** 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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