.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
//////////////////////////////
"