AMS 691-03 Fnancal Dervatves I Fnal Examnaton (Take Home) Due not later than 5:00 PM, Tuesday, 14 December 2004 Robert J. Frey Research Professor Stony Brook Unversty, Appled Mathematcs and Statstcs frey@ams.sunysb.edu Ths s a fnal take home examnaton. You can consult outsde sources for general nformaton and revew but must complete the questons themselves on your own. 08 December 2004 1. Consol Bond Valuaton A consol s a bond whch pays a fxed amount at a fxed nterval n perpetuty. Let G t denote the prce of a consol bond payng amount g at monthly ntervals. Let r denote the annual rskfree rate of return and assume t s constant for all tme. Further assume that the consol s soveregn debt wth no default rsk. Wrte a closed form expresson for the prce of the consol G t. Assume we are at tme t = 0. Also, note that 1 + x + x 2 + = 1 / (1 x) and that under contnuous nterest the dscount factor for a payment t years hence s exp[ t r ]. As an asde one could also use smple nterest n whch case the dscount would be 1 / (1 + t r). The choce depends on the conventons of the market you fnd yourself n. We wll assume the former,.e., contnuous compoundng, verson here. Frst, consder the case n whch we are due to receve the next payment mmedately. In that nstance we receve payments at monthly ntervals;.e., {0, 1/12, 2/12,..., k/12,...}. The dscounted present value of the payments s G 0 = exp@-k r ê 12D g = g exp@-r ê 12D k 1 = g ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 - exp@-r ê 12D k=0 k=0 More generally, f we are due to receve our next payment at tme t next 0 wth tme stated n years to be consstent, then we only need to dscount the sum above G 0 = exp@-r t next D k jg 1 y exp@-r t next D ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ z = g ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 - exp@-r ê 12D { 1 - exp@-r ê 12D (1) (2)
2 ams-q03-fnal-p 2. Condtonal Probabltes Let be a Posson dstrbuted random varable wth mean l. Let {Z 1, Z 1,, Z k, } be a sequence of..d. ormally dstrbuted random varables wth mean m and standard devaton s. Defne X as a random varable whch s the sum of a random number of random varables: X = k=1 Z k (3) Wrte closed form expressons for the expectaton, E[ X], and varance, Var[X], of X. Frst, we consder E@XD: É n ÉÉ E@XD = E Z k = E E Z k = n k=1 ƒ (4) ow by the ndependence of and {Z 1, Z 1,, Z k, } E Z k = n = ne@zd k=1 ƒ É (5) Thus, ÉÉ E E Z k = n = E@nE@ZDD = E@< E@ZD = l m k=1 ƒ (6) ext, Var[X]. Frst consder the computaton of E@X 2 D 2É EAX 2 E = E j y Z k z = E E j y Z k z k k=1 { k k=1 { ow for a r.v. E@X 2 D = Var@XD + E@XD 2 so n E j y Z k z k k=1 { and 2 ƒ 2 ƒ ÉÉ = n É n É = n = n Var Z k + n 2 E@ZD 2 2É E j y Z k z = E@D Var@ZD + EA 2 E E 8ZD 2 k k=1 { Recall that l s also the varance of the Posson r.v.. Thus, we have É 2É y É2 Var Z k = E j Z k z - E Z k = E@D Var@ZD + E@ZD 2 Var@D = lis 2 + m 2 M k k=1 { (7) (8) (9) (10)
ams-q03-fnal-p 3 3. Rsk eutral Prcng Let the SDE descrbng the evoluton of a stock be d S t = m S t d t + s S t d W t, 0 t T (11) where W t s the standard Wener process. Let r denote the annual rskfree rate and assume throughout that t s constant. A trader holds an opton whch at expry T allows her to receve the value of a call opton wth strke K or a zero coupon bond maturng at tme M ( T) and payng Z M. Let H t denote the prce of the opton at tme t T. Provde an economc argument for how ths opton ought to be valued and wrte a closed-form expresson for ts prce functon. Denote the Black-Scholes formula for a European call at tme t for an underlyng stock wth current prce S t and volatlty s, rskfree rate r, and opton strke K and expry T by C@t, 8S t, s<, r, 8K, T<D Let Z@tD denote the prce of the zero at tme 0 t M and note that (12) Z@tD = exp@-rhm - tld Z M (13) At expry the opton has the value whch the s max of the value of the call opton or the zero coupon bond: H T = max@max@s T - K, 0D, exp@-rhm - TLD Z M D = max@s T - K, exp@-rhm - TLD Z M D (14) ote, however, that max[a, b] = max[a b, 0] + b so we can restate the last bt as max@s T - K, exp@-rhm - TLD Z M D = max@s T - HK + exp@-rhm - TLD Z M L, 0D + exp@-rhm - TLD Z M (15) Under the rsk neutral measure P è we have H t = exp@-rht - tld E Pè @H T D = exp@-rht - tld J E Pè @max@s T - HK + exp@-rhm - TLD Z M L, 0DD + exp@-rhm - TLD Z M H t = exp@-rht - tld E Pè @max@s T - HK + exp@-rhm - TLD Z M L, 0DD + exp@-rhm - tld Z M (16) (17) In terms of (12) and (13) C@t, 8S t, s<, r, 8K + Z@TD, T<D + Z@tD (18) Thus, the prce of the opton s the prce of a call wth expry T and strke K + Z@TD plus the zero coupon bond wth current prce Z@tD. 4. Itô s Lemma Consder the stochastc process X t defned by the constant coeffcent SDE d X t = m d t + s d W t, 0 t (19) where W t s a standard Wener process. Further, let Y t be the stochastc process defned by the constant coeffcent SDE d Y t = n d t + u d W t, 0 t (20)
4 ams-q03-fnal-p where W t s another standard Wener process whch s uncorrelated to the one above. Wrte SDEs for the followng stochastc processes (a) HX t L 2 (b) exp@x t D (c) X t - Y t In each case let Z t denote the random varable n queston. Frst, note Itô s lemma for Z t = f (X t, t) gven X t n (17): d Z t = j ÅÅÅÅÅÅÅÅÅÅ f k X m + ÅÅÅÅÅÅÅÅ f t + 1 ÅÅÅÅÅ 2 2 f ÅÅÅÅÅÅÅÅÅÅÅÅ X 2 s2y { z d t + ÅÅÅÅÅÅÅÅÅÅ f X s d W t (a) ote that f / t = 0. Also, the nverse of X 2 s not a functon, so we choose the postve branch X t = è!!!!! Z t : d Z t = I2 m è!!!!! Z t + s 2 M d t + 2 s è!!!!! Z t d W t (21) (22) (b) Agan, note that f / t = 0. Ths case s just a varant of geometrc Brownan moton and llustrates the need for the usual H- s 2 ê 2L correcton term: d Z t = k jm + ÅÅÅÅÅÅÅÅ s2 y z Z t d t + s Z t d W t 2 { (c) In ths case, we note that the dfference X t - Y t s a ormally dstrbuted random varable wth mean (m n) t and varance Hs 2 + u 2 ) t. Let W t be a standard Wener process, then d Z t = Hm - nl d t + è!!!!!!!!!!!!!!! s 2 + u 2 d W t (24) (23) 5. Monte Carlo A call opton wth strke K and expry T s wrtten on a stock. Let S t denote the prce of the stock at tme t. The stock s prce s assumed to evolve accordng to a constant coeffcent Brownan moton wth mean m and volatlty s but wth one excepton: At a specfed tme 0 < d < T the stock pays a dvdend d S d, where d s unformly dstrbuted on the nterval [a, b] such that 0 < a < b < 1. Descrbe how you would construct a Monte Carlo smulaton to estmate the value of the call opton at tme t = 0. (a) Provde an economc argument for the approach you choose. (b) Detal how you would generate the sample paths. (c) Descrbe how you would produce the fnal estmate. (d) How would you estmate the relablty of your result? Settng asde the dvdend for the moment, the SDE for S t s Ths has the soluton d S t = m S t d t + s S t d W t S t = S 0 expckm - 1 ÅÅÅÅÅ 2 s2 O t + s W t G (25) (26) Immedately after the dvdend s pad at tme t = d + the value of the stock s
ams-q03-fnal-p 5 The remanng soluton to expry s then S d + = H1 - dl S d = H1 - dl S 0 expckm - 1 ÅÅÅÅÅ 2 s2 O t + s W d G S T = HS d + L expckm - ÅÅÅÅÅ 1 2 s2 O HT - dl + s HW T - W d LG = H1 - dl S 0 expckm - ÅÅÅÅÅ 1 2 s2 O T + s W T G (27) (28) It s nterestng that, for a European opton on whose underlyng dynamcs are descrbed by (25) and for whch the dvdend s expressed as a percentage of the nstant prce, you are ndfferent to when the dvdend s receved durng the nterval 0 t T. Under the rsk neutral measure dp è the current prce of the opton s the martngale C 0 = exp@-rht - tld E Pè @max@s T - K, 0DD (29) The fact that the dvdend yeld s ndependent of the prce dynamcs means that the change to the rsk neutral measure does not affect the dstrbuton of d. Recognzng ths and gven (28) and (29), the expectaton can be computed condtoned on the value of d. Wth d U@d» a, bd representng the densty of a unform dstrbuton on [a, b] we have C 0 = a b j exp@-rht - tld k - maxch1 - dl S 0 expckr - 1 ÅÅÅÅÅ 2 s2 O T + s W è T G - K, 0G d P è HW è y T L d U@d» a, bd z { (30) At ths pont the ntegral represented by (30) can be drectly estmated by Monte Carlo. For the k th experment we generate a dvdend yeld d HkL ~ U[a, b] and random prce component W è HkL è!!!! T ~ [0, T s] and use (30) to compute a C HkL 0. C HkL 0 = exp@-rht - tld maxci1 - d HkL M S 0 expckr - ÅÅÅÅÅ 1 2 s2 O T + s W è HkL T G - K, 0G After completng such experments the estmate for C 0 s (31) C` 0 = ÅÅÅÅÅÅ 1 HkL C 0 k=1 (32) It turns out that we can smplfy thngs further. Agan, denote the Black-Scholes formula for a European call at tme t for an underlyng stock wth current prce S t and volatlty s, rskfree rate r, and opton wth strke K and expry T by C@t, 8S t, s<, r, 8K, T<D (33) At tme t = 0 for a fxed dvdend rate d, the value of (30) s C 0»d = exp@-rht - tld - maxch1 - dl S 0 expckr - 1 ÅÅÅÅÅ 2 s2 O T + s W è T G - K, 0G d P è = C@0, 8H1 - dl S 0, s<, r, 8K, T<D (34) If we substture (34) nto (30), then C 0 = C 0»d d U@d» a, bd a b (35) The evaluaton of (35) s smpler than our orgnal ntegral (30); for the k th experment we generate a dvdend yeld d HkL ~ U[a, b] and use (34) to compute a C 0»d HkL. C 0»d HkL = CA0, 9I1 - d HkL M S 0, s=, r, 8K, T<E (36)
6 ams-q03-fnal-p After completng such experments the estmate for C 0 s now C` 0 = ÅÅÅÅÅÅ 1 HkL C 0»d k=1 (37) Ths s a common pattern. We buld a mathematcal model to descrbe the rsk neutral prcng of the asset, apply the analytcal smplfcatons that we have avalable to us, and then handle any remanng complcatons usng numercal technques. Fnally, t s mportant to estmate the confdence ntervals assocated wth these estmates. In both (32) and (37) ths s the standard devaton of the sample dvded by the square root of the sample sze. Runnng an experment wth both (30) and (35) wth parameters: S 0 = 95, s = 35%, d ~ U[0.02, 0.04], r = 5%, K = 100, T = 0.25 wth a sample sze of 5,000 yelds the followng results: sample sze = 5000 complex ntegral results mean = 3.86074 std err= 0.11795 smplfed ntegral results mean = 3.86809 std err= 0.0029181 Both approaches more or less agree on the prce, but, clearly, there are tremendous advantages to applyng analytcal smplfcatons where possble; n ths nstance, the smplfed ntegral s estmate has only about 2.5% the standard error of that of the complex". A tarball contanng sample Octave code to llustrate both approaches above can be found n ams-q03-fnal-o.tgz. Ths code wll also run n MATLAB. In the UIX, Lnux and Mac OS X operattng systems, ths archve can be unpacked by gunzp -dc ams-q03-fnal-o.tgz tar -xvf - If you smply unpack the tarball everythng wll happly fnd tself wthn a drectory called ams-q03-fnal-o contaned n whatever drectory you unpacked the orgnal fle. Wthn Octave cd nto that drectory and run the scrpt ams_q03_fnal_o_00 To get a feel for how thngs work, run the scrpt wth dfferent sample szes and perform multple runs at the same sample sze. I am not famlar wth the Wndows or pre-x Mac operatng systems, but f anyone can supply gudance as to how to unpack such fles n those envronments, then I would be grateful to hear about t so I can nclude such drectons n the future. 6. Bnomal Lattce At tme t = 0 a stock s prce s S 0 = 98. The stock has an annual volatlty s = 20% and the annual rskfree rate s r = 4%. We need to prce a put opton wth strke K = 100 and expry T = 0.15. Assume the stock s prce evolves accordng to a geometrc Brownan moton. Perform the followng analyses: (a) Compute the value of the put based on a three-step bnomal lattce. Show both the forward and backward lattces and all calculatons.
ams-q03-fnal-p 7 (b) Compute the value of the put usng the Black-Scholes formula and compare t wth the result above. For the bnomal opton prcng model (BOPM) the rsk neutral measure, forward tree and backward tree are Bnomal: rsk neutral measure {up, dn} = {0.511175, 0488825} 112.071 y 107.169 102.482 102.482 forward lattce = 98 98 93.7139 93.7139 89.6152 j z k 85.6958 { 0 y 0 1.49608 0 forward lattce = 3.95052 3.06669 6.53334 6.28614 10.1852 j z k 14.3042 { The value of the opton s the root of the backward tree: 3.95. For the Black-Scholes opton formula (BSOF) the value s sgnfcantly dfferent at 3.80. Black-Scholes: value = 3.80127 The more ntervals n the tree, the closer the BOPM approaches the BSOF. For example, for 100 ntervals the value of the BOPM s about a half cent off from the BSOF. The Mathematca notebook ams-q03-fnal-m.nb contans the code and computatons necessary to reproduce the above.