Question:

Assume Hewlett-Packard’s (HPQ) stock price is $50 with an annualized volatility of 40%. The riskless rate is 4%. For simplicity’s sake, assume HPQ does not pay a dividend (actually, they do). Consider a one year European call option with a strike of $50. To summarize:

Stock (S) = $50
Strike (K) = $50
Volatility (sigma) = 40%
Term (T) = 1.0
Dividend = 0
Riskless rate (r) = 4%

To perform this calculation, you will need the standard normal cumulative distribution.
Click here to view a lookup table.
You can also get N(z) in Excel with the function =NORMSDIST(z)

(i) What is the option’s delta?
(ii) Use this delta in a sentence (i.e., interpret its meaning)
(iii) If we assume (as we often do) that the stock price follows geometric Brownian motion (GBM), how are the stock’s PERIODIC RETURNS assumed to be distributed?
(iv) Under GBM again, how are the future stock PRICE LEVELS assumed to be distributed?
(v) What does the Black-Scholes option pricing model give for the price of the call option?

Answer:
(i)
A Eupean call option on a non-dividend paying stock has a delta = N(d1)
d1 = 0.3 (see spreadsheet here for detail)
N(d1) = NORMSDIST(0.3) = 0.62

(ii)
Possibilities include,
“if the stock price changes by X%, the call option price changes by 0.62X%”
“0.62 is the (instantaneous) rate of change of the option price with respect to the stock price”
“0.62 is the slope of the tangent line on the plot of stock price (x) to option price (y)”
“0.62 is the slope of the linear approximation given by the first-order derivative with respect to stock price”

(iii)
GBM assumption: periodic returns are normally distributed

(iv)
GBM assumption: future price levels are lognormally distributed
Please make sure you understand why:
the continously compounded periodic return is given by LN(ST/S0).
In other words, LN(ST/S0) is normally distributed.
That means, by definition, that ST/S0 is lognormally distributed (i.e., both the future price level and the future price as a ratio of the current price are lognormally distributed)

(v)
d1 = 0.3
N(d1) = 0.618
d2 = -0.1
N(d2) = 0.46
Black-Scholes = (S)N(d1) - (K)EXP[(-r)(T)]N(d2)
= (50)(0.618) - (50)(0.9608)(0.46) = $8.79 (approximately)[/img][/img]

David,

A Eupean call option on a non-dividend paying stock has a delta = N(d1)
d1 = 0.3.  Are you saying that all European call options have a delta of Nd1.  How was the .3 calculated
I can get to the .62 from the table

Hi Frank,

Yes, a euro call on non dividend stock has delta of = N(d1)
To recall, I visually cheat by looking at the formula: SN(d1) - (K)EXP[(-r)(T)]N(d2)
And recall our thread about first derivative? If a and b are constants, and f(c) = Sa - b,
here we have c = Sa - b, such that f’(c) = dc/dS = a = N(d1)
i.e., if the function were just c = SN(d1), then f’(c) = N(d1)
(it is a cheat, the actual first derivative is quite a few more steps as the other term is not a constant, but it ends up the same place)

I added the link to the XLS. You’ll see it is the long:
d1 = [LN(S/K) + (r + variance/2)(T)]/[(volatiltiy)(T)
I don’t have a good shortcut/intuition for d1. It is, in all likelihood, too much to test for.

N(d2) however is more interesting: it is the Merton model we see in Credit. It is the probability the option will be struck (in credit terms, the probability firm value will fall below the default threshold). N(d2) is quite worth understanding intuitively.

David

Thank you David it is clear now.  I am on to Quant 1

Frank