At any time before expiry fair premium of a European call option for a non-dividend paying stock is

where

S = spot price at valuation date such as today

t = time to expiry, at valuation date. This value is measured in years. If now our option is 2 years 6 months from maturity then t=2.5 at valuation date

K and r are all constants

sigma is also assumed constant -- see below

N() is the cumulative normal distribution function. I believe N(0) = 50%, N(g) + N(-g) = 1.0 and N() is monotonic increasing

Now, in the BS formula, sigma is treated as a constant -- the well-known and unrealistic constant-volatility assumption. If I were to get sigma scaled Up for t=2.5 years (our example) and denote it "simga_t" or σt then

...............(same as before)

Let's try to understand parts of this monster

Q2: why the (...)

**(...)**

*-*A: that comes from the simple fact that at expiration (not now), the terminal valuation (I didn't say "PnL") is in the form of "stock price at expiry - strike"

Q2b: what's the implication on delta?

%%A: Well we know the part after the "-" is independent(?) of spot price, so if we simulate a tiny change in S, that portion remains unchanged. Delta calc can safely ignore it.

Q: for a deep ITM call, how is this simplified?

A: ln(S/K)/sigma_t dominates in both d1 and d2, so d1 and d2 are approximately equal and N ~= 1.0. So

C ~= S-K*exp(-rt). In other words the European call valuation is mostly its intrinsic value.

Q: for a deep OTM and small rt i.e. drift ?

A: d1 and d2 are approximately equal, both extremely negative, dominated by ln(S/K), so

C ~= K * N(large negative value)

Q: what's the exp(-rt)?

A: simple. Discounting the strike price to present value. I believe for t below 2 years (listed options) this factor is close to 1.0 and has a minor effect. However if you ignore it a profitable deal can become unprofitable.

Estimating delta, vega, gamma, theta all requires differentiating through the normal distribution.

(use http://en.wikipedia.org/wiki/Wikipedia:Tutorial/Citing_sources/sandbox to edit equations).

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