When European Option Time Value Can Be Negative
Article Quant Q&A · Author: Alex
Summary
The document asks whether a plain-vanilla European option can have negative time value, defined in the discussion as option value minus intrinsic value. One response constructs a deep in-the-money put case: with a positive interest rate, the discounted strike value can be below the undiscounted strike used for intrinsic value, producing negative time value under that definition. Another response argues that intrinsic value should instead be based on a discounted forward-price bound, in which case a price below intrinsic value would imply an arbitrage opportunity.
Key ideas
- The conclusion depends on how intrinsic value is defined for a European option.
- A deep in-the-money European put with positive rates can have a discounted payoff below the strike-based intrinsic value.
- A forward-based intrinsic bound is presented as a no-arbitrage lower bound for option value.
- The responses disagree, so the example does not establish a universally accepted convention.
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Full text
# Negative time value european options
# 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?
## Answer by turtle_in_mind (score 7)
https://quant.stackexchange.com/a/39298
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
- $S(t)<<K.$
## Answer by q.t.f. (score 3)
https://quant.stackexchange.com/a/17866
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.
## Answer by Mark (score -2)
https://quant.stackexchange.com/a/34338
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 -0.09Shown 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.