Financial Derivatives

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2025

Definitions

In this class 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.
The value of a derivative is then the discounted value of its payoff.
- Linear payoffs are easier to price.
- Non-linear payoffs are harder to value.
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.
- Time 0 is where we are right now.
- Time T is when the derivative expires.
If S_{T} denotes the value of a stock at maturity, the payoff of the derivative is a function of S_{T} denoted as f(S_{T}).
An important question that we answer in this class is how to price this derivative.

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: f(S) = S - K
- Typically 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.

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: f(S) = \begin{cases} 0 & \text{if } S < K \\ S - K & \text{if } S \geq K \\ \end{cases}
Since the payoff is non-negative, the holder of an option must pay a premium to the seller.

Some derivatives involve 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.
Another example is credit default swaps (CDS) which involve the exchange of periodic payments in exchange for protection in case of a bond default.

It is also possible to embed derivatives to simpler assets such as bonds.
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.
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.

In theory, we could design a derivative with any payoff function f(S).
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.

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.

Buy-side traders buy or sell derivatives for either hedging or speculative purposes.
The net demand, which can be positive or negative, is balanced by sell-side traders or market makers that provide liquidity to the rest of the market.
For market makers to hedge their exposure, they might need to trade the underlying asset and risk-free bonds dynamically.

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 options, the hedging recipe depends heavily on the modelling of the stock price evolution over time.
- Time can be seen as either discrete or continuous.
- The distribution of random shocks will affect the evolution of stock prices over time.