Optimal P&L Options Rebalancing Strategies — 1

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What is the best rebalancing strategy for an options portfolio? Options models based on risk-neutral valuation (virtually all options models) implicitly assume a continuous delta neutral rebalance (and completely ignore other risks such as Vega etc). In reality this type of rebalancing is impossible, and even undesirable. Options market makers have rules of thumb, but how do we know which rule of thumb is most likely to provide the best holding period (or at least fiscal period) risk-adjusted returns. Indeed, this question extends to many more complex issues such as the alteration of strategy mid-stream to adjust for dramatically different/new market events.

An approach to choosing optimal rebalance strategies

One approach is to forward and backward test the strategies and market conditions. The forward testing is accomplished by running the position for an assumed evolution of the forward markets. Importantly, this must including all rebalances, costs, and all manner of effects that a trader would experience on a real world trading floor (see Figure 1). The holding period P&L calculations are repeated many times for the forward price possibilities, and for each trading strategy. PaR is one implementation of this type of analysis, and all of TG2RM1st — Chapter 12 is dedicated to the introduction of PaR analysis.

Each trading strategy’s P&L history is stored (see Figure 2). From the trading strategy P&L distribution it is possible to extract average holding period return/P&L and typical P&L risk. For example, the strategy in Figure 2 b) has a slightly higher average P&L on expiration than the strategy in Figure 2 a), but the distribution of that P&L is also wider, implying greater P&L risk, more on this later.

Figure 1 a) and b) two possible evolutions of forward prices: a smooth upward drifting forward market (left), and an initially upward drifting forward market that encounters a large sell-off accompanied by a large increase in volatility, but thereafter resuming a more settled state

Figure 2 a) and b) Holding Period P&L Distribution for two different rebalancing strategies (e.g. one might be delta only, while the other might be delta/gamma).

The backward testing is very similar except that now the forward prices are provided from a database of market histories (so the default condition is that valuation is taken explicitly from, say settlement prices, rather than calculated). The rebalances are performed as per the user required rebalance model (in this example, market convention implementation of Black-Scholes). Figure 3 a) plots the net P&Ls (the dots) for approximately 5000 different complete holding periods (each with full rebalancing etc) and has plotted these against two particular market factors (disguised here as Factor A and Factor B, but the plotting could be against any user desired factor such as implied vols, moving averages, or more sophisticated market measures). From these plots it also possible to extract average holding period P&L and P&L risk, as before.

It is noteworthy that it also possible to use these results to look for cheap/dear conditions as illustrated by Figure 3 b). This plot shows that whenever Factor A is approximately in the middle, and Factor B is the range of 130-220, the position is losing money, and so it should have dealt the other way around (i.e. if it was originally bought, then it should have been sold and vice versa).

Indeed many other interesting results can be extracted from such analysis.

Figure 3 a) and b) Net holding period P&Ls for a particular strategy traded against a historical database of mark to market prices (on the left, each point represents the net P&L of an entire holding period), while on the right a surface has been fit through the P&L’s, and shows that the market factors

However, the matter at hand is the selection of optimal rebalance strategy. One approach, now is to use the average holding period P&L’s and the holding P&L risk measures along the lines of an efficient frontier approach. In the normal context, this would mean plotting the different investment returns against the investment risk. Here, we can accomplish something similar by considering each trading strategy as a different investment process, and plotting the equivalent values as determined from the forward and backward tests.

In its simplest form, such a plot would be like the middle plot on the right hand side of Figure 4. This shows the expected P&L for each strategy/market scenario combination (different colours), and now it is simply a matter of choosing the strategy with the highest P&L (return) for a given level of risk. Since each trading operation will have its own risk threshold, the optimal strategy is subjective (as it should be). In this plot, each strategy/market combination has been recalculated many times for to fine tune rebalancing frequency, and so each strategy is a line with multiple point for each variant, as opposed to a singe point.

These results indicate that the red strategy, regardless of rebalance adjustment is always worse than the other methods, and so can be immediately discarded. However, the light blue, dark blue, green strategies cross each other at different risk thresholds, and generally one of these is dominate at different risk thresholds, hence the best risk adjusted strategy (for that level of risk preference)

Figure 4). Efficient Frontier for the selection of best holding period strategy on a risk adjusted basis, and including additional dimensions to account for extra market and operational factors.

But this is not the whole story if you trade in the real world? In the real world there are many other factors that impact selection of best strategy. One example is that P&Ls are hardly ever symmetric/Gaussian (as implicitly assumed by tradition Efficient Frontier analysis). Indeed, it is our job as traders to ensure that the P&L is not symmetric (i.e. eliminate drawdown while keeping upside). Another example is that of operational factors. For example rebalancing with cash bonds has balance sheet and credit implications and thus costs, and while using, say, futures would eliminate reduce such, then we are exposed to short squeeze on roles etc. So Figure 4 is shown in 3-D to account for (in this case just one more of these other factors), which then contributes to the best strategy selection process.

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