Hi Mugono,
1) Let S_t, D_t be two hypothetical prices of the underlying with all else equal. Suppose that S_t<=D_t. Then max(S_t-K,0) <= max(D_t-K,0) and so the respective intrinsic values of a european non-dividend paying call option are increasing. It is shown that (in a world of non-negative rates) the intrinsic value <= call price. To this point the argumentation is clear. However, then the core reading states that from this it is evident that the c_t(S_t)<=c_t(D_t). My point was that it may happen in theory (or maybe it may not - I do not know / it is not stated at this stage anywhere) that the time value of the option would fall and hence would outweigh the increase in the intrinsic values. Hopefully, I am not misunderstood: the intuition behind is clear to me. I am trying to just found out whether from the very first principles these effects on option prices can be rigorously derived or they are more about interpretation/intuition (the latter case would mean that in very rare cases - maybe just mathematically possible cases - the basic statements for example on Deltas mentioned in your post may not be valid).
2) In your example, you arrived at lower bounds. Mathematically [call (K1 strike) price] = 15.01<15.02=[call (K1 strike) price] is possible, isn't it?
3) Sorry for being unclear. The task is to asses the interest rate movements impact on the option price - just based on first principles (without any "hardcore" Black-Scholes assumptions). The combined material pack states that buying a call option is roughly comparable to buying a share directly/immediately. And hence the reasoning is as follows: buying a call option and investing that rest of the money (which would be required to be paid for the share) into the risk-free investment will end up in a higher investment return when interest rates increase. So the call price must increase with increasing rates. The comparison statement in bold above is something what is a bit unclear to me (I still have some intuition, but I am not that convinced).
Overall: I think it is prohibited to publish online any direct core-reading / cmp copies without the consent of the respective body, isn't it?
Ok, I think I'm beginning to understand your queries (a little) better
. An option's dynamics is complex and is affected by a myriad of factors, e.g. time to expiration, moneyness, implied volatility skew / dynamics etc.
1. Time value is an interesting animal
. The answer to your question: "[can] time value of the option fall and outweigh the increase in intrinsic value" is yes it can.
For example, stock options about to expire after earnings usually have inflated option premiums before earnings. The market may expect the stock to jump (say) 10% on average post earnings and will inflate the option premium (via the implied volatility) to capture this expectation. Let's say a bullish trader decides to buy at the money (ATM) call options on a $100 stock for $5 that they intend to hold through earnings.
Following earnings, the implied volatility will get crushed and deflate. If the stock only moves (say) to $102 [so is $2 ITM] the trader could very well be sitting on a loss despite the ITM gains. The size of the loss will depend on the residual time (and implied volatility) left in the option. Were the option to eventually expire with the stock at $102, the trader would be nursing a loss of -$3. In other words, they would have spent $105 for a stock currently trading for $102.
2. Is there a typo in your question? In practice, you'd likely need to see what price the same strike / expiration put is trading for to see if there's a potential to arb it. Remember that puts and calls (European options) are governed by put-call parity: c - p = (F - K)*exp{-r(T-t)}
where F = forward price, c = call option, p = put option, K = strike, r = risk-free rate, T-t = time to expiration.
This relationship is governed by no arbitrage.
At two different strikes, you'd check whether there is a potential for a box arbitrage (akin to a zero coupon bond).
3. Whilst I agree that the price of a call option increases with rates I personally wouldn't explain it in the way described above. A call option is synthetically equivalent to holding the underlying (F) and a put option. One way I think about the impact of rates on options is as follows:
- Options are priced off the forward [note that an option gives you the right (if you're long) or obligation (if you're short) to take possession of the underlying at the option's expiration date (European options) or earlier if option is American style]
- The relationship between a stock price and its forward price is F = S*exp{r(T-t)}
The payoff of a call option is max{F-K,0}
- Therefore, if r increases => F increases => call payoff increases => call option price increases.
With all that said, the bold text is approximately true for deep in the money options where the delta is (close to) 1. In this case, the option will move 1-1 with the underlying.
Hope that helps.
p.s: this is a complex area and it can take a while for the ideas to embed.
