.chapter21<-function(i=0){ " i Chapter 21: Black-Scholes-Merton option model - ------------------------------------- 1 What is option 2 what is a call option 3 What is a put option 4 European options vs. American options 5 Option is a zero-sum game 6 Payoff function for a call option buyer 7 Payoff function for a put option buyer 8 Hedging, speculation, and arbitrage 9 various trading strategies involving options 10 density distribution and cumulative density distribution 11 standard normal and normal distributions 12 Excel functions for a standard normal and cumulative normal distribution 13 Excel functions related to normal distributions 14 Sequence to remember the Black-Scholes-Merton model 15 Nobel Prize-winning formula 16 implied volatility 17 Put-call parity 18 Definitions of Option Greeks 19 Videos 20 Links Example #1:>.c21 # see the above list Example #2:>.c21(1) # see the 1st explanation ";.zchapter21(i)} .n21chapter<-20 .zchapter21<-function(i){ if(i==0){ print(.c21) }else{ .printEachQ(21,i,.n21chapter) } } .c21<-.chapter21 .C21EXPLAIN1<-"What is option ////////////////////////////// An option would give a option buyer a right to buy (or sell) something from (to) the option seller in the future with a pre-determined price. Example 1: John paid $1.2 today to have an option to buy one IBM's stock in 3 months by paying $150. Example 2: Mary paid $1.03 to sell City Group's share with a price of $40 in 3 months. ////////////////////////////// " .C21EXPLAIN2<-"What is a call option? ////////////////////////////// A call option buyer would have a right to buy something from the option seller in the future with a fixed price. ////////////////////////////// " .C21EXPLAIN3<-"What is a put option? ////////////////////////////// A put option buyer would have a right to sell something to the option seller in the future with a fixed price. ////////////////////////////// " .C21EXPLAIN4<-" European options vs. American options ////////////////////////////// For European options, the options could be exercised on maturity dates, while American options could be exercised any time before and on maturity dates. C(European) <= C(American) P(European) <= P(American) ////////////////////////////// " .C21EXPLAIN5<-"Option is a zero-sum game ////////////////////////////// There are two sides of the transaction: buyer and seller An option buyer pays to have a right, while the option seller receives cash flow to bear an obligation In terms of final payoff or profit/loss: Zero-sum gain: if buyer loses then seller wins. if buyer wins then seller loses. payoff (seller) = - payoff(buyer) profit/loss (seller) = - profit/loss(buyer) ////////////////////////////// " .C21EXPLAIN6<-"Payoff for a call ////////////////////////////// For a call option buyer, the payoff function is: payoff(call) = Max(sT-x,0) sT is the terminal price T is the maturity date x is the exercise price (strike price) For example, if the stock terminal stock price is $25 and exercise price is $10 then the payoff of a call option buyer is Max(25-10,0) = max(15, 0) = 15 If the stock terminal stock price is $5 and exercise price is $10 then the payoff of a call option buyer is Max(5-10,0) = max(-5, 0) = 0 Since this is a zero sum gain, the payoff of a call option seller is payoff(call seller)= - payoff(call buyer) = - Max(sT-x,0) ////////////////////////////// " .C21EXPLAIN7<-"payoff function for a put option ////////////////////////////// For a put option buyer, the payoff function is: payoff(call) = Max(x-sT,0) sT is the terminal price T is the maturity date x is the exercise price (strike price) For example, if the stock terminal stock price is $25 and exercise price is $10 then the payoff of a put option buyer is Max(x-sT,0) = max(10-25,0) = max(-15, 0) = 0 If the stock terminal stock price is $5 and exercise price is $10 then the payoff of a put option buyer is Max(10-5,0) = max(5, 0) = 5 Since this is a zero-sum game, the payoff of a put option seller is payoff(put seller)= - payoff(put buyer) = - Max(x-sT,0) ////////////////////////////// " .C21EXPLAIN8<-"Hedging, speculation vs. arbitrage ///////////////////////////// Hedging: With a position, we take an opposite option to reduce the risk. Example #1: We have many shares of IBM's stock. The risk is that the price might go down. To hedge the risk, we might buy a put options. Example #2: Our company would import equivalent from England worth 10 million pound in three months. The current (exchange risk) is that the pound might be appreciated again USD. To hedge the risk, we might enter futures contracts long pound, i.e., the company fixes the exchange rate today. speculation: Without a position, we take a position depends on our expectation. Example #1: We expect IBM's stock is going up, we buy 100 shares. Example #2: Our analysts expect British pound is going depreciate, we enter a contract to sell it with a fixed rate in 3 months. arbitrage: Buy low and sell high simultaneously to earn a profit. ////////////////////////////// " .C21EXPLAIN9<-"various trading strategies involving options ////////////////////////////// Example 1: If we expect stock A will arise, we could buy its call options. Example 2: If we expect stock A will arise, we could sell its put options. Example 3: we own stock A. To protect ourselves, we could buy its put options. Example 4: we expect company B is going through its restructuring process. Because of this, stock A might go up is the process is successfully, or going down if the restructure is a failure. Thus, we could buy one call and one put at the same time. This is called a straddle. Example 5: Same as above, If we expect stock A has a high chance for going up than going down we could buy two call and a put option. Example 6: Same as Example 4, if we expect stock A has a high change for going down than going up we could buy two puts and buy a one call. ////////////////////////////// " .C21EXPLAIN10<-"density distribution vs. cumulative density function ////////////////////////////// Cumulative density is cumulative properties. Case #1: Toss a dice with values from 1 to 6 Each number has 1/6 probability What is the prob. to get a number less than 4? 1/6 + 1/6 + 1/6 = 3/6 =1/2 1/6 is probability of each number, while 1/2 is the cumulative probability (distribution) Case #2: uniform distribution: assume density function is 1/a and choose a=5 from 1 to 5, the density is 1/5. total sum is 5*1/5 =1 what does it mean that the density function is 1/5 (0.2)? Take a value between 0 and 5, such as x=1.5 and choose a small number about called deltaX. The probability that a random number falls into this small strip = f(x) * deltaX= 0.2*detlaX ////////////////////////////// " .C21EXPLAIN11<-"standard normal and cumulative normal distributions ////////////////////////////// The standard normal distribution is defined by the following equation: 1 f(x) = --------- * exp(-0.5 *x^2) sqrt(2*pi) where f(x) is the density function, x is the input pi is 3.1415926 Manually, we could calculate a few values. x=0 -> f(x) = 1/sqrt(2*3.1415925) -> 0.3989423 x=1 -> f(1) = 1/sqrt(2*3.1415926)*exp(-0.5) -> 0.2419707 x=-1 -> f(-1) = 1/sqrt(2*3.1415926)*exp(-0.5*(-1)^2) -> 0.2419707 The normal distribution is a probability distribution that associates the normal random variable X with a cumulative probability . The normal distribution is defined by the following equation: 1 f(x) = ---------------- * exp(-0.5 *[(x-mean)/(sigma))]^2) sqrt(2*pi*sigma^2) ////////////////////////////// " .C21EXPLAIN12<-"Excel functions for density and cumulative density normal distributions ////////////////////////////// =norm.dist(x,mean,standard_dev,cumulative) x is the input value mean: is the mean of the normal distribution standard_dev is the standard deviation cumulative =TRUE is for cumulative distribution =FALSE is for probability mass function (density distribution funtion) Examples for standard normal density funtion 1 f(x) = --------- * exp(-0.5 *x^2) sqrt(2*pi) Example 1: x=0 -> f(x) = 1/sqrt(2*3.1415925) -> 0.3989423 -> =NORM.DIST(0,0,1,FALSE) -> 0.39894228 Example 2: xx=1 -> f(1) = 1/sqrt(2*3.1415926)*exp(-0.5) -> 0.2419707 -> ==NORM.DIST(1,0,1,FALSE) -> 0.241970725 Examples for cumulative normal distribution. we know that a normal distribution is symmetric. Thus, cumulative with x=0 should be 0.5 Example 3: = NORM.DIST(0,0,1,TRUE) -> 0.5 Example 4: = NORM.DIST(3,0,1,TRUE) -> 0.998650102 Example 5: = NORM.DIST(-3,0,1,TRUE) -> 0.001349898 ////////////////////////////// " .C21EXPLAIN13<-"Excel functions related to normal distributions ////////////////////////////// 1) =norm.dist(x,mean,standard_dev,cumulative) 1A) =norm.dist(x,mean,standard_dev,cumulative) 1B) =norm.dist(x,mean,standard_dev,TRUE) 2) =norm.inv(probability, mean,standard_dev) This formula and 1B) is a pair. =norm.dist(x,mean,standard_dev,TRUE) -> p =norm.inv(p, mean,standard_dev) -> x = NORM.DIST(0,0,1,TRUE) -> 0.5 = norminv(0.5,0,1) -> 0 = NORM.DIST(0.23,0,1,TRUE) -> 0.590954115 = norminv(0.590954115,0,1) -> 0.23 =NORM.DIST(0.3,0.5,2.5,TRUE) -> 0.468118628 =NORMINV(0.468118628,0.5,2.5) -> 0.3 3) =norm.s.dist() for standard normal distribution 4) =norm.sinv() inverse of the standard normal distribution 5) =normdist() the same as norm.dist() 6) =normssdist() <=> norm.s.dist() 7) =normSINV() <=> norm.s.inv() ////////////////////////////// " .C21EXPLAIN14<-"Sequence to remember Black-Scholes-Merton model ////////////////////////////// For a call option: Step 1: draw a time line |---------------------| 0 T Step 2: on maturity, the payoff of call = Max(sT-X,0) Step 3: take out the parentheses since we would consider sT-X distribution later sT s0 <-------------<--------------| X exp(-rT)X<-------<--------------| Step 4: consider present values S0 - exp(-rT)X Step 5: add distribution in front of those two values call = S0 * N() - exp(-r*T)*X* N() N() is the cumulative standard normal distribution. It will be defined soon. ////////////////////////////// " .C21EXPLAIN15<-"Nobel prize winning formula ////////////////////////////// Black-Scholes-Merton call option (European) input value: S, X, T, r, sigma S : today's stock price X : exercise price (strike price) T : maturity day in years r : continuously compounded risk-free rate sigma: volatility (standard deviation of stock returns) ln(S/X) + (r+ 0.5*sigma^2)T d1= ------------------------------ sigma* sqrt(T) d2 = d1 - sigma * sqrt(T) call = S*N(d1) - exp(-r*T)*N(d2) where N(x) =normdist(x,0,1,TRUE) put = exp(-r*T)*N(-d2) - S*N(-d1) ////////////////////////////// " .C21EXPLAIN16<-"Put-call parity ////////////////////////////// First, let's look at the logic. Portfolio 1: cash plus a call option assume the exercise price is X payoff at the maturity if sT>x we exercise our call get stock if sT<=x we let the option expires i.e., keep our money X the value of our portfolio is max(sT,X) Portfolio 2: one share plus a put option assume the exercise price is X payoff at the maturity if sT<=x we exercise our put to have X if sT>x we let the option expires i.e., keep the stock the value of our portfolio is max(sT,X) Since the final values of those two portfolios are the same, their present values should be the same as well. C + exp(r*T)*X = p + S ////////////////////////////// " .C21EXPLAIN17<-"Greeks for options ////////////////////////////// Delta is defined as the derivative of the option to its underlying security price. Thus, the delta of a call is defined: The delta of a European call on a non-dividend-paying stock is: Delta (call) =N(d1) The delta for a European put on a non-dividend-paying stock is: delta (put) = N(d1) - 1 If we sell a European call, we could hold delta shares of the same stock to hedge our position. This is named a delta hedge. Since the delta is a function of the underlying stock (S), to maintain an effective hedge we have to rebalance our holding constantly. This is called dynamic hedging. For European call and puts we have closed form-solution for many Greeks, see .showFormula(45). An easy way to define those Greek letters is given in the following table. Greek Description Using call as an example ----- --------------------------------------- ------------------------- Delta the change in an option value divided c2 - c1 by the change in the underlying stock's delta = ---------------- price S2 - S1 Gamma the change in Delta divided by the change delta2- delta1 in the stocks price Gama = ------------- S2 - S1 Theta the change in the option value divided c2 - c1 by the change in the time Theta = ------------- T2- T1 Vega the change in the option value divided c2 - c1 by the change in the volatility Vega = -------------- sigma2 - sigma1 Rho the change in the option value divided the c2 - c1 by the change in the interest rate rho = ------------- r2 - r1 ------------------------ How to remember? Delta: 1st order derivative Gama: 2nd order derivative Theta - - > T - - > Time Vega - - > V - - > Volatility Rho - - > R - - > Rate ////////////////////////////// " .C21EXPLAIN18<-"Implied volatility ////////////////////////////// 40,A1,S 40,A2,X 0.5,A3,T 0.05,A4,r 0.2,A5,sigma call -> 2.755491431 If our observed option is $2, it means that our 20% volatility is too high. why? How to estimate the implied volatility? Given: S,X,T,r and c0 Method I: a) generate n sigmas, sigma1,sigma2, ..., sigman b) apply Black-Scholes-Merton model to generate n related call (c1,c2,..., cn) c) estimate n differences, such as abs(c1-c0), abs(c2-c0), ..., abs(cn-c0) d) choose sigma that has the smallest absolute difference Method II: using solver a) choose a cell to enter any value, such as 0.2 b) click \"Data\" -> Solver c) the cell contains the call value will be our objective funtion d) minimize e) by changing the cell contains sigma ////////////////////////////// " .C21EXPLAIN19<-"Youtubes ////////////////////////////// Understanding Calls and Puts (Sasha Evdakov, 4m25s) https://www.youtube.com/watch?v=zNVgDHCqSVA Option Pricing, 2woooooo (5m48s): https://www.youtube.com/watch?v=I0Xa5wlmM_A Option Basics and Fundamentals (Sasha Evdakov, 7m8s) https://www.youtube.com/watch?v=R-0VhsmQbZ8 Run, Profit, Call Options & Put Options Explained Simply In 8 Minutes (How To Trade Options For Beginners) (7m55s) https://www.youtube.com/watch?v=EfmTWu2yn5Q Options Trading: Understanding Option Prices (7m31s) https://www.youtube.com/watch?v=MiybniIIvx0 Call vs Put Options Basics (17m) https://www.youtube.com/watch?v=uQLMSU2NNlk Buying Options vs Selling Options (18m57s) https://www.youtube.com/watch?v=8wTTq_ftZ18 Option Profit & Loss Diagrams (16m34s) https://www.youtube.com/watch?v=m2LOJRxYkRg Call and Put option for dummies (12m17s) https://www.youtube.com/watch?v=tr5T8y2Qndg Options Pricing & The Greeks (Option Alpha, 31m32s) https://www.youtube.com/watch?v=kCJcEOYuuII Khan Academy (10m23s) https://www.khanacademy.org/economics-finance-domain/core-finance/derivative-securities/black-scholes/v/introduction-to-the-black-scholes-formula ////////////////////////////// " .C21EXPLAIN20<-"Links related to options ////////////////////////////// What is a call option? ------------------------------------ http://www.investopedia.com/terms/c/calloption.asp http://www.theoptionsguide.com/call-option.aspx http://www.call-options.com/what-are-call-options.html What is a put option? ------------------------------------ http://www.investopedia.com/terms/p/putoption.asp https://www.tradeking.com/education/options/put-options-explained http://www.theoptionsguide.com/put-option.asp What is Black-Scholes Option model? ------------------------------------ https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model http://www.macroption.com/black-scholes-formula/ http://www.investopedia.com/university/options-pricing/black-scholes-model.asp https://richnewman.wordpress.com/2007/06/24/a-beginner%E2%80%99s-guide-to-the-black-scholes-option-pricing-formula-part-1/ Online calculator for the Black-Scholes Option models ------------------------------------ https://www.mystockoptions.com/black-scholes.cfm http://www.erieri.com/blackscholes http://www.danielsoper.com/fincalc/calc.aspx?id=37 http://www.calkoo.com/?lang=3&page=29 http://www.soarcorp.com/black_scholes_calculator.jsp http://www.fintools.com/resources/online-calculators/options-calcs/options-calculator/ Greeks for Options ------------------------------------ http://www.calkoo.com/?lang=3&page=29 Options Pricing & The Greeks (Option Alpha, 31m32s) https://www.youtube.com/watch?v=kCJcEOYuuII ////////////////////////////// "