Options, Futures, and Other Derivatives summary
3rd Party summary of John C. Hull’s textbook
Financial Derivatives Overview
Key Concepts:
• Derivatives are financial instruments whose value depends on an underlying asset (e.g., stock, bond, com-
modity).
• They are used for hedging, speculation, and arbitrage.
• Types of derivatives: forwards, futures, options, and swaps.
• Equity derivatives (like call/put options) and interest rate derivatives (like swaps) are key areas in your
course.
• Arbitrage-free pricing, replication, and risk-neutral pricing are foundational concepts in derivative
pricing.
Example Question: - What is the payoff of a forward contract on a stock with a forward price of
$50?
Answer: The payoff of the forward contract at maturity is:
• Long position payoff: 𝑆𝑇 − 50 (where 𝑆𝑇 is the spot price at maturity).
• Short position payoff: 50 − 𝑆𝑇 .
Fundamental Theorem of Asset Pricing (FTAP)
Key Concepts: - The Fundamental Theorem of Asset Pricing (FTAP) states that in a no-arbitrage
market, there exists a risk-neutral measure under which all securities are priced.
• It links no arbitrage to the existence of a risk-neutral world where the discounted expected value of the
future cash flows is equal to the current price.
• Replication means creating a portfolio of the underlying asset and a risk-free bond that replicates the payoffs
of the derivative.
Example Question:
Given a call option with a strike price of $50, a stock price of $52, a risk-free rate of 5%, and a
1-year maturity, show how the absence of arbitrage can lead to the existence of a risk-neutral pricing
measure.
Answer:
Using the FTAP, the price of a derivative is the discounted expected payoff under the risk-neutral probability
measure.
For a call option with strike 𝐾, the price 𝐶0 is:
𝐶0 = 𝑒−𝑟𝑇 𝔼𝑄 [max(𝑆𝑇 − 𝐾, 0)]
1
, Binomial Tree Model
Key Concepts: - The binomial tree model is a discrete-time model used for option pricing. It approximates
the underlying asset’s price movements over discrete intervals.
• The model assumes that at each step, the asset price either up or down by a fixed factor.
• The risk-neutral probabilities are used to calculate the option’s price by working backward from expiration.
Formula: The price of a derivative at time 𝑡 = 0 is given by:
𝐶0 = exp(−𝑟 ⋅ Δ𝑡) ⋅ (𝑞 ⋅ 𝐶𝑢 + (1 − 𝑞) ⋅ 𝐶𝑑 )
where:
• 𝐶𝑢 and 𝐶𝑑 are the option prices at the up and down nodes,
• 𝑞 is the risk-neutral probability,
• 𝑟 is the risk-free rate
• Δ𝑡 is the time step.
Example Question:
A stock price is $50. The stock can either go up by 10% or down by 10% over one period. The risk-free rate is 5%.
What is the value of a European call option with a strike price of $52 using a one-period binomial tree?
Answer:
Up move: 𝑆𝑢 = 50 × 1.10 = 55
Down move: 𝑆𝑑 = 50 × 0.90 = 45
Option payoffs:
𝐶𝑢 = max(55 − 52, 0) = 3
𝐶𝑑 = max(45 − 52, 0) = 0
Risk-neutral probability:
𝑒0.05 − 0.90
𝑞= = 0.75
1.10 − 0.90
Option price:
𝐶0 = 𝑒−0.05 × [0.75 × 3 + 0.25 × 0] = 𝑒−0.05 × 2.25 ≈ 2.14
Black-Scholes Formula
Key Concepts: - The Black-Scholes model is a continuous-time model used for pricing European options. It
assumes constant volatility, no dividends, and a lognormal distribution of asset prices.
• The model uses stochastic calculus and provides a closed-form solution for European options.
2
3rd Party summary of John C. Hull’s textbook
Financial Derivatives Overview
Key Concepts:
• Derivatives are financial instruments whose value depends on an underlying asset (e.g., stock, bond, com-
modity).
• They are used for hedging, speculation, and arbitrage.
• Types of derivatives: forwards, futures, options, and swaps.
• Equity derivatives (like call/put options) and interest rate derivatives (like swaps) are key areas in your
course.
• Arbitrage-free pricing, replication, and risk-neutral pricing are foundational concepts in derivative
pricing.
Example Question: - What is the payoff of a forward contract on a stock with a forward price of
$50?
Answer: The payoff of the forward contract at maturity is:
• Long position payoff: 𝑆𝑇 − 50 (where 𝑆𝑇 is the spot price at maturity).
• Short position payoff: 50 − 𝑆𝑇 .
Fundamental Theorem of Asset Pricing (FTAP)
Key Concepts: - The Fundamental Theorem of Asset Pricing (FTAP) states that in a no-arbitrage
market, there exists a risk-neutral measure under which all securities are priced.
• It links no arbitrage to the existence of a risk-neutral world where the discounted expected value of the
future cash flows is equal to the current price.
• Replication means creating a portfolio of the underlying asset and a risk-free bond that replicates the payoffs
of the derivative.
Example Question:
Given a call option with a strike price of $50, a stock price of $52, a risk-free rate of 5%, and a
1-year maturity, show how the absence of arbitrage can lead to the existence of a risk-neutral pricing
measure.
Answer:
Using the FTAP, the price of a derivative is the discounted expected payoff under the risk-neutral probability
measure.
For a call option with strike 𝐾, the price 𝐶0 is:
𝐶0 = 𝑒−𝑟𝑇 𝔼𝑄 [max(𝑆𝑇 − 𝐾, 0)]
1
, Binomial Tree Model
Key Concepts: - The binomial tree model is a discrete-time model used for option pricing. It approximates
the underlying asset’s price movements over discrete intervals.
• The model assumes that at each step, the asset price either up or down by a fixed factor.
• The risk-neutral probabilities are used to calculate the option’s price by working backward from expiration.
Formula: The price of a derivative at time 𝑡 = 0 is given by:
𝐶0 = exp(−𝑟 ⋅ Δ𝑡) ⋅ (𝑞 ⋅ 𝐶𝑢 + (1 − 𝑞) ⋅ 𝐶𝑑 )
where:
• 𝐶𝑢 and 𝐶𝑑 are the option prices at the up and down nodes,
• 𝑞 is the risk-neutral probability,
• 𝑟 is the risk-free rate
• Δ𝑡 is the time step.
Example Question:
A stock price is $50. The stock can either go up by 10% or down by 10% over one period. The risk-free rate is 5%.
What is the value of a European call option with a strike price of $52 using a one-period binomial tree?
Answer:
Up move: 𝑆𝑢 = 50 × 1.10 = 55
Down move: 𝑆𝑑 = 50 × 0.90 = 45
Option payoffs:
𝐶𝑢 = max(55 − 52, 0) = 3
𝐶𝑑 = max(45 − 52, 0) = 0
Risk-neutral probability:
𝑒0.05 − 0.90
𝑞= = 0.75
1.10 − 0.90
Option price:
𝐶0 = 𝑒−0.05 × [0.75 × 3 + 0.25 × 0] = 𝑒−0.05 × 2.25 ≈ 2.14
Black-Scholes Formula
Key Concepts: - The Black-Scholes model is a continuous-time model used for pricing European options. It
assumes constant volatility, no dividends, and a lognormal distribution of asset prices.
• The model uses stochastic calculus and provides a closed-form solution for European options.
2
