Real EV: Discrete Random Variable
To briefly summarize what we’ve learned about random variables in Understanding Alpha and Expected Value, Real EV involves a more nuanced approach by considering the entire range of potential outcomes rather than simplifying them to maximum profit/loss, and/or a single point between those two extremes.
Profit/Loss diagram of a defined-risk options trade with partial profit region
This method treats each option strategy as a discrete random variable and calculates the EV based on the actual probability density function (PDF) or cumulative density function (CDF) of the underlying asset. This could be a non-parametric distribution obtained through historical data or one modeled using methods such as Black-Scholes, Binomial, or other exotic option models. For our purposes, we're using Black-Scholes probability to combine the EVs of the max loss, the max profit, and every $0.01 increment inside of the partial profit/loss region.
The expected value E[X] for a discrete random variable X taking on n different values x1,x2,…,xn with corresponding probabilities p1,p2,…,pn is calculated for a defined-risk options trade as:
E[X]=P(ML)∗ML+P(MP)∗MP+∑i=1npi×xi
Here, P(x) is the probability of event x occurring. For any payout that is not maximum profit or loss, each xi would be a possible value for the strategy payout at expiration, and pi would be the probability of that payout. Real EV considers all possible values and payouts at every possible price the underlying may expire at.
The merit of Real EV is its applicability to more complicated situations where multiple variables influence the outcome. Consequently, the method is more computationally intensive and requires a deeper understanding of stochastic calculus and probability theory.
To briefly summarize what we’ve learned about random variables in Understanding Alpha and Expected Value, Real EV involves a more nuanced approach by considering the entire range of potential outcomes rather than simplifying them to maximum profit/loss, and/or a single point between those two extremes.
Profit/Loss diagram of a defined-risk options trade with partial profit region
This method treats each option strategy as a discrete random variable and calculates the EV based on the actual probability density function (PDF) or cumulative density function (CDF) of the underlying asset. This could be a non-parametric distribution obtained through historical data or one modeled using methods such as Black-Scholes, Binomial, or other exotic option models. For our purposes, we're using Black-Scholes probability to combine the EVs of the max loss, the max profit, and every $0.01 increment inside of the partial profit/loss region.
The expected value E[X] for a discrete random variable X taking on n different values x1,x2,…,xn with corresponding probabilities p1,p2,…,pn is calculated for a defined-risk options trade as:
E[X]=P(ML)∗ML+P(MP)∗MP+∑i=1npi×xi
Here, P(x) is the probability of event x occurring. For any payout that is not maximum profit or loss, each xi would be a possible value for the strategy payout at expiration, and pi would be the probability of that payout. Real EV considers all possible values and payouts at every possible price the underlying may expire at.
The merit of Real EV is its applicability to more complicated situations where multiple variables influence the outcome. Consequently, the method is more computationally intensive and requires a deeper understanding of stochastic calculus and probability theory.
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