CM2-Chapter12-p.27: Interpretation of why time value can be negative in put options and not in calls

Hi
I dont quite understand the explanation at the end of the page of why a call is different from a put, regarding the time value being bellow zero: "It also doesn't happen with the call options because they work the other way around, ie holding the call means that an investor can invest money".
I understand that for a call the intrinsic value is max(St-K,0) so the time value is ct-max(St-K,0) and for it to be negative we have ct<max(St-K,0), but due to the no arbitrage inequality ct>=max(St-K*exp(r*(T-t)), this cannot happen so the time value of a call is always equal or greater than zero for a positive risk free rate.
And I can follow all the reasoning of p.27 of ch.12 up to the above aforementioned sentence. But what is meant by an investor can invest money in the case of a call vs a put? What is the thinking behind this sentence?

Thanks

 

Time value can be negative for puts but not calls because your upside is capped on a put but not on a call. Your payoff on a put can be at most the strike K, which happens in the case that the underlying price is 0 at expiry.

For example imagine the underlying price did go to zero (or very close to zero). Then the intrinsic value of your put would be K. However, the value of the put is bounded above by K * exp(-r(T-t)), the present value of the max possible payoff, which is less than K (for r>0). Hence the time value is negative.

This can’t happen for a call because however high the underlying price goes it can still go higher, so we can’t make the same argument about the present value being capped by the intrinsic value. And, from an intuitive point of view, the value of a call should always be higher than the payoff because there is always the chance the underlying can go higher and give a higher payoff (and your downside is capped). So time value for a call is positive. This differs from holding a deep in the money put you can’t really improve much further.

The argument above also describes when a put has negative time value. This will happen with some combination of: deep in the money (payoff close to K), a high risk-free rate r, long time to maturity T-t.

 

Hi
Thank you for the insightful answer. Still however I dont quite understand what is written in the last paragraph of the CMP for CM2-ch12-p.27 with "It also doesn’t happen with the call options because they work the other way round, ie holding the call means that an investor can invest money. ". Could it mean that the investor can invest to always a higher price raising the price of the share, so effectively the call has no upper bound, whilst for a put the investor cannot sell the share below zero price to reduce the share price even further therefore we have a case where the time value can be negative for a deep in the money put? Still though I think the phrasing in CMP is not the best if this is the explanation.

Thanks

 

Hi Steve
Thanks for the reply. My question still holds though because based on my understanding holding a put or a call doesn't tie up any amount of money until expiry, where you have the right to buy the underlying share for the call or sell the underlying share for the put. So in either case the investor's money is available to earn the risk free rate and therefore I don't see any differentiation between a call and a put with regards to this.

Thanks

 

Hi Steve
Thanks for the explanation, I understand your point but still this needs two presumptions that are not mentioned in the CMP neither are necessary when you hold an option.
In any case whether you hold a call or a put, you can invest any amount of money in risk free so I don't understand how this argument helps.
I think the crucial thing is that the return in a put option has a maximum value of K, therefore when you reach the limit the fair value today has to account for discounting and thus the current price could be less than the intrinsic.
In a call if you have for any reason an upper bound to the value of the share you could have again negative time value for the same reason as in the put. In the case of a dividend, the value of the share is expected to reduce, therefore again the expectation for the price of the share given the risk free discount rate could lead to a negative time value.
In my mind, the argument of holding the cash in the call and holding the share in the put with the differences of each, produces an argument for the differences in the time value of a put and call only for the case of the two presumptions and not in the general case.

Thanks