Financial Derivatives

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2026

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.

A More Formal Definition

• 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.

Pricing Under the Physical Measure

• If \xi denotes the continuously compounded discount rate for the derivative, its value is: V = e^{-\xi T} \operatorname{E}\left[f(S_{T})\right]
• The problem: we do not generally know \xi.
  • For many derivatives, the only way to know \xi is to already know V, a circular problem.

The Replication Approach

• Black and Scholes (1973) and Merton (1973) showed that continuously trading in the underlying asset and a risk-free bond can replicate the payoff of any derivative written on the asset.
• If the payoff can be replicated, the derivative’s value must equal the cost of the replicating portfolio. Otherwise there is an arbitrage opportunity.
• Crucially, replication works for any probability measure.
  • Since the physical measure encodes investor risk preferences and expected returns, derivative prices are independent of both.

Pricing Under the Risk-Neutral Measure

• Since we are free to choose any measure, we work under the risk-neutral measure, where all assets earn the risk-free rate.
• Under this measure, the value of any derivative simplifies to: V = e^{-rT} \operatorname{E}^{*}\left[f(S_{T})\right]
• A key difference from the physical measure: we discount at r regardless of the derivative’s risk, and we need only specify the dynamics of S_T under the risk-neutral measure.

Example: Derivative with 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: 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.

Example: Derivative with 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: 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.

Derivatives with Periodic Payments

• 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.

Assets with Embedded Derivatives

• Many assets have embedded options: derivatives built into an otherwise standard instrument.
• Callable bonds: the issuer can repay principal before maturity.
  • The issuer holds the call option.
  • Callable bonds trade at a lower price than an equivalent plain-vanilla bond.
• Convertible bonds: the holder can convert the bond into shares at a fixed price.
  • The bondholder holds the call option on the company’s stock.
  • Convertible bonds trade at a higher price than an equivalent plain-vanilla bond.

Do We Need Other Payoffs?

• 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).
• With bonds, forwards, and options at every strike, it is possible to build any payoff that depends on the asset price at maturity.
• In this sense, options with different strikes complete the market.
• Only finitely many strikes trade, so options strategies, combinations of options with different strikes and maturities, approximate the payoffs traders want.

Purposes 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.

Why Firms Hedge

• Shareholders can adjust their own exposure to commodity prices, so why should the firm hedge?
• Froot et al. (1993): outside financing costs more than internal funds.
• Hedging keeps a drop in cash flow from forcing the firm to cut good investments.

The Market for Derivatives

• 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.

Pricing and Hedging of Derivatives

• 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.

References

Black, Fischer, and Myron Scholes. 1973. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy 81 (3): 637–54.

Froot, Kenneth A., David S. Scharfstein, and Jeremy C. Stein. 1993. “Risk Management: Coordinating Corporate Investment and Financing Policies.” Journal of Finance 48 (5): 1629–58.

Merton, Robert C. 1973. “Theory of Rational Option Pricing.” The Bell Journal of Economics and Management Science, 141–83.