(Analytic Formula for the European Normal Black Scholes Formula)


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1 (Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually th undrlying scurity is assumd to follow a lognormal procss (or Gomtric Brownian Motion). Howvr, thr ar som tradrs who bliv that th normalprocss dscribs th ralmarktmor closly than that of lognormal countrpart. Th drivation of th formula will b provd mathmatically by th famous noarbitrag argumnt (Hull[1]). Thn, th Grks (dlta, gamma, thta, and vga) will b computd by diffrntiation. Th ida of th thory is that Th fair valu of any drivativ scurity is computd as th xpctation of th payoff undr an quivalnt martingal masur (Karatzas [2]). Mor spcifically, th pric is th xpctation of a discountd payoff undr th risk nutral masur. This masur (or dnsity) is a solution to a parabolic partial diffrntial quation drivn by th undrlying procss (lognormal procss). Th dtaild statmnts will b shown blow. Sction following th nxt will show argumnts concrning Normal modl that dos not allow ngativ undrlir pric. Also brif xprimnts rgarding how on can approximat lognormal Black Schols with Normal vrsion will b xplord with som xprimnts. 1 Analytic Formula Thorm 1 (Analytic Formula for a Normal Black Schols Modl) Lt us assum that th currnt futur pric, strik pric, risk fr intrst rat, volatility, and tim to maturity as dnotd as F, K, r, σ, andt t rspctivly. Alt us also assum that th currnt futur pric follows th following Normal procss: df µdt + σdw t (1) whr µ is a constant drift. Thn, th fair valus of call C and put P ar xprssd as : C r(t [(F K)N(d 1 )+ σ T t d2 1 /2 ] (2) and P r(t [(K F )N(d 1 )+ σ T t d2 1 /2 ] (3) Nw York Univrsity, Dpartmnt of Mathmatics 1
2 whr d 1 F K σ T t Morovr, th Grks (dlta, gamma, vga, thta) ar computd by simpl diffrntiation of th abov formulas to giv: (Not: Dtail calculations will b shown blow for mor gnral audinc) Dlta (Call) C F r(t 1 [(F K) d2 1 /2 1 σ T t ] + r(t N(d 1 )+ r(t σ T t d2 1 /2 (F K) [ σ T t ][ 1 σ T t ] r(t N(d 1 ) (4) Dlta (Put) P F r(t 1 [(K F ) d2 1 /2 1 σ T t ] r(t N(d 1 )+ r(t σ T t d2 1 /2 (F K) [ σ T t ][ 1 σ T t ] r(t N(d 1 ) (5) Gamma (Call) Gamma (Put) 2 C F 2 ( ) r(t 1 d2 1 /2 1 σ T t r(t σ 1 d2 1 /2 (6) T t ( *Not: This is vidnt from th PutCall parity in that by diffrntiating th PutCall parity formula twic with rspct to th undrlir stablishs th quality of Put and Call for all option modls..) Vga (Call) Vga (Put) C σ r(t 1 (F K) d2 1 /2 (F K) [ σ 2 T t ] + r(t T t d2 1 /2 + r(t σ T t d2 1 /2 (F K) [ σ (F K) ][ T t σ 2 T t ] T t r(t d2 1 /2 (7) ( *Not: This is vidnt from th PutCall parity in that by diffrntiating th PutCall parity formula with rspct to σ stablishs th quality of Put and Call for all option modls..) 2
3 Thta (Call) C t r r(t [(F K)N(d 1 )+ σ T t d2 1 /2 ] r(t 1 [(F K) d2 1 /2 ( 1 F K )] 2 σ(t t) 3/2 r(t σ [ 2 2 t d 1 /2 ] r(t [ σ T t d2 1 /2 (F K) ( σ (F K) )( )] T t 2σ(T t) 3/2 r r(t [(F K)N(d 1 )+ σ T t d2 1 /2 ] r(t σ [ 2 2 t d 1 /2 ] (8) Thta (Put) P t r r(t [(K F )N(d 1 )+ σ T t d2 1 /2 ] r(t 1 [(K F ) d2 1 /2 ( 1 F K )] 2 σ(t t) 3/2 r(t σ [ 2 2 t d 1 /2 ] r(t [ σ T t d2 1 /2 (F K) ( σ (F K) )( )] T t 2σ(T t) 3/2 r r(t [K F )N(d 1 )+ σ T t d2 1 /2 ] r(t σ [ 2 2 t d 1 /2 ] (9) (Proof) W shall paralll th argumnt givn in Hull [1]. Th proof will b huristic. Lt us study th bhavior of th dlta hdgd portfolio which consists of long dlta shars of futur contract and short on drivativ in qustion. Say, call it Π. Lt us also dnot th valu of drivativ by g. Thn, th valu of th dlta hdgd portfolio is givn by: Π g g F F (10) So applying Ito s lmma using th SDE givn in (1) into th changs of th abov portfolio valu, on can gt: 3
4 Π g g F F ( g t + g F µ g 2 F 2 σ2 ) t + g F σ W t g F (µ t + σ W t) (11) ( g t g 2 F 2 σ2 ) t W want th avov quantity to b a martingal undr th discountd xpctation with risk fr rat. This is ssntially th sam as stating that th abov quantity quals th gain from th risk fr intrst rat for th portfolio valu. So, w hav: Π rπ t (12) Sinc it cost nothing to ntr into a futurs contract, on has: Π g (13) Thus w hav: ( g t g 2 F 2 σ2 ) t rg t (14) Thrfor, w obtains th following parabolic PDE: g t g 2 F 2 σ2 rg (15) So, if th trminal payoff is givn by g(t,f) f(f ), thn by th application of FynmanKac (s Karatzas and Shriv[2]), on obtains th following solution: g(t, x) E x [ r(t f(y)] r(t σ T t f(y) (xy) 2 2σ 2 (T dy (16) whr f(y) { (y K) + for Call (K y) + for Put. Thrfor, th for th formula for th abov can b simplifd by simply xpanding th xprssion insid th intgral. Th dtail will b shown for mor gnral audinc. For th call, w hav: 4
5 Call g(t, F ) E F [ r(t f(y)] + r(t σ T t r(t r(t r(t r(t (KF ) σ T t (KF ) σ T t (KF ) σ T t (y K) + (F y) 2 2σ 2 (T dy (F + σ T tx K) + x2 2 dx (F + σ T tx K) x2 2 dx (F K) x2 2 dx (σ T tx) x2 2 dx r(t [(F K)N(d 1 )+ σ T t d2 1 /2 ] (17) whr d 1 F K σ T t Similarly for th put, w hav: Put g(t, F ) E F [ r(t f(y)] + r(t σ T t r(t (K y) + (F y) 2 2σ 2 (T dy (K F σ T tx) + x2 2 dx (KF r(t ) σ T t (K F σ T tx) x2 2 dx (KF r(t ) σ T t (K F ) x2 2 dx (KF r(t ) σ T t (σ T tx) x2 2 r(t [(K F )N(d 1 )+ σ T t d2 1 /2 ] (18) dx whr d 1 is dfind similarly as in (17). This is what w nd to prov. Th Grks ar obtaind by simpl diffrntiations which ar workd out in th statmnt of th thorm. QED. 5
6 2 Normal Procss undr th Boundd Ngativ Assumption In th prvious sction, pricing schm for th Normal Procss is provd in mor gnral stting. That is th undrling prics ar allowd to bcom ngativ. In this sction, th modifid option formulas will b drivd whr undrlir pric can not fall blow crtain numbr, say R. Using th sam notations as in th prvious sction, w hav th following PDE for non ngativ undrlir procss. g t g 2 F 2 σ2 rg g(t,f) f(f ) F R (19) This is a problm constraind in th half plan in F axis. Th solution of th abov PDE can b solvd by transforming th abov PDE into a typical PDE problm for th ntir F axis. This is don by xtnding th trminal function f(f )intof R (F ) which is basically an odd xtntion of th original function f(f ). Thus, th uniqu solution xists for this PDE (John [4]), and givn by: g(t, x) E x [ r(t f(y)] r(t σ T t r(t σ T t R f R (y) (xy) 2 2σ 2 (T dy f(y)[ (xy+r) 2 2σ 2 (T (x+yr)2 2σ 2 (T ]dy (20) whr f(y) { (y K) + for Call (K y) + for Put. Thrfor th formula for th Call option is givn by xpanding th mssy xprssion in (20) as: Call g(t, F ) E F [ r(t f(y)] r(t σ T t r(t σ T t r(t σ T t r(t r(t + R R (KF R) σ T t (K+F R) σ T t f R (y) (F y) 2 2σ 2 (T dy (F y+r)2 f(y)[ 2σ 2 (T (y k) + [ (F y+r) 2 2σ 2 (T (F +yr)2 2σ 2 (T ]dy (F +yr)2 2σ 2 (T ]dy (F + σ T tx+ R K) + x2 2 dx (F σ T tz+ R K) z2 2 dz r(t [(R + F K)N(d 1 )+ σ T t d2 1 /2 ] 6
7 r(t [(R F K)N(d 2 )+ σ T t d2 2 /2 ] (21) whr d 1 R+F K σ,andd RF K T t 2 σ T t Similarly for put, w hav: Put g(t, F ) E F [ r(t f(y)] r(t σ T t r(t σ T t r(t σ T t r(t r(t + R R (KF R) σ T t (K+F R) σ T t f R (y) f(y)[ (k y) + [ (F y) 2 2σ 2 (T dy (F y+r) 2 2σ 2 (T (F y+r)2 2σ 2 (T (F +yr)2 2σ 2 (T ]dy (F +yr)2 2σ 2 (T ]dy (K F σ T tx R) + x2 2 dx (K + F σ T tz R) z2 2 dz r(t [(K F R)N(d 1 )+ σ T t d2 1 /2 ] r(t [(K + F R)N(d 2 )+ σ T t d2 2 /2 ] (22) whr d 1 R+F K σ,andd RF K T t 2 σ T t Lt us compar th rsult with th rsults from th prvious sction. Lting R 0 in th xprssion (21) givs: Call r(t [(F K)N(d 1 )+ σ T t d2 1 /2 ] r(t [(F K)N(d 2 )+ σ T t d2 2 /2 ] (23) whr d 1 F K σ,andd F K T t 2 σ T t Sinc d 2 trm is such a larg ngativ quantity, th trm r(t [(F K)N(d 2 )+ σ T t d2 2 /2 ]is avrysmallquantity. Thus, th call pric bcoms: Call r(t [(F K)N(d 1 )+ σ T t d2 1 /2 ] (24) which is th sam rsult as th on givn in th prvious sction. 7
8 Actually, th th scond trm is so small that on can rplac th abov rlation with quality in most practical cass. Th idntical argumnt can b mad for th put as wll. Th abov pric hav a vry nic thortical proprty whn σ is xtrmly larg. Th xprssions (2) and (3) blows up to infinity whn w lt σ Without th loss of gnrality, lt us assum R 0 as bfor. Thn taking th limit of th xprssion (21) givs: lim Call lim σ + σ + r(t [(F K)N(d 1 )+ σ T t d2 1 /2 ] lim σ + r(t [(F K)N(d 2 )+ σ T t d2 2 /2 ] r(t σ T t (F K) + lim r(t σ + r(t σ T t (F K) + lim r(t [ σ + [ d2 1 /2 d2 2 /2 ] (F K)2 (( 2(T ) n (F +K)2 ( 2(T ) n ) σ 2n ] n! n1 r(t (F K) (25) Similarly for put, w hav: lim σ + Put r(t (K F ) (26) Thus, as th volatility σ blows up to infinity, th pric is boundd abov by constant for both call and put. 8
9 3 Approximation Exprimnts In practic, option prics ar givn in th markt and on nds up computing implid volatility for a particular option. Suppos on is givn an implid volatility for th lognormal modl but wants to us normal modl to comput its grks (dlta, gamma, vga, and thta). This is usually dalt with approximating σ (Normal) by ltting σ (Normal) Ĝσ (Lognormal), WhrĜ is a scaling constant. What kind approximations can on mak? Th popular choic is to st Ĝ K (strik), or Ĝ F (futur pric). Thn us ths into th Normal modl. Ths approximations display rasonabl closnss to th actual lognormal pric. Brif xprimnts hav shown that taking th avrag of ths two Ĝ (F + K)/2 givs vry approximation to th lognormal pric. Sinc, this papr is intndd to b a vry short papr, brif xplanation of th xprimnts with som graphs would b shown blow. Th tst is prformd on Europan Eurodollar futur option Call whr strik is 98.25, days to xpiration is 22 days, and risk fr rat is st to %. W took a rangs of markt option prics from 0.05 to 0.13 and futur prics from 97.8 to For a givn input pric and a futur pric, Nwton s mthod was usd to comput Lognormal implid vol. Thn this implid vol is scald appropriatly and insrtd into Normal formulas. 9
10 Eurodollar Futur Option Pric Analysis (Using Spot as Scaling) RComputd Option Prics X2 Spot Yild X1 Markt Option Pric 0.1 Figur 1: Eurodollar futur Call. Th top graph is th pric surfac using Normal modl with spot as th scaling factor. Th bottom surfac is th original Lognormal pric surfac. Not that Normal pric is somwhat ovrstating th pric. 10
11 Eurodollar Futur Option Pric Analysis (Using Strik as Scaling) RComputd Option Prics X2 Spot Yild X1 Markt Option Pric Figur 2: Eurodollar futur Call. Th top graph is th pric surfac using Normal modl with strik as th scaling factor. Th bottom surfac is th original Lognormal pric surfac. Not that Normal pric is somwhat undrstating th pric in this cas. 11
12 Eurodollar Futur Option Pric Analysis (Using (Strik + Spot)/2 as Scaling) RComputd Option Prics X2 Spot Yild X1 Markt Option Pric Figur 3: Eurodollar futur Call. Th top graph is th pric surfac using Normal modl with ( strik + spot )/2 as th scaling factor. Th bottom surfac is th original Lognormal pric surfac. Not that Normal pric is much closr to th actual pric in this cas. 12
13 Eurodollar Futur Option Pric Error Analysis x Error X1 Markt Option Pric X2 Spot Yild Figur 4: Eurodollar futur Call. Th rror showing th diffrnc btwn Th Lognormal modl and Normal modl with ( strik + spot )/2 scaling. Not that th rror incrass as th option bcoms far in th mony. 13
14 Rfrncs [*] [1] Hull, J. (1993), Options, Futurs, and Othr Drivativ Scuritis, 2nd Edition, Prntic Hall. [2] Karatzas, I. and S.E. Shrv (1988), Brownian Motion and Stochastic Calculus,SpringrVrlag, Nw York. [3] Prss, W.H., Tokolsky, S.A., Vttrling, W.T. and Flannry, B.P. (1992), Numrical Rcips in C: Th Art of Scintific Computing, 2nd Edition, Cambridg Univrsity Prss, Cambridg [5] John, Fritz (1978), Partial Diffrntial Equations, SpringrVrlag, Nw York 14
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