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VIII. Quant Interview Questions · 14. November 2023

To find the probability that the barrier is breached at any time between now and a future time T, we can use the reflection principle of Brownian motion. However, without going into complex stochastic calculus, a simplified approach can consider the probability that the stock price exceeds H at time T. This probability can be expressed using the cumulative distribution function (CDF) of the normal distribution, denoted as Phi. The stock price at time T is normally distributed with mean equal to...

VIII. Quant Interview Questions · 14. November 2023

Ito's Lemma: If you have a function f that depends on time and another variable x, and if you differentiate it with respect to both time and x, you get the change in f. For our purposes, x is going to be our Brownian motion W(t). We're interested in the function f(t, W(t)) = W(t)^2 - t. For a function f(t, X(t)), the differential df using Ito's lemma is: df = (∂f/∂t) dt + (∂f/∂X) dX + 0.5 (∂^2f/∂X^2) (dX)^2 Where: - ∂f/∂t is the partial derivative of f with respect to t. -...

VIII. Quant Interview Questions · 14. November 2023

To prove that Y(t) = W(t)^2 - t is a martingale, where W(t) is a standard Brownian motion, we'll use the properties and definitions of martingales and stochastic calculus. Definition of a Martingale: A process Y(t) is a martingale with respect to some filtration if: 1. The expected value of |Y(t)| is finite for all t. 2. Y(0) is integrable and its expected value is 0. 3. The expected value of Y(t) given the history up to time s is equal to Y(s) for all 0 <= s < t. Let's prove each of...

VIII. Quant Interview Questions · 14. November 2023

Quant Interview Question: European Call Option Pricing with Different Volatility Assumptions You're tasked with pricing a European call option under two different scenarios: 1. Using a constant volatility of 20%. 2. Drawing volatility from a random distribution with an average of 20%. ________ Which option do you anticipate would generally be more expensive, and why? At a first glance, many would assume that the option with stochastic volatility would come with a heftier price tag. This is...

VIII. Quant Interview Questions · 14. November 2023

In the context of stochastic calculus, considering the term 'dX' known as a Wiener process, what does the relationship 'dX^2 equals dt' when 'dt' tends toward 0 signify? A) A summation of squared values B) The convergence of a sequence C) The limit of a function D) The behavior of a Wiener process Unveil the hidden connection with your choice! __________ The correct answer is: D) The behavior of a Wiener process Explanation: In the context of stochastic calculus and the Wiener process, dX^2 =...

VIII. Quant Interview Questions · 14. November 2023

Which of the following statements about Volga is most accurate? A. Volga is the rate at which vega changes with respect to the underlying price B. High Volga values imply that options are more sensitive to kurtosis risk of the underlying asset C. Volga becomes particularly important when trading binary options due to their vega profile D. All else being equal, an option with a longer time to maturity will always have a higher Volga than an option with a shorter time to maturity ------------- B....

VIII. Quant Interview Questions · 14. November 2023

In the intricate world of stochastic calculus, a key question arises: does the derivative dWt/dt exist for Brownian Motion Wt? The answer is intriguing.

VIII. Quant Interview Questions · 14. November 2023

Does the derivative dWt/dt exist, where Wt is a Brownian Motion? 1. Yes, it exists, and it's a fundamental concept in stochastic calculus. 2. No, it doesn't exist, and I'm curious to know why! 3. I really don't know – let's explore this together! Cast your vote and share your thoughts in the comments below! Let's unravel the mysteries of stochastic calculus together. #QuantitativeFinance #StochasticCalculus #BrownianMotion #Derivative #FinancialMath #QuantitativeAnalysis #FinanceEducation...

VIII. Quant Interview Questions · 14. November 2023

Let's model the price of a stock (S) using the following stochastic differential equation (SDE): dS = μ * S * dt + σ * S * dW Where: - S is the current price of the stock. - μ is the drift coefficient (expected return). - σ is the volatility coefficient. - dt is the time step. - dW is a Wiener process increment (Brownian motion) that follows a normal distribution with mean 0 and standard deviation √dt for each time step. For this example, you have: - S = $100 (current stock price) - μ =...

VIII. Quant Interview Questions · 14. November 2023

In risk-neutral valuation, predicting the next step in a random walk, even with real probabilities of 0.55 up and 0.45 down, is not straightforward. The expected direction—up, down, or indeterminate—depends on additional factors like the risk-free rate and the magnitude of movements.