ACFI310 - Lecture 7 - Options on futures contracts

How are futures options typically referred to?

By the maturity month of the underlying futures contract.

When do futures options usually expire relative to the futures contract?

On or a few days before the earliest delivery date of the underlying futures contract.

Are most future options American or European style

American style

When a call futures option is exercised, what does the holder receive?

1. A long position in the futures

2. A cash amount equal to (futures price at most recent settlement - strike price)

When a put futures option is exercised, what does the holder receive?

1. A short position in the futures

2. A cash amount equal to (Strike price - Futures price at most recent settlement)

If the futures position is closed out immediately after exercise, what is the payoff from a call futures option?

Payoff = F - K (Where F is futures price at exercise, K is strike price)

If the futures position is closed out immediately after exercise, what is the payoff from a put futures option?

Payoff = K - F (where K is strike, F is futures price at exercise)

A July call option on gold futures has a strike of $1800/oz. It's exercised when the futures price is $1840 and most recent settlement is $1838. One contract is on 100 ounces. What is the total payoff?

• Cash from exercise: (1838 - 1800) × 100 = $3,800

• Gain from closing long futures: (1840 - 1838) × 100 = $200

• Total payoff = $4,000

A Sept put option on corn futures has a strike of 300 cents/bushel. Exercised when futures price is 280 cents and settlement is 279 cents. One contract is 5,000 bushels. What is the total payoff?

• Cash from exercise: (3.00 - 2.79) × 5000 = $1,050

• Loss from closing short futures: (2.79 - 2.80) × 5000 = -$50

• Total payoff = $1,000

What is the put-call parity relationship for European futures options?

c + Ke^(-rT) = p + F₀e^(-rT)

In a binomial tree for futures options, what is the formula for Δ (delta) to create a riskless portfolio?

Δ = (f_u - f_d) / (F₀u - F₀d)

What is the risk-neutral probability formula for futures options in a binomial tree?

p = (1 - d) / (u - d)

What is the valuation formula for a futures option using the binomial model?

f = [pf_u + (1-p)f_d]e^(-rT)

What are the formulas for European call and put options on futures (Black's model)?

• c = e^(-rT)[F₀N(d₁) - KN(d₂)]

• p = e^(-rT)[KN(-d₂) - F₀N(-d₁)]

What are the d₁ and d₂ formulas in Black's model?

• d₁ = [ln(F₀/K) + σ²T/2] / (σ√T)

• d₂ = [ln(F₀/K) - σ²T/2] / (σ√T) = d₁ - σ√T

In Black's model, what does setting the dividend yield q equal to r represent?

It ensures the expected growth rate of the futures price in a risk-neutral world is zero.

Why can futures prices be treated like stocks paying a dividend yield of r?

Because futures contracts require no initial investment, so in a risk-neutral world the expected return should be zero, which is equivalent to treating the futures price as a stock with dividend yield equal to the risk-free rate.

A 6-month European call on spot gold: futures price = 1860, strike = 1800, r = 5%, σ = 20%. Using Black's model, what are d₁ and d₂?

• d₁ = [ln(1860/1800) + 0.2² × 0.5/2] / (0.2√0.5) = 0.3026

• d₂ = d₁ - 0.2√0.5 = 0.1611

Using the values from the previous card, what is the option value?

c = e^(-0.05×0.5)[1860N(0.3026) - 1800N(0.1611)] = $132.56

What is put-call parity for non-dividend paying stocks?

c + Ke^(-rT) = p + S₀

What is put-call parity for stock indices (dividend yield q)?

c + Ke^(-rT) = p + S₀e^(-qT)

What is put-call parity for foreign exchange (foreign rate r_f)?

c + Ke^(-rT) = p + S₀e^(-r_fT)

What is put-call parity for futures?

c + Ke^(-rT) = p + F₀e^(-rT)

What are Heating Degree Days (HDD)?

For each day: max(0, 65 - A), where A is the average of the highest and lowest temperature in °F.

What are Cooling Degree Days (CDD)?

For each day: max(0, A - 65), where A is the average of the highest and lowest temperature in °F.

What is a typical weather derivative product?

A forward contract or option on the cumulative CDD or HDD during a month.

Who typically uses weather derivatives and why?

Energy companies use them to hedge the volume of energy required for heating or cooling during a particular month.

What are the three main energy sources for derivatives?

Oil, natural gas, and electricity.

Where do oil derivatives trade?

In the OTC market (virtually all derivatives available on stocks/indices), plus on NYMEX and ICE.

What is a typical natural gas OTC contract?

Delivery of a specified amount of natural gas at a roughly uniform rate to a specified location during a month.

What makes electricity an unusual commodity for derivatives?

It cannot be stored.

What are some types of electricity derivative contracts?

5×8, 5×16, 7×24, daily or monthly exercise, and swing options.

How does an energy producer hedge risks using derivatives?

• Estimate relationship: Y = a + bP + cT + ε (where Y = monthly profit, P = energy prices, T = temperature)

• Take position of -b in energy forwards and -c in weather forwards

What are CAT bonds?

Catastrophe bonds - an alternative to traditional reinsurance issued by a subsidiary of an insurance company that pays higher-than-normal interest rates.

How do CAT bonds work?

If claims of a certain type exceed a certain level, the interest and possibly the principal on the bond are used to meet claims.

What are the potential advantages of futures options over spot options?

• Futures may be easier to trade than the underlying asset

• Exercise doesn't lead to delivery of the underlying asset

• Futures options and futures usually trade on the same exchange

• May entail lower transaction costs