The Effect of Options on Portfolio Risk - Part 2

The Effect of Options on Portfolio Risk - Part 2

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n our previous discussion on options within a portfolio, we introduced the concept of risk as the standard deviation of returns over some time period over a given set of scenarios.

To summarize, we considered three scenarios over a one month period. The three scenarios considered were 1.) stock appreciates by14% 2.) stock stays unchanged and 3.) stock appreciates by 14%.

In addition to a portfolio of just stock, we considered four options overlay strategies. Long and short positions in at-the-money calls and puts were also examined.

For our basic example, the following risk measures were calculated

Stock	Long Call	Long Put	Short Call	Short Put
40%	61%	23%	23%	61%

In two of these cases, the risk was lowered and in two it was increased. Next we illustrated for two other strategies similar results. The strategy called leverage bought 50% more stock on margin, and the strategy Divest, sold half of the stock position. These results follow:

Stock	Leverage	Divest
40%	60%	20%

From the perspective of risk, these portfolio changes can have the same affect as the option strategies discussed above.

The point of this article is to illustrate another way of calculating the risk measures, for the option strategies that does not rely upon the scenarios chosen. Ultimately, this new approach would be more accurate because it doesn't rely on the choice of scenarios, but it is also easier to calculate.

To do this, we introduce another concept of options theory called - Delta. The delta of an option is a number that needs to be calculated. It can usually be calculated by whatever means one uses to calculate the price of the option itself. This is because it is part of the option price calculation, so is usually easily obtained.

There are several definitions of delta that are really all different ways of saying the same thing. Delta refers to the sensitivity of an option to the price of the underlying stock. By convention, the delta of a long call is positive, because a call value appreciates as the stock price appreciates; and the delta of a put is negative, as the put price declines as the underlying stock price appreciates.

Let's discuss the delta of a call in a little more detail. When the stock price is at the strike price, the delta is .5. This means that for a one dollar increase in the stock price, the call will appreciate by half that or fifty cents. As the stock price appreciates, the delta will also increase. When the price gets high enough, the delta approaches 1 and goes no higher. This means that the option moves dollar for dollar with the stock. As the price declines below the strike, the delta decreases and eventually gets to zero.

Put deltas exhibit the same behavior, only negative. When a put is at-the-money, its delta is negative .5. As the price increases, the delta moves to zero, and as the price declines, the delta moves to negative one.

The graph below displays the delta of both a put and a call, as prices move.

Now we can go back to our original observations on risk. The effect of adding a long call to a portfolio was to move the risk from 40% to 63%. An increase of about 50%. This is simply because the delta of the call was .5. Similarly, selling a call lowered the risk by almost 50%. The same .5, which is the delta of the call, only this time there was a decrease in the risk, because the short call has a negative delta.

Similarly, long and short puts had comparable affects on risk, as their deltas would indicate.

Note that options positions have the same effect on risk as just levering or diluting a stock position. In point of fact, options are a way of controlling stock and realizing the economic benefit of their moves. The benefits are directly proportionate to the delta of the options.

What differentiates options from just leveraging stock is that the Deltas move as the stock moves. So when we buy a call that is at-the-money, the delta is initially .5 and so we are controlling one half the notional amount of the options contract. As the stock price increases, our exposure will naturally increase, as the delta increases. This is beneficial provided one is long the contract; if one is short they need to be aware that the options exposure, and hence one's short position, naturally becomes more short as prices rise. Conversely, if one is long a put, they are short stock, and the short position increases as prices decline. This is also beneficial, but unfortunate if one is short. This dynamic affect of options is one of their unique qualities.

As a final point, the risk calculations for portfolios with stocks and options must be updated regularly. While the risk of our stock position will always be calculated as 40%, the risk of options exposure is always changing, in a manner proportionate to its delta. This value is very dependant on the level of the stock (see our graph above) and so needs to be continually monitored.

Analyses such as this are very important in investment decisions. The goal of asset allocation is to target the risk of a portfolio and to select amongst assets in a way that achieves the desired level of risk. Options can play an important role within a portfolio context, by allowing investors different risk-return tradeoffs than offered by conventional assets, but investors might be wise to incorporate their risks within the overall asset allocation framework.