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.

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 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:
    • X_t = X_t-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.