C15: BlackScholesMerton Model
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50 Terms
What is the Black-Scholes-Merton (BSM) model?
A continuous-time model used to price European options using no-arbitrage principles
What is the key assumption about stock prices in BSM?
Stock prices follow a continuous stochastic process with randomness and constant volatility
What distribution do stock prices follow under BSM?
Lognormal distribution
Why are stock prices lognormally distributed?
Because log returns are normally distributed over time
Why is lognormal distribution important?
It ensures stock prices cannot become negative
What is geometric Brownian motion?
A continuous-time process describing stock price evolution with drift and volatility
What is the difference between simple returns and log returns?
Log returns can be summed over time while simple returns cannot
Why are log returns used in BSM?
They allow modeling of stock price distribution over time
What is volatility in BSM?
The standard deviation of continuously compounded returns
What does higher volatility imply?
Greater uncertainty and higher option values
How is volatility estimated in practice?
Using historical data or implied volatility from market prices
What is historical volatility?
Volatility calculated from past stock price movements
What is implied volatility?
The volatility implied by current market option prices
Why is implied volatility important?
It reflects market expectations of future uncertainty
What is the key difference between historical and implied volatility?
Historical is backward-looking; implied is forward-looking
What are the main assumptions of the BSM model?
Constant volatility and interest rates, no arbitrage, no transaction costs, continuous trading, no dividends (in basic form)
Why is no-arbitrage important in BSM?
It ensures consistent pricing across financial instruments
What is the key idea behind the BSM derivation?
Construct a riskless portfolio using the stock and option
What happens to the riskless portfolio?
It must earn the risk-free rate
What is the role of continuous trading in BSM?
It allows continuous adjustment of the hedge
What is delta in BSM?
The sensitivity of option price to changes in the underlying stock price
What is delta hedging in BSM?
Continuously adjusting positions to maintain a riskless portfolio
What is the BSM differential equation?
A relationship that all derivative prices must satisfy under no-arbitrage
What determines the solution to the BSM equation?
The boundary conditions of the derivative payoff
What is the key result of the BSM model?
A closed-form solution for European call and put prices
What does the BSM formula represent conceptually?
Expected payoff under risk-neutral probabilities discounted at the risk-free rate
What are the key inputs to the BSM model?
Stock price, strike price, time to maturity, volatility, and risk-free rate
What is the role of time to maturity?
Longer time increases option value due to greater uncertainty
What does the function N(x) represent?
The probability that a standard normal variable is less than x
How are d1 and d2 interpreted conceptually?
They measure moneyness and probability of exercise under risk-neutral valuation
What does N(d2) represent?
The risk-neutral probability that the option will be exercised
What does N(d1) represent?
The sensitivity of the option price to the stock (delta)
What is the intuition behind the BSM formula?
Option value equals expected payoff weighted by probabilities and discounted
What happens to call value when stock price increases?
It increases
What happens to put value when stock price increases?
It decreases
What happens to option values when volatility increases?
Both calls and puts increase in value
What happens when volatility approaches zero?
Option value approaches intrinsic value
What is the effect of interest rates on options?
Higher rates increase call values and decrease put values
Why does the expected return (μ) not appear in BSM pricing?
It is irrelevant under risk-neutral valuation
What is risk-neutral valuation in BSM?
Assuming all assets grow at the risk-free rate for pricing purposes
Why does risk-neutral valuation work?
Because prices are determined by arbitrage, not investor preferences
What is the key insight about expected return?
It is already embedded in the current stock price
What is the relationship between BSM and binomial models?
BSM is the continuous-time limit of the binomial model
How is time measured in option pricing?
Typically using trading days rather than calendar days
Why are trading days used instead of calendar days?
Most price changes occur when markets are open
What is the role of dividends in BSM?
They reduce the effective stock price used in valuation
How are dividends incorporated into BSM?
By subtracting the present value of expected dividends from the stock price
What is put-call parity in the BSM framework?
A relationship linking call and put prices to stock and bond values
How can put-call parity be used with BSM?
To derive one option price from the other
What is the key takeaway of the BSM model?
Option pricing depends on arbitrage and volatility, not subjective expectations