How closely does a Put's premium proxy the opposite Call's Time Value?

Question

What are the odds of being assigned for a long dated in-the-money call option? - Personal Finance & Money Stack Exchange

The math gets a little tricky here, but here's a neat trick to at least let you know if you should be worried: The value of the put option with the same strike and expiration is a quick and dirty proxy for the time value of the call.

1. Why's this true? I understand the following quote.

John Hull. Options, Futures, and Other Derivatives (2017 10 edn). p 216.

The excess of an option’s value over its intrinsic value is the option’s time value. The total value of an option is therefore the sum of its intrinsic value and its time value.

2. How accurate and precise is this proxy?

Answer 1

We can get insight into this by applying Put Call Parity (PCP).

The time value of a call is $TVC = C-(S-K)^+$. For an in the money call we know that $S-K>0$ so we can simplify this: the time value of an in-the-money call is $$TVCITM = C - (S - K)$$

By Put Call Parity we have $S + P = D(K) + C$. Assuming 0 interest rates (so no time discounting, $D(K) = K$) it can be written $$S + P = K + C$$

Combining these two equations we can write $$TVCITM = C - S + K = P$$

So if interest rates are zero the value of the put option with the same strike and expiration is exactly equal to the time value of the in-the-money call.

In general, if interest rates are non-zero then the error involved is $K-D(K)$. For example interest rates = 4% a year, K=50, t = 0.25 years, error = 0.50 approximately.

Answer 2

Intrinsic value by definition is just the moneyness of the option (how much the call is in the money). The price of the option is total value, which is the time value + intrinsic value. Therefore, the Time Value = Price - Intrinsic Value.

Intuitively, for deeper in the money calls, the put value is a better proxy for the time value. The deeper in the money calls will trade very close to the intrinsic value as it is very likely to get exercised. Correspondingly, the deep out of the money puts, at the same strike and maturity, will have just time value left as it is unlikely to be exercised.

For at-the-money or near the money options, longer maturity options, and in higher interest rate environments, the put approximation for time value is less accurate. Although in all of those cases, an approximate time value is more useful.

Answer 3

From your link:

The time value is close to zero and a dividend is being paid.

Generally speaking, the only time a rational actor will exercise a call option early is when a dividend is about to be paid, and the value of that dividend is higher than the call option's current time value.

As I pointed out in the linked question, this statement is incorrect. Since share price is reduced by the amount of the dividend, there's nothing to be gained there. If there is time remaining in an ITM call, exercising it throws away that time premium. If capturing the dividend is desired, sell the call and buy the stock. There's no arb here.

However, when the time premium of the ITM put is less then the dividend then an arb exists.