# Lecture 8 - Market Evaluation of Derivatives

This lecture is a continuation of derivatives. Students will learn the market evaluation of future contracts, and European call and put options using Black-Scholes formula.

## Market Evaluation of Futures

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

• 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
• 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:

## Market Evaluation of European Options

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:
• Xt = Xt-1 + 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 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