Chapter 28 – Pricing
Futures and Options
BA 543 Financial Markets and Institutions
Chapter 28 – Pricing Futures
 Pricing of Futures Based on Arbitrage
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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
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In book we have the cash and carry trade or
Reverse cash and carry trade
 Why does this arbitrage work to set prices?
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All current and future prices are locked-in
Chapter 28 – Pricing Futures
 Examples of the Spot-Future Relationship
 Formula: FT = S0 (1 + rf )T
 What does this mean…
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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 566-568, when
price too high or price too low
Chapter 28 – Pricing Futures
 Some Frictions with Spot-Future Parity
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Interim Cash Flows – dividends on stocks, margin calls on
contracts, coupon interests payments, etc…
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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
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Transaction costs
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Getting into and out of positions is not costless
Agents receive compensation for providing the
markets for futures contracts (commissions)
Short Selling
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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
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Unknown deliverables and dates
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Deliverable is a basket of goods
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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
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
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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
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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
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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 = S0 – K (e-rT) where,
 P is current Put price
 C is current Call price
 S0 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
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Discrete model – binomial pricing
Continuous model – Black and Scholes
 Some complications when the underlying
asset is a fixed income bond
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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