Intrinsic Value and Time Value of European Call Options
Summary
The document distinguishes an option’s intrinsic value from its theoretical value before expiration. Although a European call cannot be exercised early, intrinsic value is defined as the payoff it would produce if exercised immediately: the positive part of the difference between the stock price and strike. This is a convention about immediate exercise value, not a claim that early exercise is permitted.
The question compares that definition with a discounted-strike expression, which is relevant to the call’s lower bound and to its limiting value when the stock price becomes very large. The answer clarifies that the latter is not the defined intrinsic value. Before expiry, time value need not approach zero; in the stated limiting case, it approaches the gap between the strike’s current value and its discounted future value. The discussion is conceptual and focuses on a non-dividend-paying stock under the stated Black-Scholes setup.
Key ideas
- Intrinsic value is the payoff an option would produce if exercised immediately, whether or not early exercise is allowed.
- A European call’s intrinsic value is the positive part of the stock price minus the strike.
- The discounted-strike expression is associated with a lower bound, not the stated intrinsic-value convention.
- Before expiry, time value need not be zero even when the call is very deep in the money.
- The explanation concerns a non-dividend-paying stock and the stated Black-Scholes context.
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Full text
# Intrinsic Value of European Options
# Intrinsic Value of European Options
I have a question regarding the intrinsic value of an European option. I use the following notations: $S_t$ price of the non dividend paying stock at time $t<T$, $T$ is the maturity, $r$ risk-free interest rate p.a., $K$ strike price, $\sigma$ volatility p.a. The Black-Scholes formula for the value of an European call option at time $t$ is given by: \begin{align} C_t&=S_t \Phi(d_1)-Ke^{-r(T-t)}\Phi(d_2) \\ d_1&=\frac{\ln\left(\frac{S_t}{K}\right)+(r+\frac{\sigma^2}{2})(T-t)}{\sigma \sqrt{T-t}} \\ d_2&=\frac{\ln\left(\frac{S_t}{K}\right)+(r-\frac{\sigma^2}{2})(T-t)}{\sigma \sqrt{T-t}} \end{align} In the textbook from Hull I find the following definition of the intrinsic value:
> The intrinsic value of an option is defined as the maximum of zero and the value the option would have if it were exercised immediately. For a call option, the intrinsic value is therefore $\max\left\{S_t-K;0 \right\}$...
Now I wonder why it is defined that way. I can only exercise the option at the expiration date. Intuitively I would have defined it as $\max\left\{S_t-Ke^{-r(T-t)};0\right\}$, which is the lower bound for the price of an European call option. Also consider the case $S_t \rightarrow \infty$ when holding all other parameter fixed. In this case $C_t-(S_t-K)$ converges to $K-Ke^{-r(T-t)}$ which is the time value of the option. On the other hand $C_t-(S_t-Ke^{-r(T-t)})$ converges to zero.
I think that as the value of the share increases, the time value should actually go to zero, since it is virtually certain that the option will be exercised. However, if the intrinsic value is defined as in Hull then this is not the case.
I am grateful for any answers and thanks in advance !
## Answer by D Stanley (score 4, accepted)
https://quant.stackexchange.com/a/61618
Intrinsic value, by definition, is the value of the option if it were to be exercised today, so there is no time value involved, and no consideration as to if the option could actually be exercised today. If the underlying is at \$50, then a call option with a strike of \$40 has an intrinsic value of \$10 by definition - if I exercise the option today, I buy a stock that's worth \$50 for \$40 for a \$10 instant profit.
I think you are trying to apply the theoretical value of the option today and not the profit if it were to me exercised today.
What your algebra shows is that prior to expiry, the time value is not floored at zero but at the difference between the present and future value of the strike.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.