# What is the probability distribution of the changes in $\Delta$?

What are the odds that a 10 delta option will become a 30 delta option in "N" number of days?

Is that calculation possible?

For instance, I want to know the probability with which my 56 day, 10 delta option will become a 30 delta option anytime over the next 16 days.

• I adjusted your question because the probability of $\Delta$ to change between $t_1$ and $t_2$ is 1 or extremely close to 1 ($\Delta$ depends on $t$, so as $t$ changes $\Delta$ will change, all else equal).
– SRKX
Commented Nov 6, 2014 at 5:27
• Note: all else equal is important in my comment. I think you will have to look at one source of change only, because otherwise you have to look at all the partial derivatives at the same time... I'd suggest looking at Vanna : $\frac{\partial^2 \Delta}{(\partial S)^2}$.
– SRKX
Commented Nov 6, 2014 at 5:29

I am assuming you are talking about probability of becoming at least $\Delta = 30$, otherwise probability is zero. Hard to give a complete answer as quite some information is missing. As you are seeking for the probability, the outcome definitely depends on which model of underlying you are using. Moreover, even if you are using the BS model, some parameter values are important. For example, it is natural to conclude that for very high values of $\sigma$ such probability is relatively high, whereas for $\sigma \ll 1$ I would expect this probability to be almost zero.
Nevertheless, let me tell how would I approach the problem in the BS framework. We have $$\Delta = N(d), \quad d = \frac{\log(S/K) + (r+\frac12\sigma^2)t}{\sigma \sqrt t}$$ and hence we can reformulate the problem in terms of $d$: the current value is $d_0 = -1.28$ and we would like to reach the level of $d^* = -0.525$. Thus, you can formulate the problem as follows: you want $S_t$ to reach the value of $$S_t \geq K\cdot\exp\left(d^*\sigma\sqrt t - (r+\frac12\sigma^2)t\right) \tag{1}$$ at least once on the interval $[0,T]$. To find the probability of $(1)$ you need to solve the first hitting time problem:you know that $$S_t = S_0\cdot\exp\left((r-\frac12\sigma^2)t+\sigma W_t\right),$$ so in the end you get $$W_t\geq d^*\sqrt t + \frac1\sigma\left(\log\frac K{S_0}-rt\right).$$ In case of $r = 0$ this problem is a first hitting time of a square root curve by a Brownian motion, which is very likely to have been studied before, so there is a chance you get an analytical solution in that case.