Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

Derivatives

Costs and Benefits Associated with Owning the Underlying

Learning Outcome Statement:

Explain the difference between the spot and expected future price of an underlying and the cost of carry associated with holding the underlying asset.

Summary:

The learning outcome statement focuses on understanding the relationship between spot prices, forward prices, and the cost of carry for an underlying asset. It discusses how costs and benefits associated with owning the underlying asset, such as dividends, interest, or storage costs, influence the forward price relative to the spot price. The content also covers the impact of risk-free rates, including scenarios with negative rates, and how these factors integrate into pricing derivatives through no-arbitrage conditions.

Key Concepts:

Cost of Carry

Cost of carry refers to the net costs and benefits of holding an underlying asset over a period. This includes opportunity costs represented by the risk-free rate and any additional costs or benefits such as storage or dividends.

Spot vs. Forward Price

The relationship between spot and forward prices is influenced by the cost of carry. If the asset has no associated cash flows other than the risk-free rate, the forward price is calculated by compounding the spot price at the risk-free rate over the period until delivery.

Impact of Negative Risk-Free Rates

When the risk-free rate is negative, the forward price of an asset is lower than the spot price, as the compounding factor (1 + r)^T becomes less than 1.

Adjustments for Costs and Benefits

If there are additional costs (like storage) or benefits (like dividends) associated with the asset, these are factored into the forward price to prevent arbitrage opportunities. Costs increase the forward price, while benefits decrease it.

Formulas:

Basic Forward Price Formula

F_0(T) = S_0(1 + r)^T

Calculates the forward price of an asset assuming no additional costs or benefits and compounding at a risk-free rate.

Variables:

$F_0(T)$ :: Forward price at time T
$S_0$ :: Current spot price
$r$ :: Risk-free rate
$T$ :: Time until delivery in years

Units: Currency

Continuous Compounding Forward Price Formula

F_0(T) = S_0 e^{rT}

This formula uses continuous compounding to calculate the forward price, applicable when dealing with continuous rates.

Variables:

$F_0(T)$ :: Forward price at time T
$S_0$ :: Current spot price
$r$ :: Risk-free rate
$T$ :: Time until delivery in years

Units: Currency

Adjusted Forward Price with Costs and Benefits

F_0(T) = [S_0 - PV_0(I) + PV_0(C)](1 + r)^T

Adjusts the basic forward price formula to account for present value of costs and benefits associated with the asset.

Variables:

$F_0(T)$ :: Forward price at time T
$S_0$ :: Current spot price
$PV_0(I)$ :: Present value of benefits at time 0
$PV_0(C)$ :: Present value of costs at time 0
$r$ :: Risk-free rate
$T$ :: Time until delivery in years

Units: Currency

Continuous Compounding with Costs and Benefits

F_0(T) = S_0 e^{(r+c-i)T}

This formula calculates the forward price under continuous compounding, adjusting for rates of costs and benefits over the life of the asset.

Variables:

$F_0(T)$ :: Forward price at time T
$S_0$ :: Current spot price
$r$ :: Risk-free rate
$c$ :: Cost rate
$i$ :: Income rate
$T$ :: Time until delivery in years

Units: Currency

Replication

Learning Outcome Statement:

explain how the concepts of arbitrage and replication are used in pricing derivatives

Summary:

The concept of replication in derivative pricing involves creating a cash flow stream similar to a derivative using a combination of positions in underlying assets and cash borrowing or lending. This strategy is used to mirror or offset a derivative position under the law of one price, where no riskless arbitrage opportunities exist. The learning module also discusses the relationship between spot and forward prices under no-arbitrage conditions, and how replication can be used to achieve the same cash flows at future times regardless of market conditions.

Key Concepts:

Replication

Replication involves recreating the cash flow of a derivative by taking positions in the underlying asset and engaging in borrowing or lending. It is used to offset or mirror a derivative position when no arbitrage opportunities are present.

Arbitrage

Arbitrage involves taking advantage of price discrepancies in different markets or forms to earn a riskless profit. In the context of derivatives, it involves strategies like selling high in one market and buying low in another simultaneously.

No-arbitrage condition

This condition implies that the forward price should be set such that no riskless profit can be made. It is typically represented by the formula where the forward price equals the future value of the spot price compounded at the risk-free rate.

Law of one price

This economic principle states that identical assets should sell for the same price in efficient markets. It underpins the replication and arbitrage strategies by ensuring that the prices do not diverge without the possibility of riskless profit.

Formulas:

No-arbitrage forward price

F_0(T) = S_0(1 + r)^T

This formula calculates the forward price of an asset under no-arbitrage conditions, assuming no costs or benefits associated with holding the asset, other than the opportunity cost represented by the risk-free rate.

Variables:

$F_0(T)$ :: Forward price of the asset at time T
$S_0$ :: Spot price of the asset at time 0
$r$ :: Risk-free rate of interest
$T$ :: Time in years to the delivery date

Units: currency

Continuous compounding forward price

F_0(T) = S_0 e^{rT}

This formula provides the forward price under continuous compounding, which is used when calculating continuous returns or costs over time.

Variables:

$e$ :: Base of the natural logarithm
$rT$ :: Product of the risk-free rate and time

Units: currency

Arbitrage

Learning Outcome Statement:

explain how the concepts of arbitrage and replication are used in pricing derivatives

Summary:

The learning outcome statement focuses on explaining the use of arbitrage and replication in the pricing of derivatives. Arbitrage involves exploiting price discrepancies to make riskless profits, while replication involves recreating a derivative's cash flow using a combination of positions in underlying assets and cash flows. Both concepts are crucial in establishing fair pricing and market efficiency in derivatives markets.

Key Concepts:

Arbitrage

Arbitrage is the practice of taking advantage of a price difference between two or more markets, striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices. It involves buying an asset in one market at a lower price and simultaneously selling it in another market at a higher price.

Replication

Replication is a strategy used to recreate the cash flow of a derivative by taking positions in the underlying assets and borrowing or lending. It is used to mirror or offset a derivative position when no arbitrage opportunities exist, ensuring that the derivative's price accurately reflects its intrinsic value based on the underlying assets.

Law of One Price

This economic rule states that identical assets should sell for the same price in efficient markets. Arbitrage opportunities arise when this law does not hold, allowing traders to buy the asset at a lower price in one market and sell it at a higher price in another.

Risk-Free Rate

The risk-free rate is the theoretical rate of return of an investment with zero risk. It is used as the discount rate in the present value calculations for pricing derivatives, particularly when determining the fair forward price of an asset.

Formulas:

Future Value with Discrete Compounding

FVN = PV(1 + r)^N

This formula calculates the future value of a cash flow after N periods, compounding at a periodic interest rate r.

Variables:

$FVN$ :: Future value after N periods
$PV$ :: Present value
$r$ :: Interest rate per period
$N$ :: Number of compounding periods

Units: currency units

Future Value with Continuous Compounding

FVT = PV e^{rT}

This formula calculates the future value of a cash flow using continuous compounding, where compounding occurs an infinite number of times per period at intervals that are infinitesimally small.

Variables:

$FVT$ :: Future value at time T
$PV$ :: Present value
$r$ :: Continuous compounding rate
$T$ :: Time in years

Units: currency units

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