Lecture 4

Question 1

What is scenario analysis used for in risk management?

Answer 1

It is used to generate scenarios of future financial conditions (e.g. stock prices, inflation) to evaluate risks and required capital for worst-case scenarios.

Question 2

How is scenario analysis applied to option pricing?

Answer 2

By generating scenarios of stock prices, calculating the option payoff in each scenario, and discounting the average payoff.

Question 3

Why is a risk-neutral world assumed for pricing options?

Answer 3

To discount using the risk-free rate as it eliminates risk premiums from calculations.

Question 4

What does the “die analogy” illustrate in scenario analysis?

Answer 4

The concept of estimating expected values via analytical methods or simulations.

Question 5

Why is simulation often preferred over analytical computation in complex scenarios?

Answer 5

It simplifies calculations for cases like higher0order moments or non-linear payoffs.

Question 6

What characteristics should the stochastic process for stock prices match?

Answer 6

Historical mean and volatility of the stock’s returns.

Question 7

Why are returns or log returns preferred over prices in simulations?

Answer 7

Because expected (log) returns remain consistent across time, avoiding the issue of varying distributions.

Question 8

How do log returns solve the issue of negative stock prices?

Answer 8

By ensuring simulated prices are always positive through exponentiation.

Question 9

Why are log returns used for simulations?

Answer 9

Log returns are less skewed which aligns better with the normal distribution, also solves negative price problem.

Question 10

How does the step size (Delta t) influence the simulation distribution?

Answer 10

The distribution’s mean and variance must scale with the step size to ensure consistency.

Question 11

What defines a stochastic process?

Answer 11

A collection of random variables indexed by time t.

Question 12

How do discrete and continuous stochastic processes differ?

Answer 12

Discrete processes have stepwise realizations, while continuous ones have smooth paths.

Question 13

What is the Markov property?

Answer 13

Future states depend only on the current state, not past states.

Question 14

What is a Wiener Process?

Answer 14

A stochastic process where random changes are scaled by the Square Root of Delta t.

Question 15

How is variance adjusted in a Wiener process for different time steps?

Answer 15

By scaling random draws with the square root of the step size.

Question 16

Why is the Wiener process central to financial modeling?

Answer 16

It ensures variance matches the time horizon of simulations.

Question 17

What does Ito’s Lemma allow us to do?

Answer 17

Model the dynamics of functions of stochastic processes, such as log stock returns.

Question 18

How does Ito’s Lemma modify the dynamics of ln(St)?

Answer 18

It introduces a correction term -0.5Sigma in the drift to account for non-linear effects.

Question 19

How are stock prices simulated step-by-step?

Answer 19

By iteratively calculating St+1 = St * e^((Mean - 0.5 Variance) * Step size + VolatilitySQRT(Step size)Random draw from ND (0,1)

Question 20

What parameters are required to simulate stock prices?

Answer 20

Historical or assumed values for drift mean, volatility and step size.

Question 21

What changes when moving to a risk-neutral world in simulations?

Answer 21

The drift / mean is replaced by the risk-free rate r, eliminating the risk premium.

Question 22

How is the risk-neutral price of an option calculated?

Answer 22

By discounting the expected payoff under the risk-neutral measure at the risk-free rate.

Question 23

What does the Black-Scholes formula calculate?

Answer 23

The price of European call and put options.

Question 24

What is d1 in the Black-Scholes formula?

Answer 24

D1 = (ln(St / K) + (r + 0.5 Variance)(T-t) / Volatility * SQRT(T-t)

Question 25

What is d2 in the Black-Scholes formula?

Answer 25

D2 = D1 - Volatility*SQRT(T-t)

Question 26

Why is Monte Carlo simulation used in option pricing?

Answer 26

It allows estimation of option payoffs by simulation numerous price paths and averaging results.

Question 27

How does the Law of Large Numbers apply to the Monte Carlo simulations?

Answer 27

The average payoff converges to the true expected value as the number of simulations increase.

Question 28

What is the confidence interval in Monte Carlo simulations based on?

Answer 28

The Central Limit Theorem, where the interval narrows with more scenarios.

Question 29

Why are historical estimates for mean and volatility important?

Answer 29

They ensure that simulations reflect the actual risk-return characteristics of the stock.

Question 30

How does option pricing using simulations relate to the dice analogy?

Answer 30

Just as more dice rolls provide a better estimate of the average roll, more simulations improve the accuracy of option pricing.