Test 2 Stock Option Pricing Math 115a

Formulas

Compound Interest P dollars invested at an annual rate r, compounded n times per year, has a value of F dollars after t years. Discrete Compounding

Compound Interest (cont.) P dollars invested at an annual rate r, compounded continuously, has a value of F dollars after t years. Continuous Compounding

Compound Interest (cont.) Investments with different rates and frequencies of compounding can be compared by looking at the values of P at the end of one year, and then computing the annual rates that would produce these amounts, without compounding. Such a rate is called an effective annual yield, annual percentage yield, or just the yield.

Compound Interest (cont.) Discrete: Interest at an annual rate r, compounded n times per year has yield y. Continuous: Interest at an annual rate r, compounded continuously has yield y.

Compound Interest (cont.) Under continuous compounding, the ratio, R, of future to present value is given by This allows us to convert the interest rate for a given period to a ratio of future to present value for the same period.

Excel Functions Sort (under Data) Histogram (under Tools – Data Analysis) MIN/MAX (Statistical) COUNT (Statistical) BINOMDIST (Statistical) RANDBETWEEN (Math & Trig) VLOOKUP (Lookup & Reference) COUNTIF (Statistical) RAND (Math & Trig) IF (Logical)

Probability Distribution Finite Random Variables Take on only finitely many values Ex.: X=0, 1, 2, or 3 (you can list all values) Probability Mass Function (p.m.f.), fX(x), where fX(x)=P(X=x)=area of the x’s rectangle. The p.m.f. is a histogram. The area under the p.m.f. is equal to one. 0  fX(x)  1 for all x.

Probability Distribution Finite Random Variables (cont.) Cumulative Distribution Function (c.d.f.), FX(x), where FX(x)=P(X  x). 0  FX(x)  1 for all x. FX(x) = 1 at the largest value of X. Expected Value

Probability Distribution Special Finite Random Variable Binomial Random Variable Must satisfy the following You have n repeated trials of an experiment. On a single trial, there are two possible outcomes, success or failure. The probability of success, p, is the same from trial to trial. The outcome of each trial is independent. Expected value: E(X)=np

Probability Distribution Continuous Random Variables The possible values of X form an interval Ex.: X=any real number (you cannot list all values) Probability Density Function (p.d.f.), fX(x), where fX(x)  P(X=x) The height of the function at x does NOT give the probability at x. P(X=x) = 0 for all continuous random variables. The area under the p.d.f. is equal to one. fX(x)  0 for all x.

Probability Distribution Continuous Random Variables Cumulative Distribution Function (c.d.f.), FX(x), where FX(x)=P(X  x). 0  FX(x)  1 for all x. FX(x) = 1 at the largest value of X, or approaches 1 in the case of an exponential R.V. Expected Value There is no general form of E(X) for all continuous random variables. However, for some forms of R.V.’s, it is known.

Probability Distribution NOTE: The fact that P(X=x)=0 is not implying that X cannot take on the value x. In reality, there are an infinite number of choices for X. The chance it equals one particular value is just very very small—essentially zero.

Probability Distribution ALSO NOTE: Because there is no practical interpretation of fX(x), other than the height of the function at x, it does not mean that fX(x) is not important. It not only shows the relative frequencies of values of X occurring, but it also gives the form for which FX(x) may find the area under.

Probability Distribution Special Continuous Random Variables Exponential Random Variables Usually describe the waiting time between consecutive events

Probability Distribution Special Continuous Random Variables Uniform Random Variables X is uniform on the interval (0,u)

Random Samples In the real world, most R.V.’s for practical applications are continuous, and have no generalized form for fX(x) and FX(x). We may approximate the density functions by taking a random sample, with a large enough sample size, n, and plot the relative frequencies within the sample. Then,

let a part represent the whole. Random Samples The whole idea behind random sampling is to let a part represent the whole.

Simulations We do repeated experiments to get a large enough random sample, such that the sample mean may approximate E(X), is not always practical. (Try flipping a coin 80 times, right?) Computers may be used to simulate these experiments, making it much more time- efficient to create the random sample.