This may seem like a very dumb question, but if the underlying stock price is greater, then why should a call option be worth more.

My reasoning is that, if the option price is not affected by the drift of the return from our stock, then this implies we are not bothered whether the stock price increases or decreases on average in the future, due to the hedging strategy we have set up in the derivation of the Black Scholes equation.

Now people will say that a higher stock price means we have more chance of being on the desirable side of the strike price, implying a higher option value, but from the above, we are assuming we do not care on whether the option has more chance of lying above or below the strike price. So surely then a higher underlying stock value shouldn't affect the call option value.

  • $\begingroup$ Delta hedging neutralizes the drift but it does not (and cannot) neutralize the initial stock price. $\endgroup$ – Antoine Conze Jan 31 '19 at 15:11

Instead of talking about an option you should apply your reasoning to the simpler example of the forward contract to see the flaw in your argument.

Suppose the spot is a martingale process and suppose the spot has value today $S_0$ and the forward which expires at $T>0$ has value $F_0^T$.

Suppose tomorrow the spot goes higher to $S_1>S_0$. Should the value of the forward $F_1^T$ be the same ?

Of course not for the simple reason that the forward is a conditional expectation of $S_T$ so

$F_1^T =E[S_T|S_1]=S_1 >S_0 = E[S_T|S_0]=F_0^T$

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  • $\begingroup$ could you please point out the flaw explicitly because I am unfamiliar using martingale properties. $\endgroup$ – math111 Jan 29 '19 at 21:53
  • $\begingroup$ In layman terms you can assume for now that « martingale » means « no drift » $\endgroup$ – Ezy Jan 29 '19 at 21:54
  • $\begingroup$ okay, I understand your example, but I can't see how it highlights a flaw in my argument. $\endgroup$ – math111 Jan 29 '19 at 22:25
  • $\begingroup$ @DLB so do you understand how a higher spot implies a higher forward value ? $\endgroup$ – Ezy Jan 29 '19 at 22:26
  • $\begingroup$ Logically I think that we have our price closer to the strike price, increasing our chance of the spot eventually being higher than the strike price. $\endgroup$ – math111 Jan 29 '19 at 22:32

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