Introduction to the Mathematics of Finance. Take-Home Midterm.
Due March 10, XXXXXXXXXXp.m.
Please write a pledge that you did not copy solutions from the work of other students. You can consult TAs
if you have any difficulties. This midterm uses content of lectures in the last two weeks of Fe
uary. It also
uses base matlab models that can be downloaded from Courseworks.
1.Matlab option model. Download from Courseworks matlab option model files BlackScholesStocks.m and
BlackScholesGraph.m and put them in the same directory. BlackScholesStocks.m contains the function that
calculates the Black Scholes price for options on nondividend paying stocks. BlackScholesGraph.m is a
script that makes a graph of option price as a function of stock price. Type at Matlab prompt
>BlackScholesGraph XXXXXXXXXXand the script will be executed and the graph will appear.
Now modify the file BlackScholesStocks.m so that the function now calculates the price of options on
stocks paying continuous dividends at a rate q. Modify the file BlackScholesGraph.m so that it now plots
graph of a call with the same parameters as before but with the dividend yield q=3% and the new strike
price 11. Submit printouts of code and graph.
2 Download matlab Brownian motion model from the courseworks. Modify it to Geometric Brownian
motion with starting value Xo=100, growth rate µ = 0.11, volatility σ = 0.25 and 5000 trajectories. Check
that the code works. Try out 50,000 trajectories. Try out 100,000 trajectories. Submit the code printout and
the graph printout for 5000 trajectories.
3. Using a
itrage arguments explain why the price of an American call option on a stock paying no
dividends should be the same as the price of a co
esponding European call. Why American calls on a
nondividend paying stock should not be exercised early.
4. Why when the stock pays dividends the argument of the problem No.3 can not be used. Give a numerical
example (choosing x, k, r, T −t, σ) in which it is obvious (without any formulas) that American put price on
a nondividend paying stock is larger then the co
esponding European put price.
5. (a) The stock price is 80 the volatility of the stock is 20%. Assuming that the time to expiration is 3
months and the interest rate is 1% per annum calculate the price P of the
European call option with strike 81.
(b) Calculate Δ, Γ, ρ, Vega using formulas for these parameters. Calculate the same parameters
approximately using the options calculator.
(c) Check that following relationship holds
Pxrx =Γ+Δ+Θ 22
2
1
σ
6. What are the parameters affecting prices European and American calls and puts. How do the prices
change when one of the parameters changes with all the others remaining the same?
7. Suppose that we have three European calls with strikes 60, 65, and 70 and the same maturity 1 month.
Their prices are 9.00, 7.00, XXXXXXXXXXIs it possible to do an a
itrage?
8. Suppose that cu
ent stock price is 100 $. Its annualized volatility is 25 % and annualized return 10 % i.e.
we assume that the stock price follows dXt = 0.1 Xt dt+0.25 Xt dWt. Write the probability density function
for the stock in 1 year. What is the mean and standard deviation of the terminal stock price? (standard
deviation of price, not of return)