## Sunday, August 16, 2015

### Systematic Strategies: HIgh Probability Trading Under Uncertainities

Stop-loss and take profit orders are powerful tools to manage an underlying trading view. And there are some misconceptions about them. Many people believe take-profit and stop-loss limits define the risk-reward ratio of a strategy. Far from it.

If a strategy has 100 on the upside as take-profit and 50 on the downside as stop-loss it is preposterous to estimate the risk-reward as 2:1. The most important input missing here are the probabilities of hitting the take-profit and stop-loss limits. For example if the probability of hitting the take profit in the trading horizon is 20% and hitting the stop-loss is 50% (i.e. the underlying can be within the range with a probability of 100% - 20% - 50% = 30%), then expected upside is 20%*100 = 20 and downside is 25. And the real risk-reward is 20:25 i.e. 4:5, far from the 2:1 before.

We can use this powerful tools and this concept of probabilities to devise profitable strategies under uncertainties. For example, suppose the underlying view is long an asset. Let's assume the asset follows a (geometric) Brownian motion (in real world) as below

dS/S = drift*dt + vol*dW

Where dt is differential of time, and dW is the standard Brownian Motion. So according to our view, here the drift is a positive value (the underlying view is long). One way to position for this is to go long and unwind at the end of the trading time horizon (let's say 1 day). The probability of our position being in profit can simply be calculated from a Black-Scholes like digital call option price.

However, another way is to put a take-profit order. This order gets executed whenever the underlying breaches the target from below. The probability of this execution is the probability of the underlying Brownian motion breaching the barrier any time up to our trading horizon. There are standard approximate solutions to this problem. For example, see here (opens PDF). As an example I have plotted the ratio of this probability to the original Digital probability (i.e. being in the money at the end of trading horizon, irrespective of what happens in between) for a range of volatility and Sharpe ratios (ratio of the drift term above to volatility) in the chart below.
As you can see, the probabilities of hitting our profit target intraday is much higher than being in the money end of the day. Additionally, higher the uncertainties about the underlying trend (lower the Sharpe) this ratio works better in out favor. This shows clearly that if your view is not with high conviction, it is better to use a take-profit target than a buy-and-hold approach.

This gives rise to an interesting way to implement a short-dated view using options. The strategy is to buy an option (say a ATM call option to implement a long view on the underlying) and put a close take-profit target, and a wide stop-loss. The worst-case loss is the option premium. And we will hit out target with 1.2x to 2x more frequently, depending on the strength of the trend. By design it may appear we have a skewed risk to reward ratio here. But given the concept of probabilities (than just the width of the stops and targets) and the convexity of a long option position this is much less risky than it appears. None-the-less it is a skewed strategy, with high probabilities of hitting our profit targets regularly, and occasional large losses.
The chart above shows the expected profit ranges for such a strategy on the NIFTY index (the flagship index of National Stock Exchange, India). The Spot level assumed to be 8500. The profit-target is 1x the daily vol move and the stops are 2x of that. (Note this is approximated, i.e. I ignored the order of stop-loss and target hitting, which is valid for large enough stops). As you can see if you are somewhat certain about the direction of the underlying trend, this is quite profitable (the break-even here is daily trend is 40% or more of the daily vol) under high vol. The profit distribution as below (click to enlarge)
This ignores two realities -1) the convexity of the option position, which goes in our favor and 2) the jumps in stock moves (price moves are rarely Brownian), which goes against us (the probabilities should get affected symmetrically, but the size of loss makes it asymmetric).

Those interested in the underlying codes (in R) can find it here.