【正文】 :58:1400:58Feb2312Feb23 1越是無能的人，越喜歡挑剔別人的錯兒。 :58:1400:58:14February 12, 2023 1他鄉(xiāng)生白發(fā)，舊國見青山。 ?2C/?S2 ?2(S,t)]dt [?C/?S ?(S,t)]dz + ?C/?S[?(S,t)dt + ?(S,t)dz] Collecting terms involving dt and dz together we get dP = [?C/?S ?(S,t) + ?C/?t + 189。 ?2)(Tt), ?(Tt)1/2) c=SeqTN(d1)XerTN(d2) p=XerTN(d2)SeqTN(d1) where d1=[ln(S/X)+(rq+?2/2)T]/(?T1/2), d2=d1?T1/2 Example: X=$70, Maturity date = June 27 (Evaluate on May 5: T=53/365 = ) ______________________________________________ S= X= T= r= . = q = European option prices: Call = Put = ? Implied volatility The volatility that makes the model price equal its market price. Assume that the call and put options in the above example are traded at and , respectively. Call implied volatility: Put implied volatility: Stock Price Part A maturity 22 days, r=% Type of options Strike price Mean option price Implied volatility(%) Call Call Call Put Put 90 95 100 90 95 4 3/4 1 3/8 1/2 1/2 2 7/8 Part B maturity 50 days, r=% Type of options Strike price Mean option price Implied volatility(%) Call Call Call Put Put 90 95 100 90 95 5 3/8 2 3/4 3 3/4 The prices are midpoint prices. The implied volatility seems to depend upon whether the option is in/out or atthemoney. The implied volatility for calls seems to differ from the implied volatility for puts. There are many reasons why the implied volatility estimates differ. (why?) ? Option Greeks Delta: With respect to an increase in stock price ?c=eqTN(d1) ?p=eqT[N(d1)1] Gamma: Delta39。 , February 12, 2023 很多事情努力了未必有結(jié)果，但是不努力卻什么改變也沒有。 2023年 2月 上午 12時 58分 :58February 12, 2023 1業(yè)余生活要有意義，不要越軌。 2023年 2月 上午 12時 58分 :58February 12, 2023 1少年十五二十時，步行奪得胡馬騎。s ? is its partial derivative with respect to a change in the continuously pounded interest rate. Specifically, the call option pricing formula (Black and Scholes) is c=SN(d1)XerTN(d2) where d1=[ln(S/X)+(r+?2/2)T]/(?T1/2), d2=d1?T1/2 It follows that ?c/?r = XTerTN(d2) and (?c/?r)/c = (X/c)TerTN(d2)0 The total differential of the call option can be written as dc = ?c/?r dr + ?c/?S dS Dc=(dc/dr)/c =(?c/c)/?r ?c/?S (dS/dr)= = (X/c)TerTN(d2)(S/c)N(d1)Ds Consider a call option with strike price of $70 and maturity of 53 days (53/365= years). The current stock price is $, the annual standard deviation of %. The risk free rate of interest () is10%. d1=, N(d1)=, d2=, N(d2)= c= Thus Dc = (70/)* +()**Ds =+*Dc If Ds, then Dc is positive. 靜夜四無鄰，荒居舊業(yè)貧。Lecture 9: BlackScholes option pricing formula ? Brownian Motion The first formal mathematical model of financial asset prices, developed by Bachelier (1900), was the continuoustime random walk,