Negative time value european options

I have a basic question for which I feel like I should have found the answer by googling it, but I didn't get a definitive answer, so here I am:

Can the time value for a plain vanilla (European) option be negative? I've read it can be (without an satisfactory explanation), while my professor said it cannot, not even for deep-in-the-money options and now I'm confused. Could someone please explain this to me?

• What about stochastic dividends? May 19 '15 at 5:09

Yes it can be negative.

Let us consider a deep in the money European put option. Suppose the stock price goes to $$0$$, then you know a european put will always be exercised at $$K$$, the strike at maturity $$T$$. This can be verified using put-call parity since the call will be valued at $$0$$ when $$S(t)$$ reaches $$0$$. Hence the value today must be $$P(t,T)K$$ where $$P(t,T)$$ is the discount factor from now till maturity $$T$$. However, the option premium is equal to the intrinsic value plus the time value. Letting $$TV$$ be the time value, we have: $$P(t,T)K = K + TV(t).$$ Hence: $$TV(t) = \left(P(t,T) - 1\right)K.$$ So we have some conditions, at least for the European case, for puts:

• $$r > 0$$, and
• $$S(t)<
• Needs a definition of r. Dec 23 '20 at 20:40
• r is interest rate from the P(t,T) the discount factor
– rajn
Jan 6 at 17:09

I don't know where you would have read that, but no, time value cannot be negative. Time value is option value minus intrinsic value. Intrinsic value is a model-imdependent no-arbitrage bound on option value. For an out-of-the-money payoff, intrinsic value is zero, and since the call or put payoff is non-negative this is a clear lower bound. For an in-the-money payoff, intrinsic value is $\pm e^{-r T}(F-K)$ where $F$ is the forward, $r$ the risk-free rate, $K$ the strike, and $T$ the maturity, with $+$ for a call and $-$ for a put. This is the price of a forward struck at $K$, which has a payoff less-or-equal to the corresponding option payoff. So negative time value would mean option price below the intrinsic value, which means one could buy the option, hedge with the forward (if in the money) and have an arbitrage: initial cost negative but final payoff non-negative.

• But what if $r<0$? May 19 '15 at 5:11
• Thanks, makes sense. I have a follow-up question, though: Does the value of a plain vanilla option always go up if the time to maturity is longer? Or asked in a different way, can theta be positive for long calls/puts?
– Alex
May 19 '15 at 8:37
• Some measure intrinsic value using the spot price, which is why you can see negative time values for some deep ITM european put options, though it's somewhat spurious. You will tend to see a positive theta in those cases, though it's not a hard and fast rule. May 19 '15 at 12:13
• The intrinsic value of an option is defined as $(\pm(S-K))_+$ where $S$ is the spot stock price at the current time rather than $\pm e^{-r T}(F-K)$ as you write in your answer. So your answer is not entirely correct. @rajn's answer is correct.
– Hans
Nov 13 '18 at 1:50

My understanding is it can be a negative for European options (US can ONLY be positive) From what I found a negative can happen when price insurance can be purchased against the underlying commodity, for example

OIL Jan 19 2018 4 Call 242 Days to Expiration
Bid 1.50 Ask 1.70 Bid/Ask Size 2054 X 1108 Open Interest 3,085 Implied volatility 29.59 Time value -0.09

• This doesn't make sense, it doesn't matter where the option is traded. Downvoted. May 22 '17 at 15:32
• You're probably confusing American options with options traded in the US. If you measure intrinsic value using the spot price and no discounting, that is $IV=(S_t - K)^+$ and time value as the difference between the option value and the IV, then the time value for American options is always positive, but as pointed by ocstl the time value for European options can be negative. This of course regardless of where the option is traded. May 23 '17 at 11:13
• Right but I would argue that it makes no sense to measure intrinsic of a European option using spot, since that intrinsic cannot be obtained. I favor measuring intrinsic using the forward price versus the strike, discounted back to today.
– dm63
Nov 13 '18 at 1:03