It's quite useful to develop a feel for how much option valuation moves when underlier spot doubles or halves. Also, what if implied vol doubles or halves? What if TTL (time to expiration) halves?

For OTM / ITM / any option, annualized i-vol multiplied by √TTL is the real vol. For example, If you double vol and half TTL twice, valuation remains unchanged.

If you compare a call vs a put with identical strike/expiry (E or A style), the ITM instrument and the OTM instrument have identical time value. Their valuations differ by exactly the intrinsic value of the ITM instrument. (See http://www.cboe.com/LearnCenter/OptionCalculator.aspx.) -- Consistent with European option's PCP, but to my surprise, American style also shows this exact relationship. I guess it's because the put valuation is computed from a synthetic put (http://25yearsofprogramming.com/blog/20070412.htm).

---

For **ATM **options, theoretical option valuation is proportional to both vol and √TTL, i.e. time-to-live. http://www.cboe.com/LearnCenter/OptionCalculator.aspx and other calculators show that

- when you change the vol number, valuation changes linearly

- when you double TTL while holding vol constant, valuation grows quadratically.

For OTM options? non-linear

For ITM options, it's approximately the OTM valuation + intrinsic value.

## Saturday, March 31, 2012

### option valuations - a few more intuitions

Labels: _fuxi, intuit_math

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