we have a algorithm that pumps out trade recommendations. I am trying to infer probabilities per below.
Assume we have set a particular order where it is trying to sell shares at a favourable price, placed with
1) X % (relative price points away from mid-price)
2) given algorithm signal is positive = probability, P at any time-slice (i.e. if signal is positive, keep the order at X% away placed). When the signal = negative, start countdown of order-expiry time.
3) Expected order-expiry time = Texp
4) Share price volatility (relative volatility per 1 sec) at sigma
5) mean = drift of stock
What is the Probability the order-placed (eventually) gets the algorithm signal = negative AND order does not get hit (touched by market) within the order-expiry time Texp.
I thought through it, and it seems to me that
Probability(reach signal = negative at time Ti; and order does not get hit until order-expiry time Texp)
= Summation [(1- P) * P^(Ti)] * P[ (X - mean)/sigma*sqrt(Ti)] * P[ (X - mean)/(sigma * sqrt(Texp)]
= where at time Ti (i units of time has passed),
(Probability signal = negative only at time Ti) * (Probability stockprice doesn't hit X within time Ti) * (Probability stockprice doesn't hit X within time Texp)
My brain is swimming.... any thoughts?