Future and Options affect both the risk and the ultimate return of the distribution of assets in an equity portfolio. The systematic and unsystematic risk of equity portfolios is definitively modified by using futures and options, and they obviously can change the level of cash inflow and outflows from a portfolio as differing hedging strategies are adopted. Typically, a dollar-for-dollar correlation exists between the changes in the underlying value of a security and the ultimate price of a relevant futures contract. This relationship effectively implies that being long in futures is the same as reducing the amount of cash from a portfolio. If we reverse this logic, then being short in a future contracts is the same as adding cash to the portfolio being managed. Evidently, a portfolio manager who has taken long futures positions will have done this in order to increase the exposure the portfolio to the underlying asset, while conversely, if a portfolio manager has taken short future positions then they are decreasing the portfolios exposure.

Therefore, a long position in a futures contract has the effect on the portfolios underlying assets by increasing the portfolios exposure to any resulting price changes of the asset. Shorting futures pretty much has the reverse effect in that it will decrease the exposure of the portfolio to any underlying assets. Of course, it really depends what the portfolio managers strategy is and what they are trying to achieve when managing the investment.

If we take a look at option contracts – they give the owner the right but not the obligation to buy or sell the underlying asset at a particular time in the future. Since options always provide this option whether or not the option holder decides to ultimately exercise the option, it infers that they do not have a proportioned impact on the returns of the portfolio – that is, they are not overtly exposed to the changes in futures prices and the underlying assets as we saw above. To illustrate, its pretty easy to see that buying a call option limits losses as you are buying the call with the hope that the stock price will rise, on the flip size, if you buy a put then when you also own the underlying security then you are effecting trying to mitigate any downside risk in the portfolio.

All in all there are some pretty clear differences and if you have some more questions in relation to this – then perhaps I will post some information on passive and active portfolio management. Either way drop a comment if you want more information.

This means that the lower bound is equal to the current stock price minus the options strike price multiplied by the natural e, to the power of negative risk-free interest rate multiplied by the options time to expiry.

Example:

Say that the current stock price = $20 (So)
And that the strike price = $18 (K)
The risk-free interest rate = 10% (r)

And the time to time to expiry = 1 year (T)

If you have done the calculations correctly then according to this rule the lower bound for this call option on a non-dividend paying stock is $3.71.

Let’s say that the market is quoting the European call option at $3.00. Such a price is less than the lower bound or “theoretical minimum”. What would happen is that an arbitrageur would short the stock (i.e. sell some of the shares underwritten this stock to buy them back at a future date), and then buy the call option.

This will provided the arbitrageur with a cash inflow of $20.00 – $3.00 = $17.00.

If this amount is then invested for 1 year at the market risk-free interest rate of 10% per annum, then the $17.00 will grow to 17е0.1 = $18.79. Therefore at the end of the year, when the option will expire if the stock price is greater than $18.00 (the strike price), then the arbitrageur will be able to exercise the option for $18.00. By doing this, it will enable them to close out the short position to make a profit of $18.79 – $18.00 = $0.79.

This means that the arbitrageur will buy back the stock at a cheaper rate than what he/she originally shorted (sold stock he/she didn’t own) and then invested that money at the risk-free rate, and then finally met the shorting obligations by buying the stock back with the aid of the option at the strike price.

What happens if the stock price is less than the strike price after 1 year (at time of option expiry?)

Then the arbitrageur will make an even GREATER profit. This is because he/she will be able tot buy back the stock at the market price which will be even cheaper than that of the strike price to close out the shorted stock position.

Say for example the stock price after 1 year = $17.00

The arbitrageur’s profit will then = $18.79 – $17.00 = $1.79

Which is $1 more than the previous example where the stock is greater than the strike price.

The additional difference in profit is the difference between the strike price and the stock price.

In a more formal approach this investment is demonstrated under 2 portfolios

• Portfolio A: 1 European call option plus an amount of cash equal to Ke-rT
• Portfolio B: 1 Share

Under portfolio A, the cash will be invested at the risk-free interest rate, and will then grow to the K in time T. If ST > K, then the call option will be exercised at the maturity of the option and as a consequence portfolio A will be worth ST.

However if on the other hand ST < K, then the call option will not be exercised at the maturity of the option, but instead the option will become worthless. The portfolio as a consequence will be worth K.

Therefore at time T, the portfolio A is worth:

Max (ST, K)

Under portfolio B on the other hand, it will be worth ST at the time T. Therefore because of this, portfolio A is always worth as much as, and can be worth more than, portfolio B at the time the option matures (expires). Hence, the worse thing that can happen to a call option is that it expires worthless, the value cannot be negative.

This means that the value of any American or European call option must be less than or equal to that of the current stock price. If the above boundaries were not as they appear, then any arbitrageur would be able to easily make a riskless profit simply by purchasing the stock and then immediately selling the call to option.

The upper bound for the purposes of both American and European put options is a little different. As for a put option, regardless of the fact that it is an American or a European call option, the option will give the holder the right to sell 1 share of a stock for a certain price (the strike price). However, regardless of how much the stock price falls, the price of the option can never exceed that of the strike price (as this is the price at which you will be able to sell the underlying stock for). Since it is the stock or share price which the option basis its own price on. The right can never be worth the strike price at which you will be selling the underlying stock or share.

Therefore the upper bound for put option prices is the options strike price. The means that the value of any American or European put option must be less than or equal to that of the options strike price.

Also, it must be noted that for European options at maturity can not be worth more than the strike price. This is because of the fact that the option cannot be worth more than the present value of the strike price today.

This means that the value of any American or European put option must be less than or equal to that of the options strike price multiplied by the natural e, to the power of negative risk-free interest rate multiplied by the options time to expiry.

If the above boundaries were not as they appear, then any arbitrageur would be able to easily make a riskless profit simply by writing the option and investing the proceeds of the sale at the risk-free interest rate.

Full assumptions about the calculation of option prices

Some of the assumptions which must be made include:

• There are market participants such as large investment banks
• There are no transactions costs
• That all trading profits (the net of trading losses) will be subject to the same tax rate (meaning there are no marginal tax rates). (This will mean that every investor’s gains or losses will have the same effect regardless if the figure is $1 or $1million).
• That borrowing as well as lending is both possible at the risk-free interest rate.
• That all market participants will be prepared to take advantage of arbitrage opportunities as and when they arise. (Meaning that such opportunities will disappear quickly as the market efficiently corrects itself).
• Therefore there are no arbitrage opportunities.
• The interest rate is the nominal rate of interest and not the real rate of interest.
• Also that such a rate must be greater than 0 (i.e. a rate exists). Otherwise if this rate was 0 then there would be no advantage that the risk-free investment would provide over cash.

Typical Options notation:

• S0: Current stock price
• K: The strike price of the option
• T: The time until the option will expire
• ST: Stock price at maturity
• r: The continuously compounded risk-free rate of interest for an investment which will mature in time T
• C: The value of an American call option to purchase 1 share
• P: The value of an American call option to sell 1 share
• c: The value of an European call option to purchase 1 share
• p: The value of an European call option to sell 1 share

Properties Of Stock Options

There are a number of different factors which affect the price of a stock option.

The 6 main factors include:

• The current underlying stock price (So)
• The strike (exercise) price (K)
• The time until the option expires (T)
• The volatility of the stock price (σ)
• The risk-free interest rate (r)
• The expected dividends during the life of the option (if any).

The Stock Price (So) &Strike Price (K)

The first 2 factors which affect stock options are arguably the 2 most important ones. They have already been discussed earlier in the options basics and are the fundamental bearings on all options. As an option is written on an underlying asset which in this case is a stock, the price and the possibility for exercise depend heavily on the these 2 factors. In the case of a call option the higher the price of the stock price against the strike price the more valuable the option will become. In addition to this, if the option has a lower strike price, then the option will also become more valuable. Whereas for a put option the lower the price of the stock price against the strike price the more valuable the option will become. However, in this case the, for the put option to become more valuable the higher the strike price the better for the option holder. Therefore, put options will behave in the opposite way to which call options behave.

The Time To Expiration (T)

The third key component of options is the time or life span of the option until it expires. All options have a limited lifespan, unlike the underlying stock by which it is written on. When this date arrives the holder of the option has the right to exercise the option (if it is in the money), if he or she does not elect to do this, or it is pointless to exercise, then the option will be terminated and cease to exist any longer. For the case of American options, both call and put options will become more valuable the greater the time to expiry. This is because there is a larger chance that the option will become in the money, and therefore have a larger chance of making its holder a profit. It is because of this reason that an option with a greater lifespan or time to expiry will always be worth more than one which has a shorter time to expiry.

Take the following example.

2 call options written on Google shares.

Both with the same strike price of $100.

However Option A expires in 6 months, where as Option B will expire in 1 year.

Now even though the options have exactly the same strike price, because of the extra time which Option B has (6 months extra) compared with Option A, it will be worth more. The reason for this price difference is due to possibility. Option B has a far greater possibility of being in the money, or even further in the money than that of Option A. It is because of this possibility that gives the option holder B the greater likelihood and therefore the probability of a more favorable outcome.

This probability or possibility of a favorable outcome is also time value. Time value is how much investor’s value time, or how much time remains in the life of the option. Ordinary shares do not have a time value as they have an infinite lifespan (as long as the company exists). However because of the fact that options have limited life spans, a large proportion of their value will be contributed to the time value of the option. As an option approaches its expiry date, the less time it has and therefore the less time value it will hold.

Therefore an option with a longer time frame should generally at least as much as the one with the shorter life.

Will the option with the longest life be always worth more? Generally speaking, with all things being equal this should be the case. However there are some situations where this statement will not hold.

The following example highlights such a case:

• 2 European Call options
• Option A expires in 1 month
• Option B expires in 2 month
• Both have exactly the same strike price
• Both are written on the same stock
• There is a very large and significant dividend to be paid in 6 weeks

What will happen?

Because there is a large dividend, this will cause the stock price to decline. Because of this declined the option with the shorter-life could therefore theoretically be worth more than that of the longer life option with 2 months to expiry.

Volatility

Volatility is an extremely important feature with regards to all options. Why you might ask? Doesn’t volatility infer that there is uncertainty and with that lower prices? Yes this is correct. This is the view which all investors will hold for stocks and shares. HOWEVER, with regards to options, the picture is very different.

Volatility is defined as a measure the amount of uncertainty which exists when speaking about the future price of something. Volatility for most investors is not a good thing. When seeking to invest in any asset or project, investors look for things which offer certainty and security. If a stock or share has a high level of volatility this would mean a high level of uncertainty. Therefore such a stock or share may perform very well, or may also perform very badly. High levels of volatility will reflect badly with respect to such assets.

However, because options have a limited life, high levels of volatility becomes a good feature. As options are short term assets which expire in a limited time frame, the investment horizon is very narrow. The holder of an option only has a limited time to generate a positive return; otherwise the option will expire and become worthless. Because of this limited time frame, the greater the chance of any movement the better it will be for the option holder.

Why is this the case? Doesn’t high volatility mean that the asset will perform badly? No, not necessarily.

Because the investor has this limited horizon to hopefully get a favorable price for the option they have purchased, they will want the option to have a high rate of volatility. This is because it will give the option holder a larger possibility, or greater chance that the option will become “in the money”. Therefore the option with the higher volatility will be worth much more than another option with very little volatility.

But what about the equally likely chance of the stock price moving in the opposite direction to what you want it to as a result of this volatility? Wouldn’t this be very bad for me as an investor? Yes, in such a case it would not be good. However, you must remember that all option holders gain the benefit of limited liability. When you purchase an option the most you can loose from the transaction is however much you have paid for that option. Therefore even if the stock price crashes significantly as a result of this high level of volatility and you are holding a call option in the stock, the most you will loose is the amount you have paid for it. Unlike the stock holders who will bear the full face of such a fall.

Risk-Free Interest Rate (r)

The risk free interest rate is also another important factor in determining the price of stock options. The risk-free interest rate means exactly what it is called. It is the rate of interest, (or return) than an investor will be able to receive, without having to bear any amount of risk. Generally speaking the risk-free interest rate is taken from the current market rate of return offered by government issued bonds. Because of their minimal level of default, such investments are deemed to be “risk-free”.

The way in which the risk-free interest rate affects the price of stock options is in a much less clear cut way. This is because the interest rate in the economy increases, the expected return required by investors from the stock will also tend to increase. In addition to this the present value of any future cash flow received by the holder of the option will decrease. This is because the market is now offering a greater interest rate to what it previously did and as such the previous option will be delivering a return below such a rate. The combined impact of these 2 effects will produce the effect of increasing the value of call options and decreasing the value of put options.

What we must recognize in this analysis however is the fact that the assumption is that while the interest rates will change, all the other variables will be held constant. Where particular emphasis must be placed on the stock or share price itself remaining constant. This is because of the fact that when interest rates rise, the price of shares will generally tend to fall, or if interest rates fall, the price of shares will rise. If the stock price can remain constant then the effect of the change in interest rates can be properly examined. Otherwise it will simply produce a net effect where 1 will cancel out the other. The net effect of an interest rate increase and the accompanying stock price decrease can be to decrease the value of a call option and increase the value of a put option. Similarly, the net effect of an interest rate decrease and the accompanying stock price increase can be to increase the value of a call option and decrease the value of a put option.

Dividends

Dividends also have a significant effect on the pricing of stock options. This is because the price of the stock or share which is issuing a dividend will fall on the ex-dividend date (to compensate for the dividend being paid). This will mean that put option holders will benefit, because the value of the stock price has decreased, where as the call option holders will suffer, because call options rely on the stock price increasing.

Therefore it can be concluded that call options are negatively correlated to the size of any anticipated dividends and the value of a put options are positively correlated to the size of any anticipated dividends.