###### Exotic Options

After completing this reading, you should be able to: Define and contrast exotic... **Read More**

After completing this reading, you should be able to:

- Explain the key differences between commodities and financial assets.
- Define and apply commodity concepts such as storage costs, carry markets, lease rate, and convenience yield.
- Identify factors that impact prices on agricultural commodities, metals, energy, and weather derivatives.
- Explain the basic equilibrium formula for pricing commodity forwards.
- Describe an arbitrage transaction in commodity forwards and compute the potential arbitrage profit.
- Define the lease rate and explain how it determines the no-arbitrage values for commodity forwards and futures.
- Describe the cost of carry model and illustrate the impact of storage costs and convenience yields on commodity forward prices and no-arbitrage bounds.
- Compute the forward price of a commodity with storage costs.
- Explain how to create a synthetic commodity position and use it to explain the relationship between the forward price and the expected future spot price
- Explain the relationship between current futures prices and expected future spot prices, including the impact of systematic and nonsystematic risk.
- Define and interpret normal backwardation and contango.

Throughout this chapter, we will assume the daily settlement of futures. This implies that futures and forward contracts will be treated as one and the same thing.

With the exception of a few commodities like gold, most commodities are held as consumption assets and not just as investment assets. Commodity assets are held for the purposes of being used in some way, after which they cease to be available for sale.

$$ \begin{array}{l|l} \textbf{Commodities} & \textbf{Financial Assets} \\ \hline \text{Storage costs are present.} & \text{Negligible storage costs.} \\ \hline \begin{array}{l} \text{Commodities are costly to} \\ \text{transport. Prices may reflect the} \\ \text{cost of transport.} \end{array} & \begin{array}{l} \text{No transport costs as they are} \\ \text{transported electronically.} \end{array} \\ \hline \begin{array}{l} \text{A higher lease rate when} \\ \text{commodities held for} \\ \text{investment purposes are} \\ \text{borrowed.} \end{array} & \begin{array}{l} \text{Lower fees charged when} \\ \text{financial assets are borrowed} \\ \text{for shorting.} \end{array} \\ \hline \text{Returns do not reflect the risk.} & \text{Returns reflect the risks.} \\ \end{array} $$

Agricultural commodities are difficult to store. There is an observable interdependence among agricultural commodities, i.e., livestock feed on plants. As such, they have seasonal prices – low prices at harvest time and high prices as storage costs of the products increases. That is, the prices of agricultural products are seasonal.

The prices of agricultural commodities are influenced by:

**Political considerations****Market factors: F**or example, the presumption of a good harvest may lower the prices**Weather conditions:**Extreme weather, e.g., strong winds, can cause destruction to crops, resulting in a decrease in supply, which may lead to an increase in the prices of agricultural commodities.

Commodities under this category include copper, aluminum, zinc, lead, nickel, platinum, gold, silver, and palladium.

As compared to agricultural commodities, their prices are not seasonal, and metal prices are not affected by the weather. Also, the cost of storing metals is relatively cheaper as compared to that of storing agricultural commodities. Most metals are held purely for investment purposes.

The prices of metals depend on:

- The rate at which new sources of extracting metals are discovered.
- Exchange rates
**:**Applicable in metals that are discovered in one country and sold in another country. - The number of uses of a specific metal.
- Changes in the methods of extraction of the metals
- Government actions.
- Environmental regulations.
- Recycling processes can, at times, affect metal prices.

Futures contracts are traded on crude oil (which is considered the largest commodity market in the world) and crude oil extracts, natural gas, and electricity.

**Crude oil**: Available in many grades and has a high global demand. Transportation of crude oil is expensive, making the prices vary regionally.**Natural gas**: Used for either heating or generating electricity. Since it is stored below or above the ground, the storage costs are high. The prices of natural gas are seasonal depending on demand. Demand is high during cold seasons and low during hot seasons.

Future contracts on electricity are traded in both the OTC and exchange-traded markets. One party of the futures contract receives a specific number of megawatts for a specified period in a specified location at a specified time. Even though futures contracts on electricity exist, they are not traded as actively as the futures contract on crude oil and natural gas.

Electricity differs from other commodities since it is almost not possible to store it. Due to its non-storability, electricity is prone to huge fluctuations in price. The price of electricity mainly depends on:

The price of electricity mainly depends on:

- The price charged at each of the generating stations; and
- High demand. For example, as electricity will be needed for air conditioning and heating in hot or cold seasons, the prices go up.

Future contracts on weather are traded in both the OTC and the exchange-traded markets.

We have two important weather variables which can be defined as:

HDD (Heating Degree Days) = \(max(0, 65 – A)\)

CDD (Cooling Degree Days) = \(max(0, A – 65)\)

Where \(A=1/2(\text{Highest + Lowest}\) temperature in a day at a specific weather station

Despite some commodities having industrial uses, they may be held strictly for investment. Traders owning metals for investment can substitute physically owning the metals to owning futures and forward contracts on the commodities. Such metals have negligible storage costs. They can also be borrowed at a lease rate.

Ignoring lease rates,

$$ F=S(1+r)^T $$

Where \(T\)=Time to maturity, and

\(r\) = Risk-free rate.

If \(F>S(1+r)^T\), to maximize profits, a trader can buy the investment commodity at the spot prices \(S\) and at the same time enter into a forward contract to sell it at maturity \(T\).

If \(F<S(1+r)^T\), to maximize profits, a trader who owns an investment commodity can sell it at a spot price \(S\) and enter into a forward contract to buy it at maturity \(T\).

A lease rate can be defined as the interest rate charged for borrowing the underlying asset.

In the previous chapter, we looked at the forward price formula for the known-yield case, which is given by:$$F=S\left( \frac{1+R}{1+Q} \right)^{T}$$Where \(F\) is the forward price, \(S\) is the spot price, \(R\) is the risk-free rate (with annual compounding), and Q is the annual yield.

Now let \(L\) be the lease rate so that we have:

$$F=S\left( \frac{1+R}{1+L} \right)^{T}$$

Solving for \(L\), we get:

$$L=\left( \frac{S}{F}\right)^{\frac{1}{T}}(1+R)-1$$

Assume that the spot price of petroleum is USD 1,200 and the 2-year futures price is 1280, and the annually compounded risk-free rate is 5% per year. What is the implied lease rate?

$$\begin{align*}L&=\left( \frac{S}{F}\right)^{\frac{1}{T}}(1+R)-1\\&=\left( \frac{1200}{1280}\right)^{\frac{1}{2}}(1.05)-1\\&=0.01665 \end{align*}$$

**Convenience yield** is the additional value that comes with holding the asset rather than having a long forward or futures contract on the asset. A good example of a consumption asset that has a convenience yield is oil. If you hold oil, you’ll have the convenience of selling it at a higher price during a shortage. Convenience yield can be considered as the rate of borrowing or the rate that would have been received with physical possession of the asset. It is, thus, arguably, the rate that should be charged to borrow it.

Convenience yield, \(Y\), should satisfy the equation:

$$ F=(S+U)×\left(\frac{1+R}{1+Y}\right)^T$$

So that,

$$Y=\left( \frac{S+U}{F}\right)^{\frac{1}{T}}(1+R)-1$$

Where \(U\) is the present value of storage costs,\(F\) is the forward price, \(S\) is the spot price,\(R\) is the risk-free rate(with annual compounding) and \(Y\) is the convenience yield.

Assume that the spot price of petroleum is USD 120 per barrel and the 2-year futures price is 100 per barrel, the present value of storing petroleum for 2 years is USD 5, and the annually compounded risk-free rate is 5% per year. What is the implied convenience yield?

Convenience yield, Y, should satisfy the equation:

$$ F=(S+U)×(\frac{1+R}{1+Y})^T $$

So that,

$$Y=\left( \frac{S+U}{F}\right)^{\frac{1}{T}}(1+R)-1=\left( \frac{120+5}{100}\right)^{\frac{1}{2}}(1.05)-1=0.1739 or 17.39\%$$

A readily available asset will have **zero convenience yield** as delivery can be made almost immediately. Thus its future price will be obtained by:

$$ F=(S+U)×(1+R)^T $$

In the presence of delivery delays/shortages, convenience yield will be high and:

$$ F<(S+U)×(1+R)^{T} $$

From example 1 above, assume that the forward price is unknown and that the convenience yield is 17.39%.

Then, the forward price can be determined using the formula:

$$ F=(S+U)×(\frac{1+R}{1+Y})^T =(120+5)×(\frac{1.05}{1.1739})^{2} =USD 100$$

Storage costs are a negative income. Traders incur storage costs of \(U(1+R)^T\) for a present value of \(U\).

Cost of carry encompasses the costs of storage, the costs of financing, and the income to be earned on the asset. Remember that financial assets lack storage costs.

Assuming that financial costs are R and the yield Q, the cost of carry will be \(\frac{1+R}{1+Q}-1\) which is approximately equal to \(R-Q\) (if R and Q are continuously compounded).

As such, the future value of the asset will be the spot price, S, continuously compounded by the difference between the financial costs R and the yield Q multiplied by the time to maturity of the contract:

$$ F=Se^{(R-Q)T} $$

$$ \text{for continuously compounded R and Q} $$

In the presence of storage costs,

$$ F=Se^{(C-Y)T} $$

$$\text{where C is the cost of carry and Y is the convenience yield (both expressed with continuous compounding)} $$

Futures prices reflect the spot prices of a commodity in the future. As the maturity of the contract approaches, the futures price converges to the spot prices. Traders take long futures positions to maximize profits if the spot price at maturity is greater than the current spot price and short futures positions if the spot price at maturity is lesser than the current spot price.

However, to ensure that these profits are realized, traders should close out the futures contracts as the time to maturity nears.

Systematic risk is defined as a risk that is dependent on market factors and cannot be diversified. Unsystematic risk, on the other hand, is a risk that can be diversified.

The Capital Asset Pricing Model (CAPM) argues that the return on investment should exceed the risk-free interest rate provided the systematic risk on a portfolio is positive (positive correlation between the assets returns and the market returns)

In the presence of a negative correlation between the asset and the market returns, the returns on the asset will be less than the market returns.

If there is no correlation between the asset and the market returns, the portfolio is considered to be a well-diversified portfolio and will be considered to have no risk.

Assume that:

P = Present value of the futures time discounted from T to 0 at the risk-free rate

R = Risk-free interest rate compounded annually

T = Time to maturity

F = Futures price of an asset

S = Spot price of an asset

Then,

$$ P=\frac{F}{(1+R)^T} $$

A trader should invest P at the risk-free interest rate so as to get F upon maturity.

To create a long futures position, the trader can invest P at the risk-free interest rate and at the same time enter into a long futures contract to buy F at maturity. The cash flows from this strategy will be –P at time 0 and +St at time T, assuming that St is the spot price at time T.

Suppose E denotes the expected value and X the expected returns compounded annually, the expected cash flow at maturity T is, therefore, E(St):

$$E(S_T)=P(1+X)^T$$

and we have seen earlier that,

$$ P=\frac{F}{(1+R)^T} $$

Therefore,

$$ E(S_T)=F \frac{(1+X)^T}{(1+R)^T} $$

This shows that the systematic risk of an investment depends on the correlation between the asset and the market returns.

If the correlation is positive, \(X>R\) and thus \(E(S_T)>F\).

If the correlation is negative, \(X<R\) and thus \(E(S_T)<F\).

If there is no correlation, the futures price will equal the expected future spot price.

*Note*: These results apply to Fx forwards and futures, financial forwards and futures, and commodity futures.

Suppose that the dividends obtained from an index are reinvested in the index, the index will grow at a rate of Q, giving the value of the investment at maturity T as:

$$ F=S \frac{(1+R)}{(1+Q)}^T (1+Q)^T = S (1+R)^T $$

The investor’s return will be greater than the risk-free rate since the index is positively correlated to itself. Thus, the expected value of the index at T>F.

**Backwardation** refers to a situation where the futures price is **below** the spot price. It occurs when the benefits of holding the asset outweigh the opportunity cost of holding the asset as well as any additional holding costs. A backwardation commodity market occurs when the lease rate is greater than the risk-free rate.

**Contango** refers to a situation where the futures price is **above** the spot price. It is likely to occur when there are no benefits associated with holding the asset, i.e., zero dividends, zero coupons, or zero convenience yield. A contango commodity market occurs when the lease rate is less than the risk-free rate.

## Question

The current spot price of a bag of \(corn\) is \($10\). There exists an active lending market for corn, where the annual lease rate is equal to \(8\%\), the effective annual risk-free rate is equal to \(10\%\), and the \(1-year\) forward price for corn is \($10.35\) per bag. Does arbitrage exist? What’s the risk-free profit up for grabs if indeed an arbitrage opportunity is available?

- No; risk-free profit = $0
- Yes; risk-free profit = $0.35
- Yes; risk-free profit = $0.08
- Yes; risk-free profit = $0.15
The correct answer is

D.An arbitrage position exists if the forward price

is not equivalentto the expected spot price.$$ \text{Expected spot price in 1 year}=S_{ 0 }{ \left(\frac{1+R}{1+δ} \right)^T }$$

Where:

\(S_{ 0 }\)=commodity spot price

\(r\)=riskfree rate

\(\delta\)=lease rate

\(T\)=time between today and the future date at which the transaction will occur, i.e, maturity

$$ =10 \left(\frac{1.10}{1.08} \right)^1=10.19$$

Since 10.35 is greater than 10.19, arbitrage exists.

To take advantage of this opportunity, an arbitrageur can make the following moves:

At initiation,

- Borrow \($10\) at the rate of \(10\%\)
- Buy a bag of corn at \($10\)
- Go short on a corn futures contract
- Lend the bag of corn at \(8\%\)
At maturity,

- Take back the bag of corn plus proceeds from the lease amounting to \($0.8(=10 × 1.08-10)\)
- Deliver the bag of corn; receive \($10.35\)
- Repay borrowed funds amounting to \($11(=10 × 1.10 )\)
- Net profit = \(10.35 + 0.8 – 11= $0.15\)