Math 489/889 Final Name:________________________________

Thursday, December 14, 2010

Problem | 1 | 2 | 3 | 4 | 5 | Total |

Possible | 35 | 15 | 20 | 20 | 30 | 120 |

Points | ||||||

- (5 points each) “Short Answer”, use a single sentence or “True or False” and if false, give a
reason why it is false in a single sentence. (If false, 1 point for the answer, 4 points for the
reason.)
- Short Answer: Why is Geometric Brownian Motion a better model of the stock market than Brownian motion with drift, where the drift parameter is the rate $r$ of market growth?
- True or False: The Black-Scholes pricing equation is based on the model that the underlying stock price follows a Brownian Motion.
- True or False: The Black-Scholes pricing equation values an option by taking the present value of the expected return on the option.
- True or False: The closed form solution of the partial differential equation that we call the Black-Scholes formula represents the final word in financial theory.
- True or False: The volatility of a stock price can be estimated from the Black-Scholes Formula if the option values are known from the market.
- True or False: European puts cannot be valued by solving the Black-Scholes equation, only European calls can be valued by solving the Black-Scholes equation.
- Short answer: What mathematical property of the Black-Scholes equation allows you to write the formula for the value of a strap (a portfolio consisting of one put and two calls, all with the same strike price) in terms of the value for a call and a put other solutions?

- (15 points) What is the price of a European put option on a non-dividend-paying when
the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per year
(continuously compounded), the volatility is 35% per year, and the time to maturity is 6
months.
- (20 points) Use the put-call parity relationship to derive the relationship between
- The Delta of European call and the Delta of European put. (The Delta of an option is the rate of change of option value with respect to $S$.)
- The Theta of European call and a European put. (The Theta of an option is the rate of change of option value with respect to $t$.)

Show your complete work.

- (20 points) Find a numerical approximation at
$t=0.2,0.4,0.6,0.8,1.0$
to the solution of the Stochastic Differential Equation:
$$dX=\left(1-X\right)dt+dW,\phantom{\rule{1em}{0ex}}X\left(0\right)=0.5$$
(Remark: With some general parameters, this stochastic differential equation is a model of a “mean-reverting process” called the Ornstein-Uhlenbeck process, a useful model in physics and mathematics.) Use $dt=0.2$, and $N=100$ in the table of net totals of randomly generated coin flips below.

$j$ ${X}_{j}$ $\left(1-{X}_{j}\right)$ $\left(1-{X}_{j}\right)\phantom{\rule{3.26288pt}{0ex}}dt$ dW $\left(1-{X}_{j}\right)\phantom{\rule{0em}{0ex}}dt+\phantom{\rule{3.26288pt}{0ex}}dW$ ${X}_{j+1}$ 0 1 2 3 4 5 $n$ 0 10 20 30 40 50 60 70 80 90 100 ${T}_{n}$ 0 0 2 -4 -6 -8 -10 -6 -10 -12 -12 - (25 points) A company’s cash position, measured in millions of dollars, follows
a general Brownian motion with a drift rate of 0.1 per month, and a volatility rate
of 0.16 per month. The initial cash position is 2.0 That is, the cash position at time
$t$
follows the SDE
$$\begin{array}{llll}\hfill dX& =0.1\phantom{\rule{3.26288pt}{0ex}}dt+0.16\phantom{\rule{3.26288pt}{0ex}}dW\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill X\left(0\right)& =2.0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
Read the problem and the SDE carefully!

- What are the probability distributions of the cash position after 1 month, 6 months, and 1 year?
- What are the probabilities of a negative cash position at the end of 6 months and one year?