Calender spread

derswap07 · March 9, 2020, 4:32pm

Vol 3 p. 267
A short calendar spread is created by purchasing the near-term option and selling a longer-dated option. Thetas for in-the-money calls may provide motivation for a short calendar spread. Assume a trader purchases the XYZ SEP 40 call with a theta of –0.007 for a price of 5.15. The trader sells the OCT 40 call with a theta of –0.011 for 5.47 to offset the cost of the SEP 40 call. The position nets the trader a cash inflow of 0.32 (= 5.47 – 5.15), and the initial position theta is slightly positive –0.007 – (–0.011) = +0.004.

If the stock price of XYZ remains at 45 (above the strike of 40) at the SEP expiration, the XYZ OCT 40 call will lose time value more rapidly than the SEP 40 call. The trader may close the position at the SEP expiration and make a profit of 0.17 = 0.32 + (5 – 5.15). Note that the profit consists of the 0.32 initial inflow plus the net cost of selling the SEP 40 call (at 5.00) and buying the OCT 40 call (at 5.15)*

I do not understand why oct call will lose time value more rapidly then sep 40 call. Shorter options have higher volatility-??

Dreamliner · March 10, 2020, 12:48am

Someone else ask the same question some days ago and he hasn’t replied to me yet,so l also want to share my opinion with you, because l always have no idea whether it is true or not.

For Theta, it means the sensitivity to a decrease in time to expiration,it is always negative, but for the put option that is deep in the money, it may be positive because the long side eager to exercise the option as soon as possible to earn more money (since maybe the underlyng price can’t drop further)

So, if the time decay is good for the long side, Theta will become larger than before, for example, it may change from -2 to -1 or even positive, if the time decay is bad of the long side (for example,for the option which is deep out of money) the Theta will become smaller, for example, -2 to -3

Therefore, if the call option is just in the money, so the long side may be more eager to exercise the option now, so, it is easy to suppose currently,the Theta of the call option matures in SEP is -1, the Theta of the OCT option is -2 (because the earlier the option matures, the better)

So, it is easy to see that an ITM OCT 40 call will lose time value more rapidly than an ITM SEP 40 call, because the Theta for the OCT is smaller than the Theta in SEP

Short calendar spread means that we buy the SEP option while sell the OCT option, so, the net theta we can calculate approximaterly like this, use our suppose mentioned: -1+2=1, so we gain the time value.

Tomiisinoyee · March 10, 2020, 1:02am

Theta relates to time decay - October call has a higher theta and thus will lose time value quicker than sep call

Dreamliner · March 10, 2020, 1:11am

So, my understanding why theta for SEP is larger than the theta in OCT can also be concluded as:

Since the option is in the money, the price may be fallen in the next few days, if the time to maturity decrease, we might be more likely to gain money

For example, if the price decrease, the price in SEP can also be 42, so we can gain the profit of 2-premium, the OCT might be drop to 35, so we will lose money equals to the premium. So, with time passes, although both option will lose value, but the SEP contract might be perform better than OCT option.

Your opinion l think may only refer to one single option, time value will decrease faster with the time decay.

Dreamliner · March 10, 2020, 1:29am

If l use the number to express my idea:
Suppose currently is June, we use the theta as with time passing for another 1 months, the time value can be approximately calculated like:

the time value of OCT call will be greater than the SEP call at the beginning
For OCT call, time value is 10, 9, 7, 4, 0
For SEP call, time value is 5, 4.1, 2.5, 0

For OCT call, theta is: -1, -2, -3, -4
For SEP call, theta is: -0.9, -1.6, -2.5

So, at the first three months, the net theta is
0.1, 0.4, 0.5

Dreamliner · March 10, 2020, 1:31am

This problem also confuse me for a few days, this is the only solution l can get, although the correctness can’t be guaranteed.

S2000magician · March 10, 2020, 2:20am

Theta is quite interesting. I have a Black-Scholes-Merton model I created in Excel so I added a theta graph to it. The specifics are:

S₀ = 45
X = 40
r_rf = 2.0%
σ = 30%

Over the range 0 < T ≤ 1, both θ_c and θ_p increase until T ≈ 0.145 (about 53 days), then they decrease. So, it’s not necessarily true that theta is higher the closer the option is to expiry.

Similar results occur when the strike price is greater than the spot price. The only case I saw where theta was always larger the closer the option is to expiration is that of an at-the-money option.

fino_abama · March 10, 2020, 4:59am

There are two main factors which influence theta value:

Moneyness
Length of time to expiration

Moneyness
The theta value is usually at its highest point when an option is at the money (ATM), or very near the money. As the underlying price moves further away from the strike price, option is going into the money or out of the money, so the theta value gets lower (in absolute terms).

A deep in the money option would have less time value to diminish, because the price would be made up of mostly intrinsic value, so the rate of decay tends to be slower. A deep out of the money option would also have less time value, but for a different reason. The further out of the money it is, the less chance there is of it finishing in the money.

Length of time to expiration
The length of time to expiration also impacts the theta value, as the effect of time decay typically increases (in absolute terms) as an option gets nearer to expiration. Theta value will usually get higher (in absolute terms) the less time there is until expiration, although the exception to this is for deep out of the money options (and deep in the money options).

If you refer to the diagram attached, at S = 45 and X = 40 (\sigma = 30%; r = 2%; S = 45; X = 40), the 60-day call theta is higher (in absolute terms) compared to the 30-day call theta.

biwa7636 · March 10, 2020, 5:04am

I am using an online BSM calculator and plotted out the theta vs time to expiration.
It is pretty clear that theta increases in its magnitude as t becomes smaller but there is indeed a comeback when it is really close.

I guess you can also take a deritivative using the BSM itself with respect to t and you can find it from wikipedia.

However, good luck explain the why… My take away is that it is convex and mostly negative. And calendar spread is to long the smaller theta and short the large theta…
https://goodcalculators.com/black-scholes-calculator/

S2000magician · March 10, 2020, 6:19am

Regarding the last two posts, it’s interesting that theta is depicted as negative. Because T is the time to expiry, as you move forward in time, T is becoming smaller, so theta should be positive (i.e., more time value when there’s more time to expiry). It’s weird that they depict it as negative.

fino_abama · March 10, 2020, 6:51am

You sure?

If that’s the case, then I guess it is weird too that the authors of this reading in the CFA syllabus show option theta as negative in the textbook.

Theta for long option positions are negative, as it shows the rate of decay in the time value of the option. I agree that time value will reduce as you are nearer to expiration, but the Option Theta shows the rate of decay in time value per day given the time to expiration.

If you are referring to short option positions, then I would agree the theta is positive.

Dreamliner · March 10, 2020, 9:38am

So, this is a diagram l found online, now l become more and more confused now By the way, it’s a very interesting question.

Now maybe we can think the Theta can be seen as a instantaneous condition and its curve can be determined by theses factors:
Factor1. For a single option, Theta will become more and more negative with time passes by
Factor2.Whether the move of the price is good or not to the long side.
Factor3. Whether the long side can exercise the option when the option is in the money.

If the stock price will be in the money in the next few days but we can’t exercise early because of the European option, so with the time passes by, it will make the long side lose its advantageous, make the Theta negative for the long side, but next, the stock will goes up, which move to the long side’s favor, so with time passes by, the Theta will become less negative, such as from -2 to -1, the factor 2 outweigh the factor1, so the downward then upward curve

For the out of money option, the stock price might be increase, the factor2 outweigh factor1, thus the upward curve.

For the at the money option, factor1 is the determined factor, so the curve is always downward, with time passes by, the Theta become more negative.

S2000magician · March 10, 2020, 1:35pm

I see where I had the disconnect: I was thinking of theta as:

\theta = {\partial V \over \partial T}

rather than:

\theta = {\partial V \over \partial t}

I’m fine now. Thanks!