# Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is

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1 Normal (Gaussian) Disribuion Probabiliy De ensiy ? The Black-Scholes Shl Ml Moel... pricing opions an calculaing Greeks (c) , Gary R. Evans. May be use for non-profi eucaional uses only wihou permission of he auhor. e Cumulaive Probabiliy Concepually calculaing wha a 0 OTM call opion shoul be worh if he presen price of he sock is () We know how o calculae he probabiliy ha he price will be above 00. () Wha, hough, will be he value of ha omain?

2 How Black-Scholes works... The Black-Scholes moel is use o price European opions (which assumes ha hey mus be hel o expiraion) an relae cusom erivaives. I akes ino accoun ha you have he opion of invesing in an asse earning he risk-free ineres rae. I acknowleges ha he opion price is purely a funcion of he volailiy of he sock's price (he higher he volailiy he higher he premium on he opion). Black-Scholes reas a call opion as a forwar conrac o eliver sock a a conracual price, which is, of course, he srike price. The Essence of he Black-Scholes Approach Only volailiy maers, he mu (rif) is no imporan. The opion's premium will suffer from ime ecay as we approach expiraion (Thea in he European moel). The sock's unerlying volailiy conribues o he opion's premium (Vega). The sensiiviy of he opion o a change in he sock's value (Dela) an he rae of ha sensiiviy (Gamma) is imporan [hese variables are represene mahemaically in he Black-Scholes DE, nex lecure]. Opion values arise from arbirage opporuniies in a worl where you have a risk-free choice.

3 The Black-Scholes Moel: European Opions 365 C SN( ) Ke N( ) C = heoreical call value S = curren sock price N = cumulaive sanar normal probabiliy is. = ays unil expiraion K = opion srike price r = risk free ineres rae aily sock volailiy r S K r 365 S K r 365 Noe: Hull's version (3.0) uses annual volailiy. Noe he ifference. Breaking his own C SN ( ) Ke N ( ) r This erm iscouns he price of he sock a which you will have he righ o buy i (he srike price) back o is presen value using he risk-free ineres rae. Le's assume in he nex slie ha r = 0. S K r 365 Diviing by his erm (he sanar eviaion of sock's aily volailiy ajuse for ime) urns he isribuion ino a sanar normal isribuion wih a sanar eviaion of. 3

4 ... or simplifying i some CP SP STR This is he absolue log growh hifference... assume ha r beween he srike is 0 an is : price an he sock price. We are calculaing he cumulaive probabiliy o his sanar normal poin. SP STR This normalizes i o sanar normal (he numeraor is now number of sanar eviaions. μ is zero so his is he log-normal zero mean ajusmen C S N( ) K N( )... an some more (assuming r o be 0) This erm, our x of wo slies ago, represens he sprea in coninuous growh erms beween he sock price an he srike price, an when normalize by he enominaor, he sprea as he number of sanar eviaions. For example, if S = 0 an K = 00 an volailiy = 0%, hen his erms equals 9.5%, or abou one sanar eviaion. x > 0 for im calls an om pus an x < 0 for om calls an im pus. S K r 365 This erm has he effec of removing he bias. S K r 365 4

5 Using he Black-Scholes Moel There are variaions of he Black-Scholes moel ha prices for ivien paymens (wihin he opion perio). See Hull secion 3. o see how ha is one (easy o unersan). However, because of wha is sai below, you really can' use Black- Scholes o esimae values of opions for ivien-paying American socks There is no easy esimaor for American opions prices, bu as Hull poins ou in chaper 9 secion 9.5, wih he excepion of exercising a call opion jus prior o an exivien ae, "i is never opimal o exercise an American call opion on a nonivien paying sock before he expiraion ae." The Black-Scholes moel can be use o esimae "implie volailiy". To o his, however, given an acual opion value, you have o ierae o fin he volailiy soluion (see Hull's iscussion of his in 3.). This proceure is easy o program an no very ime-consuming in even an Excel version of he moel. For hose of you ineres in anoher elegan implie volailiy moel, see Hull's iscussion of he IVF moel in 6.3. There you will see a role playe by ela an vega, bu again you woul have o ierae o ge he value of he sensiiviy of he call o he srike price. Calculaing implie volailiy wih B/S: SP STR Very easy o o: Once Black-Scholes Shl is srucure, you can use an ieraive echnique o solve for σ. Name: Dae: Gary R. Evans Ocober 7, 0 Pu Opion Implici Daily Volailiy (IDV) Calculaor Sock Symbol: DIA Price:.60 Monh: Dec Pu Srike: 0.00 Price: 3.50 Expiraion ae: /7/0 Ineres rae: Days o mauriy oay: Days o mauriy overrie: Implie aily volailiy: One-ay ime ecay: Version 3.4 Aug 6, Calculae 5

6 VBasic ieraive echnique use in IDV maser 'Below is he acual calculaion of implie volailiy. 'The Ringer is for esing emporary values in consrucion only. ' Do CIPD = CIPD DeNom = Log(SockPR / SrikePR) + ((InRR / 365) + (CIPD ^ ) / ) * DTMR DurVol = CIPD * DTMR ^ 0.5 DND = WorksheeFuncion.NormSDis(DeNom / DurVol) DND = WorksheeFuncion.NormSDis(DeNom / DurVol - DurVol) Ringer = Exp(-InRR * DTMR / 365) TempCallPR = SockPR * DND - SrikePR * Exp(-InRR * DTMR / 365) * DND Loop Unil TempCallPR >= CallPR 'Comman below wries a value back o a name esignae cell Range("CIPD").Value = CIPD Calculaing IDV for srangles (V. 3.3) Name: Gary R. Evans Dae: March 30, 0 Srangle Implie Daily Volailiy Calculaor Sock Symbol: DIA Ineres rae: Sock Price: CALL PUT Monh: Apr Apr Srike: Expiraion: 4// 4// Price: Days o mauriy: DTM overrie: Implie aily volailiy: One-ay ime ecay: Version 3.3 Augus 6, 0 One Year: Average DGR: Sanar Deviaion: Average ABS DCGR: ay: Average DGR: Sanar Deviaion: Average ABS DCGR: Calculae Example: March 30, 0 weeken srangle 6

7 =LN(SP/KP)+(IR+(DV*DV)/)*(DTM/365) =LN(SP/KP)+((IR/365)+(DV*DV)/)*DTM An example... Consier an im opion wih 0 ays o expiraion. The srike price is 05 an he price of he sock is 00 an he sock has an aily volailiy of 0.0. Assume an ineres rae of 0.0 (% annual). 05 r C 00N e N Using an Opion Value Calculaor o Calculae his same Value Call Opion Price Calculaor (Daily Volailiy) Sock symbol: Trial Call opion: May Dae Toay: 4/6/0 Expiraion Dae: 5/6/0 DTM: 0 Sock Price: Srike Price: Daily Volailiy: Ineres Rae: 0.00 Time: 0 Numeraor: Duraion Volailiy: Dela N(): N(): Opion Price:.70 Opion Premium:.70 NUM =LN(SP/KP)+((IR/365)+(DV*DV)/)*DTM (( ) ( ) ) DUV =NORMSDIST(NUM/DUV) 7

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