Chapter 28 – Pricing Futures and Options BA 543 Financial Markets and Institutions

Chapter 28 – Pricing Futures Pricing of Futures Based on Arbitrage Arbitrage is the process whereby one makes more than the risk-free rate on an investment with a guarantee (cannot lose on this investment) In Futures it is known as Spot-Futures Parity In book we have the cash and carry trade or Reverse cash and carry trade Why does this arbitrage work to set prices? All current and future prices are locked-in

Chapter 28 – Pricing Futures Examples of the Spot-Future Relationship Formula: F T = S 0 (1 + r f ) T What does this mean… The Futures price is based on the current spot or cash price and the current risk-free interest rate Violations of this formula provide and arbitrage opportunity Arbitrage opportunity exploited until formula is meet Examples in the text, pages , when price too high or price too low

Chapter 28 – Pricing Futures Some Frictions with Spot-Future Parity Interim Cash Flows – dividends on stocks, margin calls on contracts, coupon interests payments, etc… Example, dividend payments – date unknown and amount unknown but can be reasonably estimated using historical payments. But they are estimates…not guaranteed and interim dividend payments (and interest payments) can only be reinvested at the current rates not rates known at time of contract Example with margins on futures, the cash flows happened during the holding period and the price is that of a forward contract (no margin accounts with forward contracts) Difference between lending and borrowing rates (bid-ask spreads) thus we have upper and lower pricing boundaries

Chapter 28 – Pricing Futures Some Frictions with Spot-Future Parity Transaction costs Getting into and out of positions is not costless Agents receive compensation for providing the markets for futures contracts (commissions) Short Selling In model assume short procedure funds are available and reinvested In practice funds are not received and a short margin is required With stocks, shorting must meet up-tick rule (thus cannot guarantee execution of short at current price)

Chapter 28 – Pricing Futures Some Frictions with Spot-Future Parity Unknown deliverables and dates Some futures contracts (Treasury Bonds) do not have a specific underlying asset and has a month long delivery date Thus short position has a delivery option Deliverable is a basket of goods With indexes the entire set of stocks or bonds must be purchased or shorted…but often a “sample” of the index is constructed to track the index Uncertainty arises as the sample may not track the index exactly Tax Treatment Formula and example ignored taxes but they can be different across cash and futures transactions

Chapter 28 – Pricing Options Options Price composed of two distinct parts Intrinsic Value – cash flow at time of exercise If exercised immediately there is the set cash flow from the option, for example call S – K (in-the-money) If there is no value in immediate exercise then holder does not exercise (out-of-the-money) Time value or time premium – value in waiting The excess of the current market price over the intrinsic value of the option Market price is higher than S – K for call option Market price reflects the potential increase in intrinsic value by waiting for a change in S

Chapter 28 – Pricing Options Arbitrage between Puts and Calls when Same underlying asset Same maturity date Same strike price Formula: P – C = S 0 – K (e -rT ) where, P is current Put price C is current Call price S 0 is current price of the underlying asset K is the strike price of the option e -rT is the continuous rate with e the exponential function, r the interest rate and t the time

Chapter 28 – Pricing Options Finding the Price of a Call or Put Option Discrete model – binomial pricing Continuous model – Black and Scholes Some complications when the underlying asset is a fixed income bond There is a limited price upside – the present value of all future cash flows Change in interest rates changes value of the underlying asset – cannot assume fixed interest rate over the life of the asset