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Unformatted text preview: ns to estimate σ. For
example, if the option expiration date is in 90 days, we will collect the daily stock prices of the past 90 days and
then compute the SD of CCR. To convert the SD of CCR to an annualized figure, we do:
√( )( ). T HE MEANING OF XE -R T
Xe-rt is the present value of the exercise price of an option, where e-rt is the present value factor that uses a
continuously compounded discount rate.
An investor who would like to defer the purchase of a stock may buy a call option today. By further setting
aside a deposit which will guarantee an amount of $X at maturity, the investor is guaranteed to have sufficient
funds in the future to purchase the stock. If S*>X, the investor can use $X in the deposit account to purchase
the stock. If S*<X, the investor can purchase the stock at S* and have money to spare.
Similar, in terms of a put option, an investor who would like to defer the sale of a stock may buy a put option
today. By borrowing an amount which $X must be paid off at maturity, the investor is guaranteed to have
sufficient funds to pay back. If S*>X, then the investor can sell the stock into the market at $S* and have
money to spare. If S*<X, then the investor may exercise the right to sell the stock at $X and have money to
repay the debt of $X. T HE UNIT NORMAL DISTRIBUTION
A normal distribution is a continuous bell shaped probability distribution that is symmetrical at its mean. The
bell shaped curve is called the normal density function. For a unit normal distribution, the mean is 0 and the
SD is 1. N(x) is the area under the curve from minus infinity to a given standardized value of x. Since the
probability of observing all possible values is 1, the probability of observing an outcome with a value greater
than x is 1 – N(x).
̅ To standardize and find x, use the formula - T HE MEANING OF D 2
Black and Scholes showed that the natural logarithm of stock price is normally distributed with:
Mean = ln S + (r – 0.5σ2)t Standard Deviation = σ√ Hence, to find the probability of seeing a call option finish in the mone...
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