Derivative Pricing Principles and Cost of Carry | CFA Level I Derivatives

In this lesson, we’ll explore the basics of derivative pricing, starting with the principles for underlying assets.

Underlying Asset Pricing Factors

Derivatives derive their value from the performance of their underlying assets. To understand how derivatives are priced, let’s first look at how underlying assets are priced. While specifics may vary, all underlying asset pricing is based on four factors: expectations, risk, benefits, and costs. Let’s examine each of these factors.

1. Expectations

Assuming no benefits or costs are associated with holding a particular asset, an investor only needs to estimate the expected future price based on the current spot price. There is uncertainty in the expected future price, but the investor makes their best prediction at time 0 for the spot price at time T.

For an asset with negligible holding costs or benefits, its expected future price is:
E(S_T) = S₀ x (1+k)^T
k: discount rate

Rearranging, S₀ = E(S_T) / (1+k)^T

2. Risk

What discount rate should be applied? At a minimum, it should include the risk-free rate of interest (RF). The challenge is determining the risk premium. To do that, let’s discuss the concept of risk aversion briefly. In finance theory, investors can be risk-averse, risk-neutral, or risk-seeking.

Risk-neutral investors are indifferent between a risk-free and risky asset with the same yield. Risk-seeking investors prefer the risky asset, despite the same expected return. Risk-averse investors prefer the risk-free asset since there is no incentive in taking on risk.

For our exercise, we’ll assume that all investors are risk-averse.

To entice a risk-averse investor to take risk, the risky asset should be priced at a discount relative to the risk-free asset. This means the investor expects additional return in the form of a risk premium, which must be sufficient to compensate for the risk. Since we assume investors are risk-averse, this risk premium must be greater than zero.

For an asset like a zero-coupon bond without interim costs and benefits, the current price should be the expected future price discounted by the sum of the risk-free rate and the risk premium.

3. Benefits of Holding an Asset

Assets with interim benefits or costs must consider these factors in valuation. Benefits can be monetary (e.g., dividend or interest payments) or non-monetary (convenience yield).

If the potential benefits are certain, the monetary value of these benefits should be discounted back to time zero using the risk-free rate plus the risk premium. The present value of the benefits should be added to the spot price.

4. Costs of Holding an Asset

Costs of holding an asset, particularly for commodities, can include storage costs and costs incurred in protecting and insuring the asset. These costs can be significant. Similarly, the monetary value of such costs should be discounted back to time zero. The present value of the costs should be subtracted from the spot price.

The net cost of holding an asset, considering both the costs and benefits of holding the asset, is known as the net cost of carry.

Considering all costs and benefits, we can describe the relationship between the current spot price of an asset with these four elements:

• Expectations
• Risk
• Benefits
• Costs

Linking Derivative Pricing to Underlying Asset Pricing

Understanding how underlying assets are priced in the spot market is critical to understanding how derivatives are priced. To grasp derivative pricing, we must establish a connection between the derivative market and the spot market of the underlying. That linkage occurs through arbitrage.

Arbitrage and the Law of One Price

Arbitrage is based on the law of one price. This law states that two securities or portfolios with identical future cash flows should have the same price, regardless of future events. If this isn’t the case, an arbitrage opportunity exists.

EXAMPLE
Imagine securities X and Y, which are identical, trading at different prices—X at $5 and Y at $4. An arbitrageur could sell X for $5 and buy Y for $4, making an immediate $1 profit. The actions of arbitrageurs will eventually cause the prices to converge, eliminating the arbitrage opportunity.

Derivatives, Hedged Portfolios, and Risk-Neutral Pricing

Derivatives allow investors to construct hedged portfolios by taking a position in the underlying asset and the opposite position in the derivative. This creates a portfolio with no uncertainty about its value at a future date.

For example, consider an asset trading at $100 and a forward contract expiring in one year priced at $103. An investor could long the asset at $100 and simultaneously short the forward contract at $103, earning a 3% return without taking any risk other than counterparty risk.

This example demonstrates that while investor risk aversion is relevant to pricing assets, it isn’t relevant to pricing derivatives. As a result, derivative pricing is sometimes called risk-neutral pricing.

No-Arbitrage Derivative Price

Based on the principles of arbitrage and risk-neutral pricing, there can only be one price for a derivative: the no-arbitrage derivative price. This price must satisfy the following equation:

Position in the underlying asset + Opposite position in the derivative = PV of the net payoff at time T

Note that this is based on the assumption of no benefit or cost associated with holding the underlying asset.

Replication

Replication is the process of creating an asset or portfolio from another asset, portfolio, and/or derivative. In our example, we showed how a long forward position in an asset can be replicated by borrowing at the risk-free rate to buy the asset.

At the initiation of a forward contract, the investor doesn’t have any cash outflow. In contrast, the investor can replicate the long forward position by borrowing $100 at the risk-free rate and using the money to purchase the asset, also resulting in a net cash flow of 0 at time 0.

At expiration, the net cash flows for both portfolios (forward contract and replicated portfolio) are identical, illustrating how a long forward can be replicated by buying an asset and borrowing at the risk-free rate.

Conclusion

In this lesson, we covered some fundamental concepts of derivative pricing, including arbitrage, replication, and the cost of carry. We’ll apply these concepts in the following lessons on pricing forwards, swaps, and options.