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# intraday time decay?

Discussion in 'Options' started by leorc, Sep 9, 2009.

1. ### leorc

how does theta work intraday?

i know that if theta of an option is X then after one day has past the option will lose X in value, but does this losing process gradually happening during the intrday or is it occur after the market close for the day?

thx a lot for ur input~

3. ### dagnyt

It depends on how you set the computer that provides the data - is programmed.

If you want to know how your source of information is doing that, you will have to ask them.

Mark

4. ### slickpick

It depends on a whole gamut of factors and as you should recall theta is different for calls and puts. If you really want to see the effects of theta on the price of an option, why don't you write a simulation?

5. ### dmo

Why is theta different for calls and puts? Actually, the call and the put at any given strike are identical except for their delta, which is the most superficial property of an option. That's why you can turn a call into a put or a put into a call in a flash simply by buying or selling the underlying. Theta doesn't change.

In other words, if you have a 90 call and you sell one unit of underlying, you have a (synthetic) 90 put.

As to whether or not to calculate intraday time decay - it just depends on how much precision you need. Your pricing model actually uses "percentage of a year remaining until expiration." For most people, (days remaining until expiration)/365 is good enough until you get within the last few days before expiration. If you're a fussy type, or within the last few days until expiration, you can use (minutes remaining until expiration)/525,600. That's how the VIX calculates time remaining until expiration.

6. ### slickpick

Mathematically, the way puts and calls are priced is different, this point is also illustrated in the put-call parity which should show that they're not the same! Now what theta is, is essentially the price sensitivity of the option to time; mathematically it's the partial derivative of the function that prices the option in respect to time.

Because, the functions are inherently different you actually get a different theta for calls an puts.

7. ### dmo

To the contrary, put-call parity is based precisely on the fact that a put and a call at the same strike have the same vega, the same theta, and the same gamma. They are the same, except for their delta. That is the basis for the fact that a long call short underlying = a long put, long put long underlying = a long call, and a long call short put = a long underlying. It is one of the fundamental truths of options.

Don't believe me, see for yourself if the theta, vega and gamma of the put and call at a given strike are the same. Cost of carry calculations can alter things slightly, so try it using an interest rate of zero.

8. ### slickpick

If we're just looking at a basic black-scholes model for a generic underlyer. We have:

C(S,t) = S*N(d1) - k*exp(-r*t)*N(d2)
P(S,t) = k*exp(-r*t)*N(-d2) - S*N(-d1)

d1 = (ln(s/k)+(r+si^2/2*)*t/(si*sqrt(t))
d2 = d1 - si*sqrt(t)

C(S,t): Price of a call
P(S,t): Price of a put
S: underlyer price
k: strike
r: risk-free rate (e.g. libor)
t: time until expiration
si: standard deviation
N(): normal cdf
N'(): normal pdf

Now when you take the negative partial in respect to t of both function you get:

dC/dt * (-1) = -S*N'(d1)si/(2*sqrt(t)) - r*k*exp(-r*t)*N(d2)

dP/dt * (-1) = -S*N'(d1)si/(2*sqrt(t)) + r*k*exp(-r*t)*N(-d2)

Where clearly dC/dt does not equal dP/dt.

You're right about the others but not theta.

9. ### spindr0

Once upon a time, the math of BS pricing wasn't beyond me. Either it is now or my attention span is gone (g). So let's talk on my level

In put-call parity, calls are priced higher due to carry cost (assume no dividend). If there was no carry cost, the theoretical values of puts and calls would be the same.

If theta is points lost per day (assuming all other inputs are constant) and if calls have higher values than puts, wouldn't it make sense that call theta is slightly higher than put theta?

N'est ce pas?

10. ### slickpick

You're correct in that if the risk-free rate is 0 then theta is equal, however that is rarely ever the case!

#10     Sep 11, 2009
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