Estimating the Chance an Option Delta Reaches a Target
Summary
The discussion outlines how to estimate the probability that an option’s delta reaches a target at least once during a specified period. The probability depends on the underlying price model and parameters; under Black–Scholes, volatility and other inputs matter. The response assumes the intended event is reaching or exceeding the target delta.
For a call option, delta is expressed as the standard normal cumulative probability of d, so the target delta can be translated into a target level for d and then into a boundary for the underlying price. With geometric Brownian motion for the underlying, the resulting probability is a first hitting time problem: the price must cross that boundary at some point in the interval. In the zero interest rate case, the transformed problem becomes the crossing of a square root curve by Brownian motion. The response does not provide a general numerical answer or a closed-form solution for the stated example; further model inputs and a method for solving the hitting problem are needed.
Key ideas
- The probability of a delta reaching a target depends on the underlying price model and its parameters.
- Under Black–Scholes, option delta can be translated into a target level for the model variable d.
- Reaching a target delta during an interval is a first hitting time problem, rather than a terminal probability calculation.
- With geometric Brownian motion, the target can be expressed as a boundary for the underlying price.
- The zero interest rate case transforms into a Brownian motion boundary crossing problem, but the discussion gives no general numerical solution.
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Full text
# What is the probability distribution of the changes in $\Delta$?
# 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.
## Answer by Ulysses (score 2)
https://quant.stackexchange.com/a/15341
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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