Chapter 28

PRICING OF FUTURES AND OPTIONS

CONTRACTS

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

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

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)

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.

Theoretical Futures Price

The effect of carry on the difference between the futures price and the cash price can be shows as follows:

Carry Futures price

Positive carry(y > r)

Will sell at a discount tocash price (F < P)

Negative carry(y < r)

Will sell at a premium tocash price (F > P)

Zero (r = y) Will be equal to the cashprice (F = P)

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

Assumptions Underlying the Arbitrage Arguments

Interim cash flowsDifferences between lending and borrowing

ratesTransaction costsShort sellingKnown deliverable asset and settlement dateDeliverable is a basket of securitiesDifferent tax treatment of cash and futures

transactions

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

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

Put-Call Parity

Put-call parity for European options with cash distributions on underlying asset:

Sr

DXCP t

f

t

)1(

where: P = Put option priceC = Call option priceX = Strike priceDt = Cash distributionS = Price of underlying assetrf = Riskfree rate

Factors That Influence the Options Price

Current price of the underlying assetStrike price Time to expiration of the optionExpected price volatility of the underlying

asset over the life of the optionShort-term, riskfree interest rate over the

life of the optionAnticipated cash payments on the

underlying asset over the life of the option

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

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

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

r

C

du

ru

r

C

du

drC du

1

1

1

1

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

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