.

A simple calculation shows that

,

where is the cumulative distribution function for the standard normal variable.

The matlab function below calculates the Black-Scholes European call price:

where is the current underlying (non-dividend paying) stock price, is the strike price, is the time to maturity, and

EuCall_BS.m

function price=EuCall_BS(S, K, T, r, sigma)
% EuCall_BS(S,K,T,r,sigma) 
% calculates Black-Scholes call price (no dividends)
% S0=spot asset price
% K=strike price
% T=time to maturity
% r=risk-free rate
% sigma=volativity

d1=(log(S./K)+(r+0.5*sigma^2).*T)./(sigma*sqrt(T));
d2=d1-sigma*sqrt(T);
N1=0.5*(1+erf(d1/sqrt(2)));
N2=0.5*(1+erf(d2/sqrt(2)));

price=S.*N1-K*exp(-r.*T).*N2;

AmPut_CRR.m

function price = AmPut_CRR(S0, K, T, r, sigma, n)
% AmPut_CRR(S0, K, T, r, sigma, n)
% S0=spot price
% K = strike price
% T = time to maturity
% r = risk-free rate (per annum continuously compunded)
% sigma = volatility
% n = number of time-steps in the CRR binomial model

dt = T/n;
u = exp(sigma*sqrt(dt));
d=1/u;
p = (exp(r*dt)-d)/(u-d);
q = 1-p;
D=exp(-r*dt);
% Initialization
V = zeros(n+1,1);

% Stock price dynamics at maturity
S=S0*d.^([n:-1:0]').*u.^([0:n]');

% Option values at time T
V = max(K-S,0);

% Backward induction
for i = n:-1:1 % time index
for j = 1:i
S(j)=S(j)*u; 
V(j) = max((p*V(j+1) + q*V(j))*D, K-S(j)) ;
end
end

price = V(1);

function price = EuPut_CRR(S0, K, T, r, sigma, n)
% EuPut_CRR(S0, K, T, r, sigma, n)
% S0=spot price
% K = strike price
% T = time to maturity
% r = risk-free rate (per annum continuously compounded)
% sigma = volatility
% n = number of time-steps in the CRR binomial model

dt = T/n;
u = exp(sigma*sqrt(dt));
d=1/u;
p = (exp(r*dt)-d)/(u-d);
q = 1-p;

% Initialization
V = zeros(n+1,1);
S(1) = S0;

% Option values at time T
V = max(K-S0*d.^([n:-1:0]').*u.^([0:n]'),0);

%Backward induction
for i = n:-1:1
for j = 1:i
V(j) = p*V(j+1) + q*V(j);
end
end
V(1) = exp(-r*T)*V(1);

price = V(1);

1. How should be chosen?
2. What is the value of my long position in the contract?
3. What is the difference between the current forward price and the delivery price ?

The answers to the above questions are related.

• If , then the value of my (long) position in the contract is . This is true by definition of the forward price; it is the delivery price such that it costs nothing (for both the buyer and the seller) to enter into the contract.
• If , then my payoff would be less than that in the 1st case since . The positive difference is exactly . Discounting it back to the present, this amount is exactly . I should be compensated by this amount today for the lesser amount at maturity (compared to the 1st case). So I should receive  (positive cashflows) when entering into the position, so the value of this position is the negative of this amount, which is . This agrees with the formula in the lecture for the value of long forward contract.
• If , then my payoff would be more than that in the 1st case since . The positive difference is exactly . Discounting it back to the present, this amount is exactly . I should pay this amount to the seller today for the extra amount at maturity (compared to the 1st case). So I should pay (negative cashflows) when entering into the position, so the value of this position is the positive of this amount, which is . Again, this agrees with the formula in the lecture for the value of long forward contract.

2. Open Matlab and change the current directory to C:\Desktop.

3. At the Matlab Command >> type

>>[hist_date, hist_high, hist_low, hist_open, hist_close, hist_vol] = stock_data(‘GOOG’);

Here, GOOG is the stock symbol for Google Inc.

4. Among others, the above will return hist_close as a (column) vector of daily closing stock prices starting from the 1st trading day of 2005 to the present day. Depending on your needs, you may wish to change the start year on Line 06 below. We can later use hist_close to analyse the continuously compounnded returns of the stock.

stock_data.m

% Script to Retrieve Historical Stock Data from Yahoo! Finance
% modified from LuminousLogic.com
function [hist_date, hist_high, hist_low, hist_open, hist_close, hist_vol] = stock_data(stock_symbol)

% Define starting year (the further back in time, the longer it takes to download)
start_year = '2005';

% Get current date
[this_year, this_month, this_day, dummy, dummy, dummy] = datevec(date);

% Build URL string
url_string = 'http://ichart.finance.yahoo.com/table.csv?';
url_string = strcat(url_string, 's=', upper(stock_symbol));
url_string = strcat(url_string, '&d=', num2str(this_month-1));
url_string = strcat(url_string, '&e=', num2str(this_day));
url_string = strcat(url_string, '&f=', num2str(this_year));
url_string = strcat(url_string, '&g=d&a=0&b=1&c=', start_year);
url_string = strcat(url_string, '&ignore.csv');

% Open a connection to the URL and retrieve data into a buffer
buffer = java.io.BufferedReader(...
         java.io.InputStreamReader(...
         openStream(...
         java.net.URL(url_string))));

% Read the first line (a header) and discard
dummy   = readLine(buffer);

% Read all remaining lines in buffer
ptr = 1;
while 1

% Read line
buff_line = char(readLine(buffer));

% Break if this is the end
if length(buff_line)<3, break; end

% Find comma delimiter locations
commas    = find(buff_line== ',');

% Extract high, low, open, close, etc. from string
DATEvar   = buff_line(1:commas(1)-1);
OPENvar   = str2num( buff_line(commas(1)+1:commas(2)-1));
HIGHvar   = str2num( buff_line(commas(2)+1:commas(3)-1));
LOWvar    = str2num( buff_line(commas(3)+1:commas(4)-1));
CLOSEvar  = str2num( buff_line(commas(4)+1:commas(5)-1));
VOLvar    = str2num( buff_line(commas(5)+1:commas(6)-1));
adj_close = str2num( buff_line(commas(6)+1:end));

% Adjust for dividends, splits, etc.
DATEtemp{ptr,1} = DATEvar;
OPENtemp(ptr,1) = OPENvar  * adj_close / CLOSEvar;
HIGHtemp(ptr,1) = HIGHvar  * adj_close / CLOSEvar;
LOWtemp (ptr,1) = LOWvar   * adj_close / CLOSEvar;
CLOSEtemp(ptr,1)= CLOSEvar * adj_close / CLOSEvar;
VOLtemp(ptr,1)  = VOLvar;

ptr = ptr + 1;
end

% Reverse to normal chronological order, so 1st entry is oldest data point
hist_date  = flipud(DATEtemp);
hist_open  = flipud(OPENtemp);
hist_high  = flipud(HIGHtemp);
hist_low   = flipud(LOWtemp);
hist_close = flipud(CLOSEtemp);
hist_vol   = flipud(VOLtemp);

Explain why it is not optimal to exercise the American option EXCEPT POSSIBLY at maturity date or immediately before .

On a side: Let and denote the time immediately before and after . The interval between and is so small such that we can assume and . Also, we will not talk about ; instead, we speak of and . If the dividend paid at is , then .

1. and in ALL possible outcomes;

2. and in ALL possible outcomes.

In fact, the two are equivalent up to time value of money. Here is how we can `normalize’ the 2nd portfolio so that its value becomes zero at initiation: since the value of the second portfolio is negative, we receive money when entering into the portfolio; we can invest this amount in the money market so that the value of the resulting portfolio becomes zero. Consequently, the value of the resulting portfolio at maturity will be positive in all possible outcomes, which is the case of the 1st portfolio.

The above portfolios are sufficient but not necessary for generating arbitrage. This is because arbitrage, based on the definition given in the lecture, is weaker than what we have just seen. Recall the following definition of arbitrage:

• and in ALL possible outcomes, and in SOME possible outcome.

If the initial value is not , an equivalent way of expressing arbitrage is that

• in ALL possible outcomes, and in SOME possible outcome.

Here, is the profit generated by the portfolio at maturity . Therefore,

Arbitrage is making positive profit with some positive probability without any risk of loss (i.e. negative profit occurs with zero probability)!