I.1  Finance, the Finance System and Financial Assets

Finance is defined as the practice of manipulating and managing money.  It is concerned with the process, institutions, markets, and instruments involved in the transfer of money among and between individuals, businesses, and governments. Financial decisions are with two distinctive features: spreading over time, and with uncertainty.
To implement their financial decisions, people make use of the financial system, which is defined as the set of markets and other institutions used for financial contracting and the exchange of assets and risks. The financial system includes the markets for stocks, bonds, and other financial instruments; financial intermediaries (such as banks and insurance companies); financial-service firms (such as financial advisory firms); and the regulatory bodies that govern all of these institutions.    
The basic types of financial assets are bond, stock, etc and derivatives - forward, futures and option, etc. They are categorized into underlying assets and derivative assets.
• Underlying Assets
The underlying assets can be stocks, bonds, currency, commodities, and other financial assets, or combinations of these.  The traditional stock and bond markets raise necessary capital for corporations and governments, and the foreign exchange market facilitates international trade and investment.
• Stocks
Stocks represent the claim of the owners of a firm. Stocks are issued by corporations and can be traded in the stock market. Common stock usually entitles the shareholder to vote in the election of directors and other matters. Preferred stock generally does not confer voting rights but it has a prior claim on assets and earnings: dividends must be paid on preferred stock before any can be paid on common stock.
• Bonds
Bonds are issued by anyone who borrows money - firms, governments, etc.  They are fixed-income instruments because they promise to pay fixed sums of cash in the future. Bondholders have an IOU (I owe you) from the issuer, but no corporate ownership privileges, as stockholders do.
• Derivative Assets
Derivatives are financial instruments that derive their value from the prices of one or more other assets such as stocks, bonds, foreign currencies, or commodities. Derivatives serve as tools for managing risks associated with these underlying assets. The most common types of derivatives are options and futures.
• Forwards and Futures
A forward is a financial contract in which two parties agree to buy and sell a certain amount of the underlying commodity or financial asset at a prespecified price at a specified time in the future. The specified time is called the time-to-maturity of the forward contract and the price specified in the contract is called the forward price.
A futures contract is a standardized forward contract traded in an exchange. To avoid the shortcomings of the forwards that each party cannot change his/her mind to reverse the position specified in the contract, in a futures contract, both the time to maturity and the amount of the underlying asset to be delivered in each contract are standardized so that the futures contract can be traded in a market place. Thus, if one party wants to change position, he/she can buy or sell in the futures market.
• Options
An option is a financial contract, which provides the holder with the right to buy or sell a certain amount of the underlying asset at a prespecified price at or before a specified time in the future. Similar to the forward and futures contracts, the time specified in an option contract is called the time-to-maturity of the option. The price specified in the contract is called the exercise price or the strike price of the option.
Unlike a forward or futures contract, an option contract gives its holder the right, but not the obligation to buy or sell the underlying asset. The two most common types of options are calls and puts. A call option is an option to purchase the underlying asset, while a put option is an option to sell.
Everyone would like to hold options because they provide a positive likelihood for the holder to make a profit. But he/she has to pay the price. The cost for this likelihood is the money paid by the buyer of option to the seller to compensate the latter's possible losses. This money is called the premium of the option, or simply the option price.
The following example may help you to understand some of the concepts of derivatives:
You decide to buy a new car and you go to the dealer's showroom. You select a certain type of car, and the dealer tells you that if you place the order today and place a deposit, then you can take delivery of the car in 3 months time. If in 3 months time the price of the model has decreased or increased, it doesn't matter. When the agreement between you and the dealer is reached, you have entered into a forward contract: you have the right and also the obligation to buy the car in 3 months.
Instead, suppose the car you selected is on offer at $ 30,000 but you must buy it today. You don't have that amount of cash today and it will take a week to organize a loan. You could offer the dealer a deposit, for example $200, if he will just keep the car for a week and hold the price. During the week, you might discover a second dealer offering an identical model for a lower price, then you don't take up your option with the first dealer. At the end of the week the $ 200 is the dealer's whether you buy the car, or not. In this case, you have entered an option contract, a call option here. It means that you have the right to buy the car in a week, but not the obligation. The expiration time is one week from now, the strike price is $30,000.          
In this example, for both the forwards and option contracts described, delivery of the car is for a future date and the prices of the deposit and option are based on the underlying asset - the car.

I.2  Important Financial Concepts

I.2.1. Time Value of Money
Time value of money refers to the fact that money in hand today is worth more than the expectation of the same amount to be received in the future. Money has a time value because of the opportunity to earn interest or the cost of paying interest on borrowed capital.
We begin our study of the time value of money with the concept of compounding - the process of going from today's value, or present value (PV ), to future value (FV ). Future value is the amount of money an investment will grow to at some date in the future by earning interest at some compound rate. If i is the interest rate and n is the number of years, the future value of a present value is given by:

          
Calculating the present values of given future amounts is called discounting, the reverse of compounding.    
The time-value-of -money concept provides a tool for making investment decisions. The most common decision rule is the net- present- value ( NPV ) rule, which is a way of comparing the value of money now with the value of money in the future. The NPV is the difference between the present value of all future cash inflows minus the present value of all investments. The rule says that accept any project if its NPV is positive; reject a project if its NPV is negative.

I.2.2. Risk and Return
In the most basic sense, risk can be defined as the chance of financial loss or, more formally, the variability of returns associated with a given asset. Most financial decision-makers are risk-averse because they require higher expected returns as compensation for taking greater risk.
Risk can be assessed from a behavioral point of view using sensitivity analysis and probability distributions. Sensitivity analysis uses the range of estimated return to obtain a sense of the variability among outcomes. The greater the range for a given asset, the more variability, or risk, it is said to have. Although the use of sensitivity analysis is rather crude, it does provide decision-makers with a feel for the behavior of returns. This behavioral insight can be used to assess roughly the risk involved.
In addition to the range, the risk of an asset can be measured quantitatively using statistics.
Probability distributions of outcome provide a more quantitative, yet behavioral, insight into an asset's risk.   Here we consider two statistics: the standard deviation and the coefficient of variation CV, which measures the dispersion around the expected value of a return .       

       
where      = return for the ith outcome
         = probability of occurrence of the ith outcome
      n = number of outcomes considered   
                    

         CV=     
The two statistics can be used to measure the risk (i.e., variability) of asset returns. In general, the higher the standard deviation and the coefficient of variation, the greater the risk.        
The return on a portfolio is calculated as a weighted average of the returns on the individual assets from which it is formed. Letting equals the proportion of the portfolio's total dollar value represented by asset j, and equals the return on asset j, the portfolio return, is

             

I.2.3. Valuation
Asset valuation is the process of estimating how much an asset is worth, and is at the heart of much of financial decision-making.
The book value of an asset or a liability as reported in a firm's financial statements often differs from its current market value. In making most financial decisions, it is a good idea to start by assuming that for assets that are bought and sold in competitive markets, price is a pretty accurate reflection of fundamental value. This assumption is generally warranted precisely because there are many well-informed professionals looking for mispriced assets who profit by eliminating discrepancies between the market prices and the fundamental values of assets. This process is called arbitrage, the purchase and immediate sale of equivalent assets in order to earn a sure profit from a difference in their prices.
Valuation process links risk and return to determine the worth of an asset. The key inputs to the valuation process include cash flows (returns), timing, and the required return (risk compensation).
Because risk describes the chance that an expected outcome will not be realized, the level of risk associated with a given cash flow can significantly affect its value. In general, the greater the risk of a cash flow, the lower its value. Turning to the opposite point of view, to compensate the risk, the holders of the assets require greater return rate, or discount rate in terms of present value calculation.
Here we show the basic valuation model. Simply stated, the value of any asset is the present value of all future cash flows it is expected to provide over the relevant time period. The value of an asset is therefore determined by discounting the expected cash flows back to their present value, using the required return as the appropriate discount rate. Utilizing the present value techniques, we can express the value of any asset at time zero, , as

  
where          = value of the asset at time zero
       C= cash flow expected at the end of year t
         k = appropriate required return (discount rate)  
  n = relevant time period   
The principle of this basic valuation model can be applied into the valuation of bond and stock:
• Bond Valuation
Bonds are long-term debt instruments used by business and government to raise large sums of money, typically from a diverse group of lenders. The value of a bond is the present value of the contractual payments its issuer is obligated to make from the current time until it matures. The appropriate discount rate would be the required return, , which depends on prevailing interest rates and risk. The basic equation for the value, , of a bond is given by

         
where        = value of the bond at time zero
           I = annual interest
            n = number of years to maturity
   M = par value of the bond
    = required return on a bond
The value of a bond in the marketplace is rarely equal to its par value. A variety of forces in the economy as well as the passage of time tend to affect value, which are really in no way controlled by bond issuer or investors. When the required return is greater than the interest rate on the bond, the bond value will be less than its par value. In this case, the bond is said to be sold at a discount. On the other hand, when the required return falls below the coupon interest rate, the bond value will be greater than par. In this situation the bond is said to be sold at a premium.
• Common Stock Valuation
Like bonds, the value of a share of common stock is equal to the present value of all future benefits it is expected to provide. Simply stated, the value of a share of common stock is equal to the present value of all future dividends it is expected to provide over an infinite time horizon:
       
                                                         
where      = value of common stock at time zero
   = per share dividend expected at the end of year t  
        = required return on a common stock   
From the formula, we can see that any action of increasing the level of expected return without changing risk should increase share value, and vice versa. Similarly, any action that increases risk (required return) will reduce share value, and vice versa.

I.3  Derivatives and Risk Management

I.3.1  The Functions of Derivatives
The most obvious function of derivatives is to facilitate the reallocation of exposure to risk among market participants. Basically there are two strategies to implement the risk reallocation: hedging and speculating.  
Hedging can be generally understood as a financial activity to either reduce or eliminate the risk of an underlying asset by making the appropriate offsetting derivative transaction. Such activities may involve certain costs, i.e. the cost for buying the futures or options.
Let us take the above car-purchasing example. If during the week you discover a second dealer offering an identical model for $ 29,000 then you don't take up your option with the first dealer. The total cost of buying the car is now $29,000 + $200 = 29,200: cheaper than the first price you were offered, i.e $300,000. If you cannot find the car at a cheaper price and buy the car from the first dealer, then the car will cost a total of $300,200. If you decide not to buy at all you will lose $200 to the car dealer. But anyway, in this example, both in the forward contract case and the option one, you are hedging against a price rise in the car - it eliminates the risk of buying a car more expensive than $300,000.
Speculating is somewhat the opposite activity to hedging. Whereas the aim of hedging is to reduce or eliminate risk by sacrificing some capital, the aim of speculating is to obtain higher returns by taking higher risks.
Again take the above example. Suppose that the car you have bought a call option for is very much in demand and there is a sudden price rise to $33,000. One colleague of yours also wants the same car and hears that you have an option to buy the car for $30,000 in a week's time. You can sell the option to your colleague for $400. This means that the car dealer still gets his sale, your friend gets the car he wants and you make $200 on selling your option. In this case you have speculated on your contract and made a 200%.
In the following we present the pricing principles and methods for futures and options. Fortunately, the prices of forwards and futures contracts on the same underlying assets with the same time to expiration are generally very close to each other, although they are normally different because of taxes, transaction costs, and other factors. Thus, here we can simply regard futures as the same as forward contracts for convenience of understanding and analysis.

I.3.2.  Futures Pricing
Different from some of their underlying assets, such like copper, etc, futures can be produced and stored at very low cost, so we can ignore those costs completely in deriving parity relations between spot (current) and future prices.
Consider a futures contract (or simply futures) on a share of a stock that pays no dividend. The contract is the promise to deliver a share at some specified delivery date at a specified delivery price. Let us denote this future price by F, and the spot price at the beginning of the stock is S. The relation between F and S is

        = F /
where r is the risk-free  interest rate, such as the interest rate offered by a bank if you deposit the amount of money in it, T is the maturity of the futures contract in number of years.

I.3.3.  Option Pricing
The valuation or pricing of options is a careful balance of different market factors. However, the basic principle lies in that, the present value of a option is the expected value of the payoff, i.e the value, of the option at the expiration date at the risk-free  interest rate. For an European call option, the payoff of the option is the difference between the underlying asset price at maturity and strike price of the underlying asset. Then the pricing model for European call option is

     
where S is the spot price of the underlying stock at expiration, K the strike price, T the expiration time, t the present time, r the risk-free  interest rate and C, the present price of the option.
The payoff (or value) of the option at expiration, max (S - K, 0), can be understood in this way: if the stock price at expiration is at or below the strike price, then the payoff of the option is zero, i.e the call option will expire worthless, because nobody wants to buy a stock at a price higher than market price; if it is greater than the strike price, then the payoff is simply the difference between the spot price and the strike price. The difference between the underlying asset price at maturity and strike price is called the intrinsic value of the call option.
There are also other types of options, e.g Asian option, American option, lookback option etc. They are distinctive in different aspects: how to calculate payoff, whether an option can only be exercised on the expiration data, etc.
The term is the result of continuous compounding. To explain this concept, consider an amount A invested for n years at an interest rate of R per annum. If the rate is compounded once per annum, the terminal value of the investments is

               
If it is compounded m times per annum, the terminal value of the investment is

             
The limit as m tends to infinity is known as continuous compounding and it can be shown that

          
So, compounding a sum of money at a continuously compounded rate R for n years involves multiplying it by  . Discounting it involves multiplying by .          
Here we present the celebrated Black-Scholes formula for calculation of option price. The derivation process of it is shown in Appendix.

• Black-Scholes Option Pricing Formula
The most popular model in options pricing is the Black-Scholes model in which the underlying asset price is assumed to be lognormally distributed. As in most other theoretical models, Black and Scholes made many assumptions. We list a few important ones here:
(i).   The underlying asset price is lognormally distributed;
(ii).  The underlying asset pays no dividend;
(iii). There are no transaction costs in buying or selling the underlying asset or option;
(iv). The short-term interest rate is known and is constant through time.
With these assumptions, Black and Scholes obtained the following formula for the price of a European call option:

           C = SN () - K   

where          d =   = = d + σ
C is the price of a call option, K is the strike price of the underlying asset, r is the annual interest rate, σ is the volatility of the annual return of the underlying asset, T is the time-to-maturity in number or fraction of year(s), N(x) is the value of the cumulative function of the standard normal distribution at  x.
There is a important relationship between a call and its corresponding put option prices with the same strike price and time-to-maturity, which can be used to calculate the put option price:

       P = C - S + K.
The Black-Scholes formula is clearly a function of five factors: S, K, T, r and σ.  S and r can be observed from the market; K and T are specified in the option contract.  σ, however, is neither specified in the option contract nor directly observable from the market. We have to estimate this volatility value using historical data of the underlying asset in order to use the Black -Scholes formula.
There are a few popular terms characterizing the sensitivity of options prices to the changes of these factors. These sensitivities are often named by Greek letters. They play an important role in both trading activities and risk management in financial institutions.
Delta () measures how fast its price changes with the price of its underlying asset. Vega () measures how fast the option's price changes with the volatility of its underlying asset. Theta ()) measures the sensitivity of its price with respect to the time to maturity. Rho () measures the sensitivity of the option's value with respect to the fluctuation of the interest rate. Gamma () measures how fast the option's delta changes with the price of its underlying asset.
         
I.3.4  Risk Management     
Risk management deals with risk reduction, basing on the principle of benefit-cost tradeoff and information available. There are four basic techniques available for reducing risk:
     • Risk avoidance:  A consciousness not to be exposed to a particular risk.
     • Loss prevention and control:  Actions taken to reduce the likelihood or the severity of  losses.
     • Risk retention:  Absorbing the risk and covering losses out of one's own resources.
     • Risk transfer: Transferring the risk to others who are willing to accept it in order to earn possible profit.
One of the greatest roles of the financial system is transferring risk. Three most commonly used methods for transferring risk are hedging, insuring, and diversifying.  
One is said to hedge a risk when the action taken to reduce one's exposure to a loss also causes one to give up of the possibility of a gain. For example, farmers who sell their future crops before the harvest at a fixed price to eliminate the risk of a low price at harvest time, also give up the possibility of profiting from high prices at harvest time.
Insuring means paying a premium to avoid losses. By buying insurance, you substitute a sure loss for the possibility of a larger loss if you do not insure.
Diversification: diversifying means holding many risky assets instead of concentrating all of your investment in only one.  Its meaning is captured by the familiar saying: "Don't put all your eggs in one basket." The diversification principle is that by diversifying across risky assets, people can sometimes achieve a reduction in their overall risk exposure with no reduction in their expected return.   

• Value at Risk : A Technique for Risk Measurement
Value-at-Risk  (VaR) measures the worst expected loss under normal market conditions over a specific period, with a specified probability. For example, suppose that a portfolio manager has a one-day VaR equal to $1 million at 1%. This means that, under normal market conditions, there is only 1 chance in 100 that a daily loss is bigger than $1 million occurs.
In practice, based on historic observations, it is often assumed that, after a certain period of time, the probability density function of the portfolio value (the portfolio profit also) represents a normal distribution. We know that the probability of the portfolio's value is less than  is given by the area, say A%, of the tail to the left of  in the probability density profile. Then we can calculate the correspondent maximal loss with this probability, i.e A%VaR:

         A% VaR =              
where is the value of the portfolio at the beginning. The graph of value probability distribution below illustrates the process of VaR calculation, in which = 100, = 60.

The graph of profit probability distribution below can give a clearer illustration

VaR uses a single number to aggregates all of the risks in portfolio. It can be a tool for management used in various ways: to set their overall risk target, to determine internal capital allocation decision, to assess the risks of investment opportunities before decisions are made, as well as to evaluate the performance on them after the event, etc.