Whilst we were, er, snoozing, our friends Andrew Smith and Oliver Bentley came out with a doozy of an article in The Actuary (of all places!): ‘Taking Shape,’ The Actuary, January/February 2022, pp. 31-33. As they explain:
Brownian motion has many uses in actuarial work, but stochastic models based upon it can be complex and difficult to replicate. By their nature, Monte Carlo scenarios start from a common point but end in random places. We show how to construct paths that are similar to Brownian motion, using a deterministic method that is simple and easily replicated, and which has end points that a user can choose. Applications include the assessment of Value at Risk for dynamically hedged portfolios.
The deterministic method they propose is their Brownian Blancmange fractal ‘random’ walk and we won’t try to explain that here. Read the paper: it is only 3 pages long.
Among other uses, it allows the user to simulate a fractal (i.e., non-random) series whose frequency and starting and ending points are set by the user. Moreover, if the position being simulated is, say, an option, the user can also set the implied or predicted volatility (think of the vol parameter fed into the option price or trading strategy at inception) and the realised volatility (the volatility of the synthetic option or delta hedge).
Obvious applications are to dynamic VaR, stress testing and hedging analyses. On the latter, one particular sentence jumps out at us: ‘When implied and realised volatilities coincide, the hedge should, in theory, work perfectly with sufficiently frequent trading.’
As it happens, we had found the same result in our recent work on option pricing, and it of central importance. A lot more later on that.
An Excel workbook illustrating all eight fractals and the VBA code necessary to generate them can be found at github.com/AndrewDSmith8/Fractals-and-Hedging.
Our congratulations to Andrew and Oliver on a fine piece of work.