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 St
- 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 St

- 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:

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Financial wizards use Black-Scholes 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:
- 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 Black-Sholes 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 |
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