Option Greeks – Theta
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About In-the-money Put Option. The value of an option is composed of Intrinsic Value and Time Value. However, for an in-the-money Put Option, the option value is less than its intrinsic value, which means the time value is negative.
I wonder why the put option value could be less than its intrinsic value. I am reading L. Galitz's book Financial Engineering recently. Chapter 10 is about Option. The question is from the chapter of that book. Is this a quote from the book?
An in-the-money put option has the potential to be exercised immediately at the current intrinsic value, PV K -S, whereas the option value on the market would incorporate the potential that the option may be out of the money at expiriation and european put option in-the-money theta 0 valuethus a negative "time value".
That makes a lot of sense. When is your seminar? I'd like to addend. We could apply same logic to a call option and justify its price being less than its intrinsic value. It should be noted that the put's european put option in-the-money theta differs from its intrinsic value by a small amount so that the transaction costs do european put option in-the-money theta allow arbitrage.
A better explanation, imo, is: A call has potentially unlimited upside, while the put's upside value is limited to K. Given the same absolute spread between S and K, I would think the time value piece of the option european put option in-the-money theta would always have to be higher for the call. European put option in-the-money theta that this explains why the intrinsic value on the in-the-money put is negative No, because there is an asymmetry because calls and puts: The nonzero probability albeit small of a large, positive payoff for a call compensates for the possibility of the option expiring out-of-the-money.
Since options usually expire in less than a year, I do not think that the unlimited upside comes to play at all.
The put-call parity relation does not distinguish between them, why should we? Because of put-call parity, a call on a non-dividend paying stock is a put plus a share of stock and european put option in-the-money theta the present value of the strike price. As above, the value of the call is composed of its intrinsic value and a time value. Because of this put-call parity relationship, the time value of the call is the time value of the put plus the time value of owning a share of stock and borrowing the present value of the strike price.
Because the option is in the money, the current value of the stock minus the present value of the strike price is positive and the time value is positive.
Therefore the time european put option in-the-money theta of the call is larger than the time value of the put. Thus, all other things being equal and barring the availability of early exercisetime european put option in-the-money theta in favor of Calls and against Puts. Because of put-call parity, a put on a non-dividend paying stock is a call minus a share of stock and lending the present value of the strike price. As above, the value of the put is composed of its intrinsic value and a time value.
Because of this put-call parity relationship, the time value of the put is the time value of the call minus the time value of owning a share of stock and lending the present value of the strike price.
Because the option is in the money, the present value of the strike price minus the current value of the stock is positive and the time value is positive. Therefore the time value of the put is larger than the time value of the call. BTW you are also assuming, that for the time value function: It's still true, just to a lesser extent and the real market price is driven much more by other factors.
The time value of S t has a tendency to drift upwards as t increases and therefore the time value of -S t has a tendency to drift downwards as t increases. European put option in-the-money theta highlights the asymmetry in the argument between calls and puts that I alluded to above. Is there a formal definition of the "time value function"? As far as know we are discussing an intuitive concept the partitioning of the option value into an intrinsic value piece and a time value piece.
It is true enough for our discussion, or you may want to revize your argument for the cases it holds. European put option in-the-money theta the drift shoud be negligible for the latter. I do not think I have seen a formal definition of it. But since you conveniently assumed it's linear, I mentioned you need to justify it.
All other things being equal, the longer the time to maturity, the higher the value of put is. I assume you are talking european put option in-the-money theta a European put option. For an American option, time value cannot be negative because if the option is sufficiently deeply in-the-money you can always exercise the option, in which european put option in-the-money theta the value of the option will equal the intrinsic value, and time value would go to zero.
Now an American put option will always be worth more than a European put option, and since there are instances when the value of an American put option equals its intrinsic value, european put option in-the-money theta follows that at times the value of a European put option becomes less than its intrinsic value.
Also it is not necessarily true, even for an in-the-money European option, that time value is negative. The option has to be sufficiently deeply in-the-money for the time value to be negative. By the way, I have not figured this out! Please see John Hull, section 9. This has to be compared against K-S.
The expression outside of the square bracket is less than 1. This is the reason for the qualification that the European option be sufficiently deeply in the money for the time value to be negative. Bagheera, yes, I'm talking about European Put. The option value before maturity could be less than the intrinsic value, I can understand. While, on the date of maturity, the option value is still less than its intrinsic value, isn't it? True for American Option, but not certain for European Option.
Hull's book thinks so. My argument is with reference to a particular point in time before maturity. Hull also, in the book, explains the nuance before the options mature. At maturity, there is no concept of time value and so value of the option equals intrinsic value for both types of options, American and European. True in most cases. It said Theta is usually negative for an option with some exceptions.
One of them is an in-the-money deep? I think this is because if the price of the underlying stock is not very volatile and european put option in-the-money theta option is deep in-the-money, it is likely that the option remains in-the-money at maturity as well. However, I do not quite understand why an in-the-money European call option on a currency with a high interest rate could have positive theta as is stated in the footnote on page of Hull.
One would tend to think that with a high interest rate, which is akin to a high dividend, the price of the foreign currency would tend to fall with the passage of time causing the currency option to fall in value. Buying a call on a currency guarantees that the cost of the currency would be no greater than the current forward rate, which already is calculated based on the high interest rate. Since high interest rates generally go down with time, this will appreciate the option. I understand it may give intuition, but under BS, the interest european put option in-the-money theta is constant.
It is assumed constant. With time though it changes and this will change the value of the option, in this particular case the assumption is the rate will decline and the call will be even more in the money and therefore its value goes up, i.
My point is, when you compute theta, r is treated as a constant. European put option in-the-money theta don't see where they have made the assumption that the rate will decline. Your idea can help explain the movement of the empirical price, but not the theoretical one under BS. Theta appears to be used in two senses: To denote the instantaneous rate of change of price with respect to the passage of time with all other relevant variables being kept constant, which is also the formal european put option in-the-money theta of theta.
To capture the general direction of movement of price with respect to the passage of time over finitely large intervals of time where other relevant variables may change. Loosely speaking, one may say that the first definition of theta corresponds to a partial derivative and the second to a "total derivative" where finite differences rather than differentials are used. Car'a'carn's explanation is with respect to the second definition.
But some references of using theta in the second sense would be helpful. What I wanted to say was that if we are talking about the theoretical price under BS, european put option in-the-money theta can't use things like declining interest rates over time to explain how the theoretical price will move. And if you use the second definition, basically you can conclude nothing. Remember that the change in volatility over time can offset the effect of the change in interest rates.
I understand Car'a'carn's explanation and I just wanted to point out that it may not be appropriate to explain prices under BS that way. Some clarification is needed. Initially I responded to bagheera's post saying that high foreign currency interest rate will drop the exchange rate, so when saying interest rate I had this in mind. Sorry about the confusion. We can do also formulae to show that theta of a call on foreign currency can be positive. If an option written 3 months european put option in-the-money theta is reevaluated today, then in the first equation K is the current forward exchange rate, since the option is in the money, european put option in-the-money theta current forward exchange rate is less than the forward exchange european put option in-the-money theta 3 months ago, therefore the last term is even smaller, and theta is higher.
Oh yes, it's the foreign interest rate above which is just like a dividend yield associated with an equity option. Also, please see the last line of page of Hull: The value of a call option is therefore negatively related to the size of an anticipated future dividend, and the value of a put option is positively related to the size of an anticipated future dividend. Here we anticipate european put option in-the-money theta foreign interest rate or the dividend rate to decline, perhaps rapidly and so "instantaneously", increasing the value of call.
On a separate note, I do not see why the strike price should be equal to the current forward price for the no-arbitrage condition to hold. The strike price could be anything with the constraint that the option be in-the-money; this is not a currency forward. I guess the overall point is that expectation of a decline in the "dividend yield" causes the price of the underlying asset to increase increasing the value of the call option and so theta could be positive.