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## Tuesday, June 5, 2012

### ln(S/K) should be compared to what yardstick?

(update -- I feel depth of OTM/ITM is defined in terms of ln(S/K) "battling" σ√ t )

Q: if you see a spot/strike ratio of 10, how deep OTM is this put? What yardstick should I use to benchmark this ratio? Yes there is indeed a yardstick.

In bond valuation, the yardstick is yield, which takes into account coupon rate, PV-discounting of each coupon, credit quality and even OAS. In volatility trading, the yardstick has to take into account sigma and time-to-maturity. In my simplified BS (http://bigblog.tanbin.com/2011/06/my-simplified-form-of-bs.html), there's constant battle between 2 entities (more obvious if you assume risk-free rate r=0)

ln(S/K) relative to σ√ t         ................. (1)

Fundamentally, in BS model ln(S/K) at any time into the diffusion has a normal distribution whose
stdev = σ√ t, i.e. the quoted annualized vol scaled up for t (2.5 years in our example)

Note the diffusion starts at the last realized stock price.

Q: Why is σ a variable and t or r are not?
σ is the implied vol.
σ is the anticipated vol over the remaining life of the option. If I anticipate a 20%, i can put it in and get a valuation. Tomorrow, If i change my opinion and anticipate a much larger vol over the remaining life (say, 2 years) of the option, I can change this input and get a much larger valuation.

The risk free rate r has a small effect on popular, liquid options, and doesn't fluctuate much

As to the t, it is actually ingrained in my 2 entities in (1), since my sigma is scaled up for t.