1. Derivatives Contracts

In these notes we study the pricing, hedging and uses of financial derivatives or derivatives for short. A derivative is a financial instrument whose payoff depends on, or is derived from, the value of another financial asset such as a stock, a foreign currency, a futures, or another quantity such as volatility.

1.1. Payoffs and Pricing

As for any other financial instrument, the value of a derivative is the present value of its expected payoff. A positive payoff means that you receive money, whereas a negative payoff represents an outflow of money.

For many derivatives, the payoff is realized at maturity ^{1 1}For American type options, the payoff can be realized anytime before or at maturity. The early exercise of American options, though, makes them harder to price.. If \(S_{T}\) denotes the value of a stock at maturity, the payoff of a derivative written on the stock will be a function of \(S_{T}\) that we denote by \(f(S_{T})\). An important question that we answer in this class is how to price this type of derivative.

If we denote by \(\xi\) the relevant continuously compounded discount rate for the derivative, the value of the derivative is:

(1.1)\[\begin{equation} V = e^{-\xi T} \ev\left[f(S_{T})\right] \label{eq_physical_measure} \end{equation}\]

Even though this expression is correct, we do not know in general the right value for the discount rate \(\xi.\) As a matter of fact, for many derivatives the only way to know \(\xi\) would be to know the value of the derivative first ^{2 2}For options, for example, the relevant discount rate under the historical or physical measure depends on the moneyness and the maturity of the option..

The seminal work of Black and Scholes [1973] and Merton [1973] showed that it is possible to price derivatives by replication. That is, by continuously trading in the underlying asset and a risk-free bond, it is possible to generate the same payoffs as a derivative written on the asset. If that is the case, the value of the replicating portfolio today must be the value of the derivative asset to prevent arbitrage opportunities.

Their analysis also showed that replication works for any value of the discount rate used to price the underlying asset, that is, for any probability measure used to assign events of the underlying asset. A crucial implication of this observation then is that replication should also work in a world populated by risk-neutral individuals.

Under the risk-neutral probability measure, the value of the derivative can be computed as:

(1.2)\[\begin{equation} V = e^{-r T} \ev^{*}\left[f(S_{T})\right] \label{eq_risk_neutral_measure} \end{equation}\]

where \(r\) denotes the risk-free rate and the expectation is taken under the risk-neutral probability measure.

A crucial difference between \(\eqref{eq_physical_measure}\) and \(\eqref{eq_risk_neutral_measure}\) is the discount rate used to price the derivative. In \(\eqref{eq_physical_measure}\) we need to know both the discount rate of the underlying asset, in order to compute the expectation, and the discount rate of the derivative in order to compute its price. In \(\eqref{eq_risk_neutral_measure},\) on the other hand, we can discount the expected payoffs of any asset at the risk-free rate.

From equation \(\eqref{eq_risk_neutral_measure}\) we can see that to price a derivative it is important to understand what is a good proxy for the risk-free rate, and also how to compute the expected payoff under the risk-neutral measure.

For some derivatives such as forward contracts the payoff function is linear, which simplifies the computation of the expected payoff. For other derivatives such as options the payoff function is nonlinear and in general harder to price compared to linear payoffs.

Example 1.1 (A derivative with a linear payoff)

A forward contract is a commitment to purchase or sell an asset at maturity for a certain price \(K\). The payoff of a long forward is the difference between the price of the asset at maturity and the price agreed in the contract, that is, the payoff is a linear function of the stock price:

\[\begin{equation*} f(S) = S - K \end{equation*}\]

Because the payoff can be positive or negative depending of the sign of \(S\), it is possible for the value of the contract to be positive or negative. Usually the contract is designed so the value at inception is zero. Later on, the value of the contract will change and might become positive or negative.

Example 1.2 (A derivative with a nonlinear payoff})

An option gives the holder the right but not the obligation to purchase or sell an asset at maturity for a given price \(K\). The payoff of an option is a nonlinear function of the asset price at maturity.

For example, the buyer of a call option receives:

\[\begin{equation*} f(S) = \begin{cases} 0 & \text{if } S < K \\ S - K & \text{if } S \geq K \\ \end{cases} \end{equation*}\]

Since the payoff is non-negative, the holder of an option must pay a premium to the seller.

1.2. More Complex Derivatives

The financial engineering revolution in the 1990s revolve around the idea that we can package simpler derivatives together and build new products. The pricing of many of these more complex derivatives in general involves to know how to price the basic building blocks used to build the financial product.

1.2.1. Derivatives with Periodic Payments

A classical way to create a more complex and perhaps more useful derivative is to put derivatives with different expirations together, creating a product that involves the payment of cash flows periodically over time.

For example, interest rates swaps involve the exchange of a fixed interest rate for a floating interest rate, or vice-versa. They allow corporations to convert a loan with floating payments into a bond with fixed payments. Another example are credit default swaps (CDS) which involve the exchange of periodic payments in exchange for protection in case of a bond default.

Pricing derivatives with periodic payments is not harder than to price a derivative with a single payment. If we denote by \(p(\cdot)\) the pricing functional, we have that:

\[\begin{equation*} p(f(S_{t_{1}}) + f(S_{t_{2}}) + \ldots + f(S_{t_{n}})) = p(f(S_{t_{1}})) + p(f(S_{t_{2}})) + \ldots + p(f(S_{t_{n}})) \end{equation*}\]

This is the same logic that applies to bonds. The price of a bond must be the sum of the present value of its coupons and its face value; otherwise there would be a very simple arbitrage opportunity. In asset pricing we call this property the law of one price, which simply says that the pricing functional must be linear to prevent arbitrage opportunities.

1.2.2. Assets with Embedded Derivatives

Many assets such as corporate bonds have embedded derivatives. For example, many bonds found in financial markets are callable, that is, the issuer has the right to pay the bond holder the principal at any time before maturity. To value a callable bond involves modelling hte evolution of the whole term-structure of interest rates and analyze in which states of the world it is profitable to call the bond.

For many callable bonds the issuer has the right to call a bond at any time starting on the first date the bond is callable until its maturity. The call option embedded in these bonds is therefore an American type option. The pricing of these bonds requires using numerical techniques such as binomial trees since there is no close-form solution for the price of an American type option.

Other bonds are convertible into shares of the issuing company at a fixed price. Thus, convertible bonds contain a call option on the company stock which might be very valuable.

1.2.3. Do We Need Other Payoffs?

In theory, we could design a derivative with any payoff function \(f(S)\) that one might think off. For example, we could choose \(f(S) = S^{2}\) or \(f(S) = \ln(S)\).

It turns out that with forwards and options it is possible to build any type of payoff that a trader might want. We will see that by having options and forwards with different strikes we can complete the market. Combining options and forwards together is usually called options strategies.

1.3. Derivatives Markets

1.3.1. Uses of Derivatives

Derivatives allow investors to obtain payoffs that might be useful to achieve certain objectives. For example, some commodity producers use derivatives to hedge their future production by fixing today the price at which they will sell in the future.

Other traders like derivatives because they can obtain custom design payoffs that allow them to speculate in very specific ways.

Therefore, derivatives make both types of traders, hedgers and speculators, better off by expanding their trading opportunity set and thus increasing their utility.

1.3.2. The Market for Derivatives

Derivatives are always in zero-net supply. For every long position there is a corresponding short position. As such, the demand for derivatives can be positive or negative depending on the intentions of the traders generating the demand.

The demand for derivatives comes from buy-side traders that want to use derivatives for either hedging or speculative purposes. For example, many hedgers in commodity markets are commodity producers that want to fix the price at which they can sell their production. To hedge their exposure using futures contracts involves selling futures to whoever is willing to take the opposite side of the trade.

The net demand, which can be positive or negative, is balanced by sell-side investors or market makers that provide liquidity to the rest of the market.

1.3.3. Pricing and Hedging of Derivatives

In many cases market makers will hedge their exposure by dynamically trading the underlying asset and risk-free bonds. One of the main results in modern asset pricing is that a perfectly hedged portfolio should earn the risk-free rate of interest. Otherwise there would be an arbitrage opportunity.

Therefore, in order to price an option or a forward contract we need to learn how to hedge or replicate the position first.

For certain derivatives such as options, the hedging recipe depends heavily on the modelling of the stock price evolution over time. In modelling the evolution of the underlying asset, time can be seen as either discrete or continuous. The choice of how to model time usually depends on how difficult is to solve the model if time is assumed to be continuous.

The distribution of random shocks will affect the evolution of stock prices over time. Some models such as the geometric Brownian motion assume that the instantaneous rate of return is normally distributed. It is possible, though, to introduce more complex models such as stochastic volatility or jumps to make the modelling closer to what is observed in real markets.

1.4. References

[BS73]

Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81(3):637–654, 1973.

[Mer73]

Robert C Merton. Theory of rational option pricing. The Bell Journal of Economics and Management Science, pages 141–183, 1973.

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