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Unformatted text preview: y, we must find the probability of
observing S*>X, hence lnS>lnX. The below shows the probability of observing a call larger than lnX: ( (
√ ) ) = N( 19 ()( )
√ ) ( ) Cheryl Mew FINS2624 – Portfolio Management Semester 1, 2011 In summary, the previous outcome showed that: d2 is simply the number of standard deviations away from the mean of a unit normal distribution
N(d2) measures the probability that the call will finish in the money on the expiration date so that
Xe-rt N(d2) is the expected present value of the exercise price that the buyer of a call may have to set
aside at the time when the option is purchased to prepare for the potential exercise of the option,
which is restricted on the expiration date for a European option T HE MEANING OF D 1
There are a number ways to interpret d1: A . RISK/TIME-VALUE ADJUSTED PROBABILITY
From the previous section N(d2) is the probability that a call option will finish in the money on the expiration
date. Prior to the expiration date, however, there is also a probability that the call option will finish in the
money or further into the money. The longer the time and the higher the volatility, the larger the probability
becomes. As such, we must incorporate this time value into the price of a call option prior to maturity, by
adding σ√ to d2 to form d1.
Since d1 > d2, N(d1) > N(d2). Hence N(d1) is sometimes called the risk or time value adjusted probability that the
option will finish in the money. This implies that S N(d1) is the expected value of the stock that the buyer of the
call may receive on the expiration date if the option is eventually exercised.
As can be seen, the BS equation that C = SN(d1) - Xe-rt N(d2) not only incorporates the intrinsic value by
considering the probability that the option will finish in the money, but also the time value of the option by
adding σ√ to d2.
The definition of d1 implies that the time value of an option:
Since Increase with the time to maturity and volatility of the underlying stock
Decays to zero as th...
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