Aug 09, 2024

Hi, can’t really understand why deep in the money options have a time value of zero. Does it have to do with volatility smile ? If it’s high above the strike price, wouldn’t that mean higher volatility would be priced in which would mean higher time value? How does this reconcile with the black scholes formula ? Thanks.

When you buy a call you are paying for downside protection. If you are so deep ITM that the odds of going OTM again are near 0%, the value of the call option will be very close to the intrinsic value because the odds are almost 0 that you will actually need that downside protection. The vol surface as you mentioned does not have to do with the above concept, but it can impact the time value. For example, equity skew tends to be rather steep, with implied vols skewing high for deep OTM puts and deep ITM calls. This would make time value higher for deep ITM calls, but still nowhere near as large as the “kink” in the payoff diagram (most convexity, highest gamma).

Hey thank you - how do I think about this with respect to black scholes? Isn’t that how these options are priced ?

Also, I see your point thanks - but aren’t you also paying for upside volatility too ? If you still have time to expiration wouldn’t that mean you have more upside volatility so that should command more time value ?

(after thinking about the above, you could just buy the underlying - so it’s about the down side protection you are mentioning - that’s how the premium is priced)

Look up the CDF of the normal distribution. As S(t)/K gets very very large (deep ITM for call), d1 gets large and N(d1) approaches 1 AND N(d2) approaches 1. This would mean that the BS formula outputs a value close to S(t) - Ke^-rt. If S(t)/K is only somewhat large, N(d1) approaches 1 but N(d2) may still be significantly <1. This is especially true where sigma or t is large given that d2 = d1 - sigma*t^.5. This makes intuitive sense because if there is still a lot of time to expiry or if vol is extremely high, then the time value of your option should be higher. However, for cases where d1 is extremely large (i.e. DEEP ITM) relative to sigma*t^0.5, then both N(d1) and N(d2) will be near 1 and thus the call price is close to intrinsic value. Note: call price = N(d1)S(t) - N(d2)K*e^-rt

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