Options and Futures Faculty of Economics & Business The University of Sydney Shino Takayama

Put-Call Parity Since the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal. C - P = S 0 - X / (1 + r f ) T If the prices are not equal arbitrage will be possible.

Option Valuation Intrinsic value - profit that could be made if the option was immediately exercised. Call: stock price - exercise price Put: exercise price - stock price Time value - the difference between the option price and the intrinsic value.

Factors Influencing Option Values FactorEffect on value Stock price increases Exercise price decreases Volatility of stock price increases Time to expirationincreases Interest rate increases Dividend Ratedecreases

Binomial Option Pricing: Text Example Generalizing the Two State Approach Assume that we can break the year into two six-month segments. In each six-month segment the stock could increase by 10% or decrease by 5%. Assume the stock is initially selling at 100. Possible outcomes: Increase by 10% twice Decrease by 5% twice Increase once and decrease once (2 paths).

Expanding to Consider Three Intervals Assume that we can break the year into three intervals. For each interval the stock could increase by 5% or decrease by 3%. Assume the stock is initially selling at 100.

Black-Scholes Option Valuation C o = S o N(d 1 ) - Xe -rT N(d 2 ) d 1 = [ln(S o /X) + (r + 2 /2)T] / ( T 1/2 ) d 2 = d 1 + ( T 1/2 ) where C o = Current call option value. S o = Current stock price N(d) = probability that a random draw from a normal dist. will be less than d.

Black-Scholes Option Valuation X = Exercise price e = , the base of the natural log r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option) T = time to maturity of the option in years ln = Natural log function Standard deviation of annualized cont. compounded rate of return on the stock

Black-Scholes with Dividends The call option formula applies to stocks that pay dividends. One approach is to replace the stock price with a dividend adjusted stock price. Replace S 0 with S 0 - PV (Dividends)

Futures and Forwards Forward - an agreement calling for a future delivery of an asset at an agreed-upon price Futures - similar to forward but feature formalized and standardized characteristics Key difference in futures Secondary trading - liquidity Marked to market Standardized contract units Clearinghouse warrants performance

Key Terms for Futures Markets Futures price - agreed-upon price at maturity Long position - agree to purchase Short position - agree to sell Profits on positions at maturity Long = spot minus original futures price Short = original futures price minus spot

Types of Commodities Agricultural commodities Metals and minerals (including energy contracts) Foreign currencies Financial futures Interest rate futures Stock index futures

Trading Mechanics Clearinghouse - acts as a party to all buyers and sellers. Obligated to deliver or supply delivery Closing out positions Reversing the trade Take or make delivery Most trades are reversed and do not involve actual delivery

Margin and Trading Agreement Initial Margin - funds deposited to provide capital to absorb losses Marking to Market - each day the profits or losses from the new futures price are reflected in the account. Maintenance or variation margin - an established value below which a traders margin may not fall.

Margin and Trading Agreement Margin call - when the maintenance margin is reached, broker will ask for additional margin funds Convergence of Price - as maturity approaches the spot and futures price converge Delivery - Actual commodity of a certain grade with a delivery location or for some contracts cash settlement

Trading Strategies Speculation - short - believe price will fall long - believe price will rise Hedging - long hedge - protecting against a rise in price short hedge - protecting against a fall in price

Theory of Futures Prices Expectations Normal Backwardation Contango