The Black-ScholesModelRandomness matters in nonlinearity An call option with strike price of 10. Suppose the expected value of a stock at call options maturity is 10. If the stock price has 0! chance of endin" at 11 and 0! chance of endin" at #$ the expected payoff is 0..If the stock price has 0! chance of endin" at 1% and 0! chance of endin" at &$ the expected payoff is 1.Applyin" Itos 'emma$ we can find (herefore$ the avera"e rate of return is r)0.si"ma*%. +,ut there could -e pro-lem -ecause of the last term..dz rdtsds + =) ()21(02t zt re e S S =(he history of option pricin" models1#00$ ,achelier$ the purpose$ risk mana"ement1#0s$ the discovery of ,acheliers work1#/0s$ Samuelsons formula$ which contains expected return(horp and 0assouf +1#/1.2 ,eat the market$ lon" stock and short warrant1#13$ ,lack and Scholes(he influence of ,eat the 4arket5ractical experience is not merely the ultimate test of ideas6 it is also the ultimate source. At their -e"innin"$ most ideas are dimly perceived. Ideas are most clearly viewed when presented as a-stractions$ hence the common assumption that academics ))) who are proficient at presentin" and discussin" a-stractions ))) are the source of most ideas. +p. /$ (reynor$ 1#13.+7uoted in p. 8#.9hy ,lack and Scholes:ack (reynor$ developed ;A54 theory;A54 theory2 Risk and return is the same thin",lack learned ;A54 from (reynor. 1## .Start under"raduate in physics(ransfer to computer science=inish 5h? in mathematics'ookin" for somethin" practical:oin A?'$ meet :ack (reynor$ learn finance and economics?eveloped ,lack)Scholes4ove to academia$ in ;hica"o then to 4I(Return to industry at @oldman Sachs for the last 11 years of his life=ischer never took a course in either economics or finance$ so he never learned the way you were supposed to do thin"s. ,ut that lack of trainin" proved to -e an advanta"e$ (reynor su""ested$ since the traditional methods in those fields were -etter at producin" academic careers than new knowled"e. =ischers intellectual formation was instead in physics and mathematics$ and his success in finance came from applyin" the methods of astrophysics. 'ackin" the a-ility to run controlled experiments on the stars$ the astrophysist relies on careful o-servation and then ima"ination to find the simplicity underlyin" apparent complexity. In =ischers hands$ the same ha-its of research turned out to -e effective for producin" new knowled"e in finance. +p. /. ,oth ;A54 and ,lack)Scholes are thus much simpler than the world they seek to illuminate$ -ut accordin" to =ischer thats a "ood thin"$ not a -ad thin". In a world where nothin" is constant$ complex models are inherently fra"ile$ and are prone to -reak down when you lean on them too hard. Simple models are potentially more ro-ust$ and easier to adapt as the world chan"es. =ischer em-raced simple models as his anchor in the flux -ecause he thou"ht they were more likely to survive ?arwinian selection as the system chan"es. +p. 18. :ohn ;ox$ said it -est$ A=ischer is the only real "enius Ive ever met in finance. Bther people$ like Ro-ert 4erton or Stephen Ross$ are Cust very smart and 7uick$ -ut they think like me. =ischer came from someplace else entirely.D +p. 11. 9hy ,lack is the only "eniusE Fo one else can achieve the same level of understandin"E =ischers research was a-out developin" clever models )))insi"htful$ ele"ant models that chan"ed the way we look at the world. (hey have more in common with the models of physics ))) Fewtons laws of motion$ or 4axwells e7uations ))) than with the econometric GmodelsD ))) lists of loosely plausi-le explanatory varia-les ))) that now dominate the finance Cournals. +(reynor$ 1##/$ Remem-erin" =ischer ,lack.(he o-Cective of this course9e will learn ,lack)Scholes theory. (hen we will develop an economic theory of life and social systems from -asic physical and economic principles. 9e will show that the knowled"e that helps ,lack succeed will help everyone succeed. (here is really no mystery. Hffect of Iaria-les on Bption 5ricin" c p C P Iaria-leS0KTrDJ J>JE EJ JJ J J JJ>J>>> >J>J>JThe ConceptsUnderlying Black-Scholes(he option price and the stock price depend on the same underlyin" source of uncertainty9e can form a portfolio consistin" of the stock and the option which eliminates this source of uncertainty(he portfolio is instantaneously riskless and must instantaneously earn the risk)free rate(his leads to the ,lack)Scholes differential e7uation The Derivationof the Black-Scholes Differential Equationshares 2KJ derivative 2 of consistin" portfolio a up set e 9 K KLK KK Sz SSt SS tSSz S t S S + ++ = + = 12 222

KK-y "iven is time in value its in chan"e (he

KK-y "iven is portfolio the of value (heSStSS+ = + = The Derivationof the Black-Scholes Differential Equation continuedThe Derivationof the Black-Scholes Differential Equation continuedKKLK K

2 e7uation al differenti Scholes ) ,lack the "et toe7uations these in and K for su-stitute 9e

9hat are the market pricesE 9hyE An American call on a dividend)payin"stock should only ever -e exercised immediately prior to an ex)dividend date5ut);all 5arity6 Fo ?ividends ;onsider the followin" % portfolios2>5ortfolio A2Huropean call on a stock J 5I of the strike price in cash>5ortfolio ;2Huropean put on the stock J the stock ,oth are worth 4AS+ST , K . at the maturity of the options(hey must therefore -e worth the same today>(his means thatc + Ke -rT = p + S0 An alternative way to derive 5ut);all 5arity=rom the ,lack)Scholes formularTKe Sd SN d NrTKe d NrTKe d SN P C = = )}1( )2( { )2( )1(Arbitrage OpportunitiesSuppose that c M 3 S0M 31 TM 0.% r M 10! KM30 D M 09hat are the ar-itra"epossi-ilities when p M %.% E p M 1 E Application to corporate lia-ulities,lack$ =ischer6 4yron Scholes +1#13.. T(he 5ricin" of Bptions and ;orporate 'ia-ilities 5ut);all parity and capital structureAssume a company is financed -y e7uity and a Uero coupon -ond mature in year ( and with a face value of 0. At the end of year ($ the company needs to pay off de-t. If the company value is "reater than 0 at that time$ the company will payoff de-t. If the company value is less than 0$ the company will default and let the -ond holder to take over the company.