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Explain derivatives, YTM, and core statistics

SquarepointData Scientist · Technical Screen
MediumData ScientistTechnical ScreenStatistics & Math

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This problem tests your understanding of fundamental financial math concepts (derivatives and YTM) and your ability to apply statistical rigor when modeling financial data.

Part 1: Financial Concepts

  1. Explain the concept of a financial derivative. How does the payoff of a European call option differ from a forward contract?
  2. Define Yield to Maturity (YTM). If a bond's market price increases while the coupon rate remains constant, what happens to the YTM, and why?

Part 2: Statistical Application

You are tasked with estimating the volatility of a stock to price an option. You are given a time series of daily returns for the last 252 trading days.

  1. You decide to calculate the sample standard deviation as your volatility estimate. What statistical assumptions are you making about the return distribution? What happens to your estimate if the returns exhibit 'fat tails' (leptokurtosis)?
  2. Suppose you want to test the hypothesis that the mean daily return is zero. What test would you use, and what are the potential pitfalls if the data exhibits volatility clustering (GARCH effects)?
  3. How would you handle a situation where you have significant outliers in your 252-day sample? Would you prefer a trimmed mean, a median, or a robust estimator like the Huber loss function, and how would you justify this choice to a risk manager?

Hints

Hint 1

For derivatives, focus on the distinction between the obligation of a forward contract and the optionality (asymmetric payoff) of an option.

Hint 2

For YTM, remember the inverse relationship between bond prices and interest rates; as the denominator (price) increases, the effective discount rate must decrease.

Hint 3

When discussing volatility, consider the limitations of the normality assumption (Gaussian distribution) when dealing with real-world financial market events.

Follow-up questions

  • How does the choice of look-back window (e.g., 252 days vs 30 days) impact the bias-variance tradeoff in volatility estimation?
  • If you were to use a Monte Carlo simulation to price an option, how would you account for the non-normality observed in the historical data?
  • Can you explain the difference between realized volatility and implied volatility, and why they often diverge?