, < A plot of the pdf at the leave odes i the example above, i.e., step 0 or t =, is If we take the log of the x-values we get somethig that look very much like a discrete approximatio to a Normal distributio. The plot below has the distributio of log price with a Normal distributio overlayed o it.

3 ams-q0-lec-06-p.b As the umber of steps Ø or the size of the time step Ø 0, the distributio of log prices asymtotically approaches a Normal distributio. Whe log[x] is Normally distributed, the X is said to be log Normally distributed. 4. Further evelopmet of the Biomial Model We ll cosider a stadard biomial process t e [0, ] usig steps, each of legth = /, is BHtL = BHt - L + XHtL è!!!! where the X(t) are i.i.d. radom variables +, with prob. ê XHtL = ; -, with prob. ê Thus, X(t) is ot observed util the ed of the iterval at t. I other words, there is o lookahead to future radom evets. At time (t ) we have BHtL = BHtL - BHt - L = XHtL è!!!! E@BHtL = E@BHtL + BHt - L = BHt - L Thus, B(t) is a martigale. The B(t) are called the iovatios because they represet ew iformatio ot cotaied i the iformatio available prior to t. Note that each B is iid with mea zero ad variace. Thus, the sum of such iovatios has mea 0 ad variace = (/) =. By appealig to the Cetral Limit Theorem we ca assert that for sufficietly large (or small ) ad assumig B(0) = 0 BHk L = BHL ~N@0, k= This process is stadardized so that idepedet of ad it has a mea of zero ad stadard deviatio of oe over a uit time iterval. As Ø, as a cosequece of the Cetral Limit Theorem, it approaches stadard Normal distributio over that uit time iterval. Note more geerally, we have for B(0) = 0, a arbitrary t > 0 ad a sufficiet umber of time steps BHtL~ NA0, è!! t E 5. Stochastic Itegral Whe Ø 0 ( Ø ), we ca thik of the limit of this process as a itegral whose result is a radom variable.thus, makig the substitutio BHtL Ø d BHtL as Ø 0

4 4 ams-q0-lec-06-p.b lim Ø 0 Hor Ø L k= BHkL = d BHtL ~ N@0, 0 We have made some sese of this itegral as the limit of a discrete biomial process, but it does ot quite have the same meaig as does the correspodig determiistic case. 6. (db(t)) = dt I the determiistic case with a smooth fuctio as the stepsize Ø 0 the higher powers of are domiated by itself. However, i the stochastic case, the limit of the biomial process we examied is ot differetiable i the covetioal sese: it evers smooths out. Now for steps for t from 0 to of size = / we have BHtL - BHt - L = B(t) = ± è!!!! ad HBHtLL =. This meas that as Ø 0 we have {HBHtLL = } Ø { Hd BHtLL = d t}. To see this aother way, we already kow that k= the value of k= as Ø 0 Fially, it s clear tha the sum k= BHkL is approximately ~ N[0, ] for sufficietly large. What the is HBHkLL? The sum equals I è!!!! M = = (/) =. Thus, makig the substitutio BHtL Ø d BHtL 0 Hd BHtLL = = 0 d t fl Hd BHtLL = d t HBHkLL k quickly becomes egligible relative to for k >. Stadard calculus is based o the otio that as Ø 0 higher order effects become eglible. A stochastic calculus ca ot make that assumptio. For the class of problems we ll be lookig at the first two orders, ad, must be cosidered i order to capture both the mea ad variace effects. 7. The Wieer Process W(t) Istead of a discrete biomial process, cosider a cotiuous process W(t) such that the differeces are i.i.d. radom variables such that for > 0 WHtL WHt L = WHtL ~ NA0, è!!!! E As before, we ca divide the iterval t e [0, ] ito equal steps of size = /. Clearly, WHkL ~ N@0, k= like the biomial process B(t) i (3) which approaches a Normally distributed radom variable i the limit, the equatio above is a simple cosequece of the fact that the icremets of W(t) are iid Normal rv s whose stadard deviatios are the square root of the size of the icremet. The radom process W(t) is a called a Wieer process. As we did with B(t) above, we ca defie: lim Ø 0 Hor Ø L k= WHkL = d WHtL ~ N@0, 0 As before, we have for a arbitrary t > 0 WHtL~ NA0, è!! t E

5 ams-q0-lec-06-p.b 5 8. (dw(t)) Æ dt i Mea Square For the biomial process examied above we determied that Hd BHtLL = d t. We could do this because, for a give step size, the value of HBHtLL = determiistically. For a Weier process the value of HWHtLL is itself a radom variable; however, give that WHtL is N[0, è!!!! ], we do kow that E[HWHtLL ]. i.e., the variace of WHtL, is. As Ø 0 ( Ø ) we gai more ad more samples over ay fiite iterval so VarAEAHWHtLL E Ø 0 over that iterval. We say that (dw(t)) Ø dt i mea square as Ø 0. I other words, the limit is ot detemiistic but we ca make the variace arbitrarily small by choosig a sufficietly small iterval. 9. Simulatio Studies We ll illustrate these otios for the stadard Wieer proces W(t). For each case we ll geerate 0 samples of the sum k= of duratio = /. First, cosider the sum k= WHkL k for k = ad over the period t e [0, ] for time steps WHkL usig = 000 steps. As expected, W() ~ N[0, ]. Number of Steps = 000; Sample Size = 0 W t - - Next, cosider the sum k= WHkL usig 0, 00 ad 000 steps. As the umber of steps icreases the samples coverge more ad more closely to t. This is what you would expect with mea square covergece: The variace decreases as the umber of samples icreases. Number of Steps = 0; Sample Size = 0 W t

6 6 ams-q0-lec-06-p.b Number of Steps = 00; Sample Size = 0 W t Number of Steps = 000; Sample Size = 0 W t 0. Itô Processes Cosider the followig stochastic process, called a Itô process d XHtL = ahxhtl, tl d t + bhxhtl, tl d WHtL The ahxhtl, tl d t is ofte called the drift term ad correspods to the mea of the process ad the bhxhtl, tl d WHtL the volatility term ad correspods to the stadard deviatio of the process. d LHtL = m d t + s d WHtL The, makig the simplifyig assumptio that L(0) = 0, we have L(t) ~ N[m t, s è!! t ]. The erivative of a Smooth Fuctio Cosider the Taylor series represetig a fuctio f(t) about t 0 where ft is the first derivative, ft t is the secod ad so forth: f Ht - t 0 L = f Ht 0 L + f t Ht 0 L Ht - t 0 L + ÅÅÅÅÅ f t t Ht 0 L Ht - t 0 L + ÅÅÅÅÅ 6 f t t t Ht 0 L Ht - t 0 L 3 + For values of t sufficiet close to t 0 we ca eglect the higher order terms ad ad oly iclude the first two to produce the well-kow approximatio:

7 ams-q0-lec-06-p.b 7 f Ht + f HtL + f t HtL t This ca also be motivated geometrically by otig that the first derivative at a poit is the slope of the taget at that poit. As ca be see below, give a smooth fuctio, there is a regio about a give poit i which the taget is close to the origial fuctio: erivative As Local Approximatio The Chai Rule Cosider d x = ahx, tl dt ad a fuctio g(x, t) which is differetiable with respect to both x ad t, the if y = g(x, t) we ca write dy as a Taylor series, discardig the higher order terms d y = ÅÅÅÅÅÅ Å d t + ÅÅÅÅÅÅÅÅ x d x + ÅÅÅÅÅ i j g ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k HL Hd tl + g ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ x d t d x + g ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H dxl Hd y xl z + OAHd xl 3 E { The higher tha liear order terms i the series become eglible at sufficietly small time scales. Thus, we drop those terms ad get Substitutig d x = ahx, tl dt ito the above gives d y = ÅÅÅÅÅÅÅÅ d y = K ÅÅÅÅÅÅÅÅ d t + ÅÅÅÅÅÅÅÅ x d x + ÅÅÅÅÅÅÅÅ ahx, tlo d t x 3. Itô s Lemma Itô s Lemma Let g(x, t) be a fuctio which is at least twice differetiable with respect to x ad oce differetiable with respect to t, the Y(t) = g(x(t), t) follows a Itô process with the same Weier process W(t) d YHtL = i j ÅÅÅÅÅÅÅÅÅÅ k X a + ÅÅÅÅÅÅÅÅ + ÅÅÅÅÅ g ÅÅÅÅÅÅÅÅÅÅÅÅ X y b z d t + { ÅÅÅÅÅÅÅÅÅÅ X b d WHtL

8 8 ams-q0-lec-06-p.b i which the derivatives of g ad the coefficiets a ad b deped o the argumets (X(t), t). Proof: First, write dy(t) as a Taylor series discardig terms which are egligible d YHtL = ÅÅÅÅÅÅÅÅ d t + ÅÅÅÅÅÅÅÅÅÅ X d X HtL + ÅÅÅÅÅ i j g ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ k HL Hd g tl + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ XHtL d t d x + g ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H XHtLL d YHtL = ÅÅÅÅÅÅÅÅ d t + ÅÅÅÅÅÅÅÅÅÅ X d XHtL + ÅÅÅÅÅ g ÅÅÅÅÅÅÅÅÅÅÅÅ Hd X XHtLL Hd y XHtLL z + OAHd XHtLL 3 E { The term i red is the secod order effect from the radom process that, i cotrast to the determiistic case, ca ot be igored. Substitutig dx(t) ad (dx(t)) i the above Expadig d YHtL = ÅÅÅÅÅÅÅÅ d YHtL = ÅÅÅÅÅÅÅÅ d t + ÅÅÅÅÅÅÅÅÅÅ X d t + ÅÅÅÅÅÅÅÅÅÅ X Ha d t + b d W HtLL + ÅÅÅÅÅ Ha d t + b d W HtLL + ÅÅÅÅÅ g ÅÅÅÅÅÅÅÅÅÅÅÅ Ha d t + b d W X HtLL g ÅÅÅÅÅÅÅÅÅÅÅÅ X Ja Hd tl + a bhd tl H d WHtLL + b Hd W HtLL N Ad, agai, discardig terms which are egligible ad fially replacig Hd WHtLL with d t yields d YHtL = ÅÅÅÅÅÅÅÅ d t + ÅÅÅÅÅÅÅÅÅÅ After some simple rearragemet we have the fial form above. d YHtL = i j ÅÅÅÅÅÅÅÅÅÅ k X a + ÅÅÅÅÅÅÅÅ X a d t + ÅÅÅÅÅÅÅÅÅÅ X b d W HtL + ÅÅÅÅÅ + ÅÅÅÅÅ g ÅÅÅÅÅÅÅÅÅÅÅÅ X by z d t + { g ÅÅÅÅÅÅÅÅÅÅÅÅ X b d t ÅÅÅÅÅÅÅÅÅÅ X b d WHtL Strictly speakig, this is ot a rigorous proof; however, the approach of expadig the differetial as a Taylor series, discardig the terms other tha d t, d WHtL ad Hd WHtLL, ad the replacig Hd WHtLL with d t is oe that works well for most practical problems. 4. A Simple Stock Model We ca adapt the above to model geometric Browia motio of a stock price S(t) with costat drift ad volatility d SHtL = m SHtL d t + s SHtL d WHtL Sometimes this is expressed i terms of the istateous rate of retur of the stock. ividig both sides by S(t) d SHtL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = m d t + s d WHtL SHtL Note that our earlier experimets with the geometric biomial lattice (which i the limit is the same at the price model above) it appeared that the log price was Normally distributed. If we take g(s(t), t) = log S(t) ad apply Itô s lemma the we get (see p. 33 of Lueberger s text) d log SHtL = Km - ÅÅÅÅÅ s O d t + s d WHtL Thus, log S(t) is Normally distributed with a mea of S(0) + Im - ÅÅÅÅ s M t ad a stadard deviatio of s è!! t.

18.S096 Problem Set 5 Fall 2013 Volatility Modeling Due Date: 10/29/2013

18.S096 Problem Set 5 Fall 2013 Volatility Modelig Due Date: 10/29/2013 1. Sample Estimators of Diffusio Process Volatility ad Drift Let {X t } be the price of a fiacial security that follows a geometric