Theta of an option

[Arka Bose]

Hi all,
When I was reading Hull, he mentioned that some options e.g a european put option in the money or a call option on currencies with high interest rates will have positive theta.
can anyone give an intuitive explanation to this?

 

[QuantMan2318]

I don't know if this suffices as 'intuitive' but since no one has replied I am telling you my understanding. As you know Theta is a measure of the change in the value of the option portfolio with the passage of time, for in the money put, the passage of time makes it all the more valuable as the Strike price is above the underlying asset price at or close to maturity. I am also assuming that the theta of a put being larger than that of the call by rKe^-rT, higher value of the Strike price makes it tend closer to rKe^-rT

Similarly for call options on currencies with high interest rates makes the currencies stronger because of higher demand for that currencies, that pushes up the underlying currency rate vis a vis the Strike price of the call and makes it in the Money.

Again I may be wrong also. Please feel free to correct me
PS I don't have any experience in the options industry.

 

[[email protected]]

arkabose,

When interest rates are high, European put options with less time to expiration are more desirable so that one can sell the underlying earlier and invest the proceeds at the high interest rate.

 

[[email protected]]

arkabose,

When interest rates are high, European put options with less time to expiration are more desirable so that one can sell the underlying earlier and invest the proceeds at the high interest rate.

Replace the "earlier" with "sooner" since it may sound as if there is early exercise in European options which is not the case.

 

[Arka Bose]

@Gyilmaz @QuantMan2318 ok i just want to know one basic thing,
Portfolio of a call will be delta S- PV of K.
Now, as time increases, the portfolio value will decrease as PV of K will increase. Thus theta is negative.
But a put portfolio will contain (PV of K - Delta S)
Over here, PV increases, the portfolio value increases,so with the passage of time, is it not that theta will be positive all the time for a put option?
I am terribly confused here.

 

[QuantMan2318]

The portfolio you are talking about is a risk-less portfolio, why did you take that specifically? and again, the PV of K decreases with the passage of time ( Ke^-rt ; as t increases Ke^-rt will fall ). I am assuming that you are taking two separate issues, the above formula is for valuing a risk free option + stock portfolio for deriving its value. More specifically it means that you are writing a call option and holding delta shares and for the put, it means that you are long one put and shorting delta shares. We are mostly concerned with theta as a rate of change in the value of the option when it is held ( we are long ) on the options.

The value of call option is only S*N(d1)-K*e^-rt*N(d2), here you can see that when we value a call option at the beginning of the tenure of the options life, the PV of K is actually less than what it would be when we take the PV of K later on and hence the value of the option is the highest at inception when its in the money and slowly reduces as the price of the share makes it move out of the money. What we are seeing in the case of Theta is that we take the 1st derivative of the BSM value of the call with respect to time and that in turn gives us the change in the value of the option as time progresses and call option's value generally reduces as time progresses as the options value is inversely related to time in the BSM model and derivative.

What we are trying to show here is that as the Share price moves in a favorable direction and the option becomes in the money, the value of the option would rise and theta would rise.

I hope I have made it clear and hope to the mods that the above explanation is correct. Someone with experience in trading and hedging with options can make the point clearer.

 

[Arka Bose]

Hi @QuantMan2318 ,
First of all, thanks for taking time here to explain.

I was thinking about that risk free portfolio specifically, as if that portfolio would give me value of an option, that means the value of the portfolio must also change w.r.t time ( since theta is change in value of call to change in time). That is why I tried to connect things there.

Second, you wrote "and again, the PV of K decreases with the passage of time ( Ke^-rt ; as t increases Ke^-rt will fall )" but, here, the time is now increasing, here T is decreasing isn't it? That is why I wrote PV will increase.
This, t decrease is perhaps you tried to explain through the BSM model later on 'the PV of K is actually less than what it would be when we take the PV of K later on and hence the value of the option is the highest at inception...' (i think?)

yes, and thus you are absolutely correct, the option value will hence decrease due to the passage of time, hence theta is negative.

But, I am confused about the put option.

 

[QuantMan2318]

Hi @QuantMan2318 ,

Second, you wrote "and again, the PV of K decreases with the passage of time ( Ke^-rt ; as t increases Ke^-rt will fall )" but, here, the time is now increasing, here T is decreasing isn't it? That is why I wrote PV will increase.
This, t decrease is perhaps you tried to explain through the BSM model later on 'the PV of K is actually less than what it would be when we take the PV of K later on and hence the value of the option is the highest at inception...' (i think?)

yes, and thus you are absolutely correct, the option value will hence decrease due to the passage of time, hence theta is negative.

But, I am confused about the put option.

Ah Yes, I get it, you were meaning the reduction in time as the maturity of the option approaches, bringing it closer to K, yes that's what I have meant in the BSM model. However, the risk-less portfolio that you have taken explains one side of the coin, what will you do when you are long instead of short as in the risk free portfolio? that's the answer I have given in the previous post and again if you take the risk-less portfolio, the opposing movements are counteracted somewhat. So I feel it is better to see options and portfolios of options.

The put on the other hand has the formula, K*e^-rt*N(-d2)-S*N(-d1), everything else remaining the same, the options value would rise with the passage of time, provided the option is in the money ( K>S ) or showing strong trends of moving in the money when we buy it, so the derivative is positive w.r.t time, may be that's what Hull meant when he said theta can be positive for European Put options that are in the money. Consider an European Put that is Out of the money as they usually are, then K<S and as time progresses and S keeps on increasing we get a negative value. Further we are not helped by the BSM formula of N(-d1) and N(-d2)

 

[Dr. Jayanthi Sankaran]

Hi @arkabose

There is a numerical example explaining the above in page 77-78 of the Forum discussion of the PQ set pertaining to theta. I thought it may be of help to you.

Thanks!
Jayanthi

 

[Dr. Jayanthi Sankaran]

Hi David,

Page 93, problem 18.4 of Hull Chapter 19: The Greek Letters (Hull Text Q & A) is as follows:

What does it mean to assert that the theta of an option position is -0.1 when time is measured in years? If a trader feels that neither a stock price nor its implied volatility will change, what type of option position is appropriate?

The answer is:

A theta of -0.1 means that if dt units of time pass with no change in either the stock price or its volatility, the value of the option declines by 0.1dt. A trader who feels that neither the stock price nor its implied volatility will change should write an option with as high a negative theta as possible. Relatively short-life ATM options have the most negative theta's.

I don't understand why the trader should write an option with as high a negative theta as possible. Is it because as time to maturity increases, the option becomes more valuable? When theta is large and negative, gamma is large and positive. The portfolio declines in value if there is no change in the stock price.

Thanks!
Jayanthi

 

[Arka Bose]

Hi Mam,
I think you are correct.

Normally, when we BUY an option, we want the volatility to be high (so that there are chances of high payoff)
It is exactly the opposite in case of writing an option. When we are short an option, we want the volatility to be low (else we will have chances of a negative payoff)

Now, writing an option means a negative gamma. We know that in case of high volatility, a short gamma (negative gamma) position is bad as we will lose. I once written the exact reason here https://forum.bionicturtle.com/threads/short-convexity-gamma.8724/

Thus, we can conclude that in case of low volatility, it is best to short gamma, which means LONG Theta.
And at ATM, the theta of the option is the at the highest, thus, it is the best moment to short/write an option ATM given that the volatility is low/unchanged.

The exact opposite would be going long the gamma (that is the option) and thus short the theta in case of high volatility.

Thus the term gamma-Theta payoff

 

[Dr. Jayanthi Sankaran]

Thanks!

Jayanthi