Constructing and optimizing volatility position-sizing profiles

Position size as a schedule

Volatility position sizing sets exposure from account equity, volatility, stop distance, and a scheduled control so a loss or exposure decision stays bounded before entry and while the position is open.

A vps-profile is constructed by allowing the control U to change only at K discrete values of S, with each change stored as an independent value Xj. Those stored values are the change-vector.

One search over the change-vector

System optimization of that profile searches for the X1 through XK values that maximize the associated return-risk-ratio F. The search is one procedure that adjusts the size-schedule inputs under stated constraints.

Each iterate forms the next change-vector by adding a maximizing-parameter A(n) times a search-vector to the current X values. A(n) is chosen to maximize F along that direction. Different numerical methods correspond to different rules for choosing the search-vectors. The method used to optimize these volatility position-sizing profiles is BFGS.

Resampling the finished list

A Monte Carlo trial randomly selects trades from a fixed profit-and-loss list and keeps a running total until either a stated profit goal or a stated dollar-loss limit is reached.

A trial that hits the profit goal first is counted as a success, and a trial that hits the specified dollar-loss limit first is counted as a failure. The estimated probability of hitting the loss limit before the profit goal equals the number of failed trials divided by the total number of trials.

Draws come from an ordered list. With a list of 100 trades, a uniform draw of 0.463 selects the 47th trade and a uniform draw of 0.937 selects the 94th trade.

That probability estimate is subject to sampling-error that declines as the number of trials grows, and satisfactory accuracy may require as many as 30000 trials.