OPtion Pricing Flashcards
(22 cards)
Continous vs discrete compounding crr
time step t
delta t
what is the effect of the number of steps?
THe more steps the closer the distribution of continous return to the normal distribution,
As the number of steps increases:
The step size Δt becomes smaller.
The distribution of returns becomes finer and smoother.
By the Central Limit Theorem, the sum of many small independent steps (log returns) converges to a normal distribution.
Key Difference: American vs. European Option Valuation
Vola component and sigma in BS? Key compenents
Volatility affects two key components:
The diffusion term (variance adjustment).
The time scaling of price uncertainty.
Vola component and sigma in BS?
What would you guess about the type of asset that would mostly make up this type of portgolio?
The minimum variance portfolio is mostly made up of low-volatility, low-correlation assets that reduce overall portfolio risk.
Risk free assets or assets that have a negative correlation with each other
Convert Discrete Risk-Free Rate to Continuous (for Black-Scholes)
Black Scholes Assumptions
Black-Scholes assumes:
No arbitrage, frictionless markets, constant r and σ, continously compounded r, lognormal stock prices, no dividends, European options only, and continuous trading.
Black Scholes with dividends
BS key concept
What does d1 represent?”
Risk-adjusted probability of being in-the-money, including hedging effects.
What does d2 represent?
Risk-neutral probability of exercise.
Why is the strike discounted?
Time value of money.
Current Moneyness: ln(S/K)
Measures how “in-the-money” or “out-of-the-money” the option is right now.
Variance Adjustment: σ^2/2
What is it?
Half of the annualized variance of the asset’s returns.
What does it represent?
Corrects for the drift adjustment due to volatility.
What does the term T−t represent in the Black-Scholes formula?
Tests understanding of time to maturity.
How does increasing time to maturity (T - t ↑) affect the option price?
Call and put option values generally increase with more time (more uncertainty → higher value of optionality).
If volatility increases but time to maturity decreases at the same time, how might the option price behave?
Volatility boosts price, but effect is scaled by T−t
If T becomes very small, even large
σ has less impact.
The closer to expiry, the less time for volatility to “play out”.
Explain why Black-Scholes assumes continuous trading and why this matters.
Continuous trading allows perfect hedging (delta-hedging infinitesimally).
Without it, dynamic hedging wouldn’t eliminate risk, invalidating derivation.
In reality, discrete trading introduces hedging errors → Black-Scholes becomes an approximation.
How would an increase in dividend yield affect a call option’s value?
Higher dividend yield lowers call option value.
Because future expected stock price growth is reduced → makes calls less valuable.
Psi Greek would be negative.
