If a stock has zero vol and some positive drift $\alpha$ (in a BS-setting) and we delta hedge a long call option dynamically over a year with some positive implied volatility.... how would that work out for us?

Would the answer depend on how often we hedge, i.e. every day, month, week?

In usual circumstances, when we delta hedge, we make money everytime we get significant $\Delta S$s, due to the call options $\Gamma$ (gamma).

However in the case of positive $\alpha$, zero $\sigma$, $\Delta S$ is going to be huge, but .... do we then make huge amounts of money off of $\Gamma$?

  • $\begingroup$ When $\sigma$ is zero $\Delta$ is zero, so the Delta Hedging consists of buying and selling no stock at all ;) $\endgroup$ – noob2 Mar 24 '17 at 21:45
  • $\begingroup$ Do you mean $\Delta$ is 1? Also, the implied volatility is positive. $\endgroup$ – Jaood Mar 24 '17 at 22:10

If vol is zero, then the stock S is riskless. If it is riskless its drift should equal the risk free rate r. Therefore, if T is the expiry of your call option then at expiry value of stock should be S_T = S_0 exp(rT).

If your strike K <= S0 exp(rT) then you know for sure your call will expire worthless and your delta hedge = 0 stocks. If K > forward, then for sure you will get stock delivered at expiry. So your delta hedge is 1 stock short.

Note that your delta is binary depending on K. The gamma however is 0 because the delta never changes. (had gamma been different from 0, your vol would not be 0).


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