Why Higher Stock Prices Raise Call Option Values
Summary
The question challenges the usual explanation that a higher underlying price raises a call’s value because it makes finishing above the strike more likely. It argues that Black–Scholes hedging removes concern about the stock’s average drift, and asks whether that should also remove the effect of the current stock price.
The answer uses a forward contract to expose the flaw: under the stated martingale assumption, its value is the conditional expectation of the future spot price. If today’s spot rises, that conditional expectation rises too. This illustrates that drift neutrality does not mean a derivative’s value is independent of the current underlying price. The explanation is brief and offers no option-pricing derivation, model details beyond the martingale assumption, or discussion of dividends, rates, or volatility.
Key ideas
- A derivative’s value can depend on the current underlying price even when its valuation does not depend on expected drift.
- Under the martingale assumption, a forward’s value equals the conditional expectation of its future spot price.
- A higher current spot raises the forward value in the example, clarifying why drift neutrality does not imply price independence.
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# Why does a higher stock value imply a higher call option value # Why does a higher stock value imply a higher call option value 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. ## Answer by Ezy (score 2) https://quant.stackexchange.com/a/43780 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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