2.1.1.1 Definition of a Stock Index

The price of a stock index is a weighted sum of stock prices. The value of a stock index at time is calculated by formula

2.1

where is a constant, is the number of stocks in the index, is the number of shares of stock , and is a suitably adjusted price of stock at time , where . Note that is the market capitalization of stock . The definition of a stock index involves three parameters: constant , numbers , and values :

1. 1. The constant can be chosen, for example, to make the value of the index equal to 100 at a given past day. When the constitution of the index is changed, then the constant is changed, to keep the index equal to 100 at the chosen day.
2. 2. The numbers can equal the total number of shares of stock , but they can also be equal to the number of freely floating stocks. Float market capitalization excludes stocks which are not freely floating (cannot be bought in the open market).
3. 3. The values are calculated differently depending on whether the index is a price return index or a total return index. Price return indexes are calculated without regard to cash dividends but total return indexes are calculated by reinvesting cash dividends. The adjusted closing price of a stock is the closing price of a stock which is adjusted to cash dividends, stock dividends, stock splits, and also to more complex corporate actions, such as rights offerings. The calculation of the adjusted closing price is often made by data providers.

2.1.1.2 Uses of Stock Indexes

Stock indexes can be used to summarize information about stock markets. Stock indexes can also be used as a proxy for the market index when testing and applying finance theories. The market index is the stock index which sums the values of all companies worldwide. Stock indexes are traded in futures markets and in exchanges as exchange traded funds (ETF). Furthermore, investment banks provide financial instruments whose values depend on stock indexes.

2.1.1.3 Examples of Stock Indexes

2.1.2.1 Initial Price Data

During the opening hours of an exchange the stocks are changing hands at irregular time points. The stock exchange receives bid prices with volumes (numbers of stocks one is willing to buy with the given bid price) from buyers, and ask prices with volumes from the sellers. The exchange has an algorithm which allocates the stocks from the sellers to the buyers. The allocation happens when there are bid prices and ask prices that meet each other (ask prices that are smaller or equal to bid prices). The algorithms of stock allocation take into account the arrival times of the orders, the volumes of the orders, and the types of the orders.

The most common order types are the market order and the limit order. A market order expresses the intention to buy the stock at the lowest ask price, or the intention to sell the stock at the highest bid price. A limit order expresses the intention to buy the stock at the lowest ask price, under the condition that the ask price is lower than the given limit price, or the intention to sell the stock at the highest bid price, under the condition that the bid price is higher than the given limit price.

2.1.2.2 Sampling of Prices

The price changes at irregular time intervals in a stock exchange, but for the purpose of a statistical analysis we typically sample price at equispaced intervals.

To obtain a time series of daily prices, we can pick the closing price of each trading day. The closing price can be considered as the consensus reached between the sellers and the buyers about the fair price, taking into account all information gathered during the day. An alternative method would choose the opening price.

However, depending on the purpose of the analysis, we can sample data once in a second, once in 10 days, or once in a month, for example. Note that when the sampling interval is longer (monthly, quarterly, or yearly), the number of observations in a return time series will be smaller, and thus the statistical conclusions may be more vague. Note also, that the distribution of the returns may vary depending on the sampling frequency.

It is not obvious how to define equispaced sampling, since we can measure the time as the physical time, trading time, or effective trading time:

1. 1. The physical time is the usual time in calendar days. Assume that we want to sample data once in 20 days. If we use the physical time, then we calculate all calendar days.
2. 2. The trading time or market time takes into account only the time when markets are open. For example, when we want to sample data once in 20 days and we use trading time, then we calculate only the trading days (not all calendar days). However, information is accumulating also during the weekends (and during the night), which would be an argument in favor of physical time.
3. 3. The effective trading time takes into account that the market activity is not uniform during market hours. To define the sampling interval, we could take into account the number of transactions, or the volume of the transactions. The effective trading time is interesting especially when we gather intraday data, but it can be used also in the case of longer sampling intervals, to correct for diminishing market activity during summer or at the end of year.²

Sampling daily closing prices can be interpreted as using the trading time, because weekends and holidays are ignored in the daily sampling. Since there is roughly the same number of trading days in every week and every month, we can interpret sampling the weekly and monthly closing prices both as using the physical time and using the trading time. Discussion about scales in finance is provided by Mantegna and Stanley (2000).

2.1.2.3 Stock Returns

Let us consider a time series of stock prices, sampled at equispaced time points. We can calculate gross returns, net returns, or logarithmic returns.

1. 1. Gross returns (price relatives) are defined by
2. 
3. 2. net returns (relative price differences) are defined by
   
4. 3. logarithmic returns (continuously compounded returns) are defined by
   

where .

Gross returns are positive numbers like 1.02 (when the stock rose 2%) or 0.98 (when the stock fell 2%). Value zero for a gross return means bankruptcy. The gross returns have a concrete interpretation: starting with wealth and buying a stock with price leads to the wealth .

Net returns are obtained from gross returns by subtracting one, and thus net returns are numbers larger than . Net returns are numbers like 0.02 (when the stock rose 2%) or (when the stock fell 2%). Value for a net return means bankruptcy.

Logarithmic returns are obtained from gross returns by taking the logarithm.³ A logarithmic return can take any real value, but typically logarithmic returns are close to net returns, because when . Value for a logarithmic return means bankruptcy. The logarithmic function is an example of a utility function, as discussed in Section 9.2.2. We will consider taking the logarithm as an application of a utility function, and apply mainly gross returns. However, there are some reasons for the use of logarithmic returns. First, we can derive approximate distributions for the stock price by applying limit theorems for the sum of the logarithmic returns, which makes the study of logarithmic returns interesting. Indeed, we can write

2.2

See (3.49) for a more detailed derivation of the log-normal model for stock prices. Second, taking logarithms of returns transforms the original time series of prices to a stationary time series, as explained in the connection of Figure 5.1.

For a statistical modeling we need typically a stationary time series. Stationarity is defined in Section 5.1. For example, autoregressive moving average processes (ARMA) and generalized autoregressive conditional heteroskedasticity (GARCH) models, defined in Section 5.3, are stationary time series models. The original time series of stock prices is not a stationary time series, but it can be argued that a return time series is close to stationarity.⁴

Note that we can write, analogously to (2.2),

Thus, we can derive approximate distributions for the stock price by applying limit theorems for the sum of the price differences. See (3.46) for a more detailed derivation of the normal model for stock prices. The time series of price differences is not a stationary time series, as discussed in the connection of Figure 5.2. However, for short time periods a time series of price differences can be approximately stationary. Thus, modeling price differences instead of returns can be reasonable.

2.2.2.1 Definitions of Interest Rates

The different definitions of interest rate are discussed in detail in Chapter 18. As an example we can consider a loan where the interest is paid at the end of a given period, and the interest is quoted in annual rate. Rate conventions determine how the quoted annual rate relates to the actual payment. Maybe the most common convention is to pay , where is the principal, is the annual rate, and is the number of calendar days of the deposit or loan. Note that loan rates are either rates that apply to a loan starting now until a given expiry, or forward rates, that are rates applying to a loan starting in the future for a given period of time.

Rates are quoted in percents but they are compared in basis points, where a basis point is , that is, is 100 basis points.

2.2.2.2 The Risk Free Rate

The risk free rate is different depending on the investment horizon. For one day horizon the risk free rate could be the Eonia rate or the rate of a bank account, and for 1 month horizon the risk free rate could be the rate of 1 month government bond.

2.3.1.1 Forwards

A forward is a contract written at time , with a commitment to accept delivery of (or to deliver) the specified number of units of the underlying asset at a future date , at forward price , which is determined at .

At time nothing changes hands, all exchanges will take place at time . A long position is a commitment to accept the delivery at time . A short position is a commitment to deliver the contracted amount. The current price of the underlying is called the spot price.

2.3.1.2 Futures

A futures contract can be considered as a special case of a forward contract. An instrument is called a futures contract if the trading is done in a futures exchange, where the forward commitment is made through a homogenized contract so that the size of the underlying asset, the quality of the underlying asset, and the expiration date are preset. In addition, futures exchanges require a daily mark-to-market of the positions.

A futures exchange acts as an intermediary between the participants of a futures contract. The existence of the intermediary minimizes the risk of the default of the participants of the contract. When a participant enters a futures contract the exchange requires to put up an initial amount of liquid assets into the margin account. Marking to market means that the daily futures price is settled daily so that the exchange will draw money out of one party's margin account and put it into the others so that the daily loss or profit is taken into account. If the margin account goes below a certain value, then a margin call is made and the account owner must add money to the margin account. In contrast to futures contracts, forward contracts may not require any marking to market until the expiration day.

A futures contract can be settled with cash or with the delivery of the underlying. For example, if the underlying of the futures contract is a stock index, then the futures contract is usually settled with cash. A futures contract can be closed before the expiration day by entering the opposite direction futures contract.

On the delivery date, the amount exchanged is not the specified price on the contract but the spot value (i.e., the original value agreed upon, since any gain or loss has already been previously settled by marking to market).

The situation where the price of a commodity for future delivery is higher than the spot price, or where a far future delivery price is higher than a nearer future delivery, is known as contango. The reverse, where the price of a commodity for future delivery is lower than the spot price, or where a far future delivery price is lower than a nearer future delivery, is known as backwardation.

2.3.2.1 Calls and Puts

The buyer of a call option receives the right to buy the underlying instrument and the buyer of a put option receives the right to sell the underlying instrument.

An European call option gives the right to buy an asset at the given expiration time at the given strike price . An European put option gives the right to sell an asset at the given expiration time at the given strike price . Let us denote with the price of an European call option at time and with the price of the asset. The value of the European call option at the expiration time is equal to

Let us denote with the price of a put option at time . The value of the European put option at the expiration time is equal to

American options have a different mode concerning the right to exercise the option than the European options. American call and put options can be exercised at any time before the expiration date, whereas European options can be exercised only at the expiration day. Thus an American option is more expensive than the corresponding European option. When we use the term “option” without a further qualification, then we refer to an European option.

The following terminology is used to describe options.

• A call option is out of the money if . A call option is at the money if . A call option is in the money if . A call option is deep out of the money (deep in the money) if ().

  The moneyness of a call option is defined as . The moneyness of a put option is defined as .⁷

• Before the expiration time the price of a call option satisfies
  

  see (14.10). The difference is called the time value of the option. The value is called the intrinsic value. Thus,

  2.6

2.3.2.2 Applications of Options

Options can serve at least the following purposes:

1. 1. Options can be used to create a large number of different payoffs. Some payoffs applied in option trading are described in Chapter 17. For example, buying a call and a put with the same strike price and the same expiration creates a straddle position which profits from large positive or negative movements of the underlying.
2. 2. Options can provide insurance. With options it is possible to create a payoff which cuts the losses that could occur without using of the options. Buying a put option gives an insurance in the case one has to sell in a future time an asset one possesses. Buying a call option gives an insurance in the case when one has to buy in a future time an asset one does not possess. Examples of providing insurance with options include the following:
   • Buying a put option on a stock gives an insurance policy for an investor. If an investor owns a stock, buying a put option will cut the future possible losses.
   • Buying a put option on an exchange rate gives an insurance policy for a company receiving payments on a foreign currency in future.
3. 3. Call options can be used to give a compensation to managers, since the payoff of a call option is positive only when the stock price is larger than the strike price.
4. 4. Options make leveraging possible, since option trading requires a small initial capital as compared to stock trading.⁸

2.3.2.3 Exotic Options

We say that an option is exotic if it is not an European or an American call or put option.

Asian Options

The value of an Asian call option at the expiration is

where

with being a collection of predetermined time points. Asian options are more resistant to manipulation than European options: The value of an European option at the expiration depends on the value of the underlying asset at one time point (the expiration date), whereas the value of an Asian option depends on the values of the underlying asset at several time points.

Barrier Options

Barrier options disappear if the underlying either exceeds, or goes under the barrier. Alternatively, a barrier option could have value only if it has exceeded, or went under the barrier. Knock-in options come into existence if some barrier is hit and knock-out options cease to exist if some barrier is hit. One speaks of up-and-out, down-and-out, up-and-in, down-and-in options. For example, a knock-out option on stock , written at time 0, with expiration time , has the payoff

where is the strike price, and is the barrier. Barrier options are cheaper than the corresponding European options, which makes them useful.