Friday, June 3, 2011

My simplified form of BS-E

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
d_1=\frac{\ln\frac{S}{K}+rt+  \frac{\sigma_t^{2}}{2}  } {\sigma_t}
d_2=\frac{\ln(\frac{S}{K})+rt-\frac{\sigma_t^{2}}{2}}{\sigma_t} = d_{1}-\sigma_t
C(S,t)=N(d_1)~S-N(d_2)~K e^{-rt}\, ...............(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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