The major questions addressed in this reading are:
This is a recap of what we saw in the previous reading.
Forward contract: It is a customized over-the-counter derivative contract in which two parties agree that one party, the buyer, will purchase an underlying asset from the other party, the seller, at a later date, at a fixed price they agree upon when the contract is signed.
Futures contract: It is a standardized derivative contract created and traded on a futures exchange in which two parties agree that one party, the buyer, will purchase an underlying asset from the other party, the seller, at a later date at a price agreed upon by the two parties when the contract is initiated. There is also a daily settling of gains and losses and a credit guarantee by the futures exchange through its clearinghouse.
Swap contract: It is an over-the counter derivate contract in which two parties agree to exchange a series of cash flows whereby one party pays a variable series that will be determined by an underlying asset or rate, and the other party pays either 1) a variable series determined by a different underlying asset or rate or 2) a fixed series.
Option contract: It is a derivative contract in which one party, the buyer, pays a sum of money to the other party, the seller or writer, and receives the right to buy or sell an underlying asset at a fixed price either on a specific expiration date or at any time prior to the expiration date.
The price or value of a financial asset is the expected future price plus any benefits such as dividends or coupon interest minus costs discounted at a rate appropriate for the risk assumed. The exhibit below shows how to get the current price of an asset by discounting its expected future price , by the risk-free rate r, plus the risk premium , over the period from 0 to T.
Source: CFA Program Curriculum, Basics of Derivative Pricing and Valuation
Let us now decompose this expression and understand what each of the terms mean.
Arbitrage is a type of transaction undertaken when two assets or portfolios produce identical results but sell for different prices. Let us take the example below:
Source: CFA Program Curriculum, Basics of Derivative Pricing and Valuation
Interpretation:
Arbitrage opportunities are exploited quickly and hence do not last for long. In the above example, when the arbitrage opportunity exists, the demand for A will go up causing its price to go up. Similarly, when more people sell B, its demand goes down, leading to a decrease in its price causing the prices of both the assets to come to the same level. It is not just the difference in price that matters; one needs to consider the transaction costs as well. If the transaction costs exceed the benefit from an arbitrage opportunity, then it is not worth exploiting.
Law of one price: If two assets have the same expected return in the future, then their market price today must be the same.
Arbitrage and Derivatives
The price of a derivative is tied to the price of the underlying. For example, if the price of a stock goes up, then a call option on the stock also goes up. Therefore, a derivative can be used to hedge an underlying, or vice versa. For example, if you are long on Google stock, then you can either short the call option or buy a put option to eliminate your risk.
Since the risk is eliminated, the expected return is the risk-free return.
A derivative must be priced such that no arbitrage opportunities exist, and there can be only one price for the derivative that earns the risk-free return. If it earns a return in excess of the risk-free rate, then arbitrage opportunities exist for (underlying + derivative) position.
Arbitrage and Replication
An asset and a derivative can be combined to produce a risk-free bond in one of the ways shown below. Conversely, an asset and the risk free asset can be combined to produce a derivative.
There are two ways to replicate when the underlying asset is a stock:
Replication is the creation of an asset or portfolio from another asset, portfolio, and/or derivative. Why is replication needed? Isn’t it easier to just buy a government security to earn the risk-free rate instead of buying the asset and a derivative? There are some situations under which replication is valuable:
Risk Aversion, Risk Neutrality, and Arbitrage-Free Pricing
Risk aversion: Most investors are risk-averse and expect a compensation (risk premium) for assuming risk. As we have seen earlier, a derivative can be combined with an asset to mitigate the risk and produce a risk-free asset. We saw in the previous section what factors affect a derivative’s pricing. Since risk aversion of an investor does not impact derivative pricing, one can derive the derivative price assuming investors are risk-neutral. When pricing derivatives, use the risk-free rate, i.e. the price of a derivative can be calculated by discounting it at the risk-free rate rather than the risk-free rate plus a risk premium.
When pricing assets, a risk premium is added to the risk-free rate. Recall the first term of current price S0 of an asset is where λ is the risk premium. This means that the asset’s price in the spot market factors in the risk aversion of an investor. Since the risk aversion is already captured in the asset pricing, it is not included in a derivative’s pricing.
Risk-neutral pricing: Derivatives pricing is sometimes called risk-neutral pricing because there is only one derivative price, which combined with the underlying asset, can earn the risk-free rate.
Arbitrage-free pricing: The overall process of pricing derivatives by arbitrage and risk-neutrality is called arbitrage-free pricing. It is also called the principle of no arbitrage. Limitations to execute arbitrage transactions include:
Clearinghouses do not place any restrictions on transactions that can be arbitraged.
A hedge portfolio is one that eliminates arbitrage opportunities and implies a unique price for a derivative.
This section lays the foundation for the subsequent sections. We look at two terms, price and value, and what they mean in context to different assets/derivatives.
Stock price vs. value
Option contract price versus value
Forwards, futures, and swaps