Arbitrage  an investor profits from price differences between different markets
 Market prices differ by location, and time
 Investor buys low and sells high
 Price difference disappears, as investors move one product from one market to another
 Investor buys gold for one year
 Notation
 return is r
 Spot price in one year is S_{t}
 Spot price today is S
 Storage cost is c
 Return to investment is:
 Synthetic investment  the investor can invest in gold by buying gold futures contract
 Synthetic  investors does not hold gold
 Investor buys gold futures (no money down)
 He invests his funds in a riskless security that earns a return, r
 The return to his investment is:
 Remember, he buys gold for F, and sells it for S_{t}
 Investors look at both investment opportunities. They use arbitrage to move funds around until
 Set both rates of return equal to each other, and solve for F
 Note  this is not a forecast of the future price of gold
 We are using reasoning to place a value on a futures contract
 Example
 The riskless interest rate for a safe investment is 3% per year
 The storage cost is 2% per year
 The gold's spot price is $2,000 per ounce
 The value of a future's contract is:

Financial wizards use BlackScholes model to estimate a Europeans option's premium
 Uses stochastic calculus
 Black and Scholes model European option premiums as a random walk
 A random walk is:
 X_{t} = X_{t1} + random error
 random error is normally distributed with a mean of zeor
 Click on spreadsheet to see an example of a random walk
 Notation
 The call's premium is C
 The put's premium is P
 The spot price of the commodity is S
 The exercise price of the commodity is E
 The riskless interest rate is r
 The time to maturity of the option in years is T
 The volatility of the spot price is s
 Functions
 The natural logarithm is ln
 The base of the natural logarithm is e, where e = 2.71828...
 The probability of a random variable drawn from a standard normal distribution is N(d)
 N stands for the normal distribution
 Standard normal has mean = 0, and a standard deviation = 1
 The example is shown below:
The BlackSholes equations are:
 Example:
 The spot price, S, is $100
 The exercise price, E, is 100
 The riskless interest rate, r, is 8%
 The time to maturity, T, is 0.5 years
 The volatility, s, is 0.2
 Note:
 Use the Excel function, NORMDIST
 Example: if d = 0.85, then the N(0.5) is "=NORMDIST(0.85,0,1,1)"
 The premiums for the European call and put options are:
The following table indicates what happens to an European option's premium, if one of the factors changes:
 For example
 If the volatility of the spot prices increases, then both the premiums for European call and put options will increase
 If the spot price increases, then the European call option will increase, while the put option will decrease
 Note  the premium increases because it is more likely to be exercised, while a premium decreaes indicates a less likelihood of being exercised.
Term 
European Call Option Premium 
European Put Option Premium 
Spot Price Increases 
increase 
decrease 
Strike Price Increases 
decrease 
increase 
Riskless Interest Rate Increases 
increase 
decrease 
Option's Maturity Increases 
increase 
increase 
Volatility of Spot Prices Increases 
increase 
increase 
