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Put–Call Parity and Option Time Value with Rates and Dividends

Article Quant Q&A · Author: ashu24
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Summary

The document asks how call and put time values relate when interest rates or continuous dividends are nonzero. It begins with the zero-rate, zero-dividend case, where call–put parity implies equal time value when intrinsic value is defined from the spot price and strike. It then considers the forward price, which incorporates the rate and dividend yield, and observes that forward call–put parity has a forward-minus-strike term.

The answer clarifies that spot option prices satisfy parity after discounting the forward payoff: the call-minus-put difference is the discounted forward-minus-strike amount. Consequently, the simple equality of spot-based intrinsic values does not carry over unchanged when rates and dividends matter. The equality of residual time values depends on using consistent forward-based terms and discounting conventions. The exchange is a concise parity explanation rather than a broader treatment of early exercise, American options, or practical valuation conventions.

Key ideas

  • Spot-based intrinsic values need not yield equal call and put time values when rates or dividends are nonzero.
  • Call–put parity relates the price difference to the forward-minus-strike payoff discounted to the present.
  • Forward-based parity can clarify the relationship between call and put residual time values.
  • The discussion does not address early exercise or extend the result to all option styles.

Tags

OptionsDerivatives pricing
Full text
# Time Value of Option


# Time Value of Option












I am working on time value of option, and especially with dividend, and I have the following questions. First if the consider the Black Scholes models with no dividends and free interest rate $r = 0$ If the write the call price : $$\operatorname{Call Price }= \operatorname{Intrinsic value + Time value}.$$ Where intrinsic $$\operatorname{value} = \max( S_t - K,0)$$ then using the call-put parity. It can be shown that

$$\operatorname{Time Value of Call} = \operatorname{Time Value of Put}.$$

However when we have continuous dividend or free interest rate I don't manage to get the following identity. Or at least I manage to get it for the Forward $F_{t,T} = S_t e^{(r-q)(T-t)}$. As $$\frac{dS_t}{S_t} = (r-q)dt + \sigma dW_t \rightarrow \frac{dF_{t,T}}{F_{t,T}} = \sigma dW_t .$$

Then the call put parity but for the future can be rewrite :

$$F_{t,T} - K = \operatorname{Call}(F_{t,T},K,\sigma,T,t) - \operatorname{Put}(F_{t,T},K,\sigma,T,t)$$

$$F_{t,T} - K = F_{t,T} - K +\operatorname{ Time Value Call Forward} - \operatorname{Time Value Put Forward}$$

$$\operatorname{Time Value call forward} = \operatorname{Time Value Put forward}$$

Do I am missing something or we do not have the relation Intrinsic Value of Call = Intrinsic Value of Put when the free interest rate and the the continuous dividend yield are not equal to 0.

I may have another question on when to exercise call option but I still need to think about it.

In advance thank you very much.

## Answer by M. Jeunesse (score 1)

https://quant.stackexchange.com/a/28260

Call put parity is :

$$C(T,K) - P(T,K) = ( F_{t,T} - K ) B(t,T)$$

and with your notation :

$$C(T,K) - P(T,K) = ( F_{t,T} - K ) + \text{TimeValueCall}(T,K) - \text{TimeValuePut}(T,K)$$

Reference:

https://en.wikipedia.org/wiki/Put%E2%80%93call_parity

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.