Chapter 28

1. Chapter 28 PRICING OF FUTURES AND OPTIONS CONTRACTS

2. Arbitrage Strategies • Cash and carry trade • borrowing cash to purchase a security and carrying that security to the futures settlement date • Reverse cash and carry trade • selling a security short and investing the proceeds received from the short sale

3. Pricing of Futures Contracts • The theoretical or equilibrium futures prices is based on arbitrage arguments. • The following information is needed: • The price of the asset in the cash market • The cash yield earned on the asset until the settlement date • The rate for borrowing and lending until the settlement date

4. A Theory of Futures Pricing • The equilibrium futures price is the price that ensures the the profit from the arbitrage strategy is zero. Profit = 0 = F + yP - (P + rP) • The theoretical futures price is: F = P + P (r - y)

5. A Theory of Futures Pricing • The theoretical futures price depends on: • The price of the underlying asset in the cash market. • The cost of financing a position in the underlying asset. • The cash yield on the underlying asset.

6. Theoretical Futures Price • The effect of carry on the difference between the futures price and the cash price can be shows as follows:

7. Principle of Convergence • At the delivery date, the futures price must equal the cash price. • As the delivery date approaches, the futures price converges to the cash price. • The financing cost approaches zero • The yield approaches zero • The cost of carry approaches zero

8. Assumptions Underlying the Arbitrage Arguments • Interim cash flows • Differences between lending and borrowing rates • Transaction costs • Short selling • Known deliverable asset and settlement date • Deliverable is a basket of securities • Different tax treatment of cash and futures transactions

9. Pricing of Options • The price of an option consists of two components: the intrinsic value and the time premium. • Intrinsic value • the economic value of the option if exercised immediately, which is either greater than zero or zero • Time premium • amount by which the option price exceeds the intrinsic value

10. Put-Call Parity • Relationship between the price of a call, and the price of a put • On the same underlying asset • With the same strike price • With the same expiration date

11. Put-Call Parity • Put-call parity for European options with cash distributions on underlying asset: where: P = Put option price C = Call option price X = Strike price Dt = Cash distribution S = Price of underlying asset rf = Riskfree rate

12. Factors That Influence the Options Price • Current price of the underlying asset • Strike price • Time to expiration of the option • Expected price volatility of the underlying asset over the life of the option • Short-term, riskfree interest rate over the life of the option • Anticipated cash payments on the underlying asset over the life of the option

13. Option Pricing Models • The theoretical options price is determined on the basis of arbitrage arguments. • Option Pricing Models • Black and Scholes Option Pricing Model • Binomial Option Pricing Model

14. Binomial Option Pricing Model • Hedged Portfolio • Long position in a certain amount of the asset • Short call position in the underlying asset • Cost of Hedged Portfolio • HS - C • Payoff of Riskless Hedged Portfolio • uHS - Cu =dHS - Cd • Hedge Ratio • H = (Cu - Cd)/(u - d)S

15. Price of a Call Option • Hedged Portfolio • HS - C • One-Period Wealth • (1 + r)(HS -C) • Payoff of Hedged Portfolio • uHS - Cu • Call Option Price

16. Assumptions of Binomial Model • Price of the security can take on any positive value with some probability • Short-term interest rate is constant over the life of the option • Volatility of the price of the security is constant over the life of the option

17. Fixed-Income Option Pricing Models • Assumptions of binomial model are unreasonable for fixed-income securities • Alternative option pricing models: • yield curve option pricing models • arbitrage-free option pricing models