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
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.
I. Stochastic Models and Processes · 12. November 2023
The Merton model, essential in credit risk analysis, views a company's equity as a call option on its assets, crucial for default probability assessment. Using the Black-Scholes formula, it combines equity with zero-coupon debt for valuation. Despite its innovativeness, the model's reliance on market data and idealistic market assumptions limit its applicability. This has spurred alternative approaches like reduced form models, addressing these shortcomings in credit risk evaluation.
IV. Advanced Concepts and Theories · 12. November 2023
The Merton model, essential in credit risk analysis, views a company's equity as a call option on its assets, crucial for default probability assessment. Using the Black-Scholes formula, it combines equity with zero-coupon debt for valuation. Despite its innovativeness, the model's reliance on market data and idealistic market assumptions limit its applicability. This has spurred alternative approaches like reduced form models, addressing these shortcomings in credit risk evaluation.
IV. Advanced Concepts and Theories · 09. November 2023
Financial models use Girsanov's Theorem to shift from real-world probabilities to risk-neutral ones, crucial for derivative pricing. It ensures arbitrage-free models in quantitative finance, adjusting processes like HJM, CIR, and Hull-White to reflect risk-neutral views for fair pricing.
#RiskNeutralProbabilities #GirsanovsTheorem #RadonNikodym
IV. Advanced Concepts and Theories · 05. October 2023
Explore the roots of the martingale concept, originally a betting strategy in fair games, and its evolution in quantitative finance. Uncover the role of unpredictability and risk, dispelling "sure-win" myths. #Finance #Martingale #RiskManagement
IV. Advanced Concepts and Theories · 04. October 2023
Brownian motion bridges Martingale's unpredictability and Markov's memorylessness, essential in quantitative finance for pricing derivatives and risk management. It illustrates a random yet memoryless movement, pivotal in financial mathematical modeling. #Finance #QuantitativeFinance
IV. Advanced Concepts and Theories · 03. October 2023
Stopping time helps optimize the exercise of exotic options to maximize payoff. It aids in deciding when to exercise these complex, event-conditioned options before expiry. Specialized models beyond Black-Scholes, like Binomial Tree, address this by balancing immediate vs. future exercise values. #StoppingTime #ExoticOptions
IV. Advanced Concepts and Theories · 03. October 2023
Ocone martingales offer stable modeling for complex, dynamic financial systems due to their unique invariance, aiding in exotic option pricing. Rooted in control theory, they ensure consistent statistical properties under specific mathematical manipulations. #FinancialMath #Martingales