2
$\begingroup$

I'm trying to calculate the probability of a calendar spread resulting in a profit at expiration, when estimating it is modeled as a lognormal distribution, by getting:

P(a <= x <= b) = CDF(b) - CFA(a)

where a and b are the breakevens at expiration.

But there is something that I don't understand:

  1. Which value shall I use as variance? The IV of the ATM option for near expiration? The IV of the stock/index underneath?
  2. Does time really matter? I mean, since lognormal distribution (as defined in scipy/numpy libraries) only requires mean and variance values, time does not matter unless you consider that volatility depends on t. If I get mean and variance for 2 calendars, one with front mont expiring in a week and another one expiring in a year, time should matter somehow, making the distribution PDF wider, and therefore affecting the results of the CDF. What am I missing here?
$\endgroup$
2
$\begingroup$

If $S_t$ is stochastic process and follow geometric Brownian motion with following SDE: $$dS_t=\mu S_t dt + \sigma S_t dW_t$$ then $S_T$ follows lognormal distribution, such that: $$S_T|S_t \sim logN\left(lnS_t+ (\mu - \frac{\sigma^2}{2})(T-t), \quad \sigma^2(T-t)\right)$$ or $$lnS_T|S_t \sim N\left(lnS_t+ (\mu - \frac{\sigma^2}{2})(T-t), \quad \sigma^2(T-t)\right)$$

As you may see, more you will go into the future, both the drift and volatility increase directly in proportionate to $(T-t)$ for log of stock price. This is natural phenomenon. You may think like this, the variability shown by stock price in one year(ie $T-t=1$) is much more than variability shown in one minute or one day(ie $T-t=\frac{1}{365}$).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.