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Explain left-skewed returns and Black–Scholes terms

Qube Research & TechnologiesSoftware Engineer · Technical Screen
HardSoftware EngineerTechnical ScreenSoftware Engineering Fundamentals

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This question tests the ability to interpret statistical properties of return distributions and explain the financial intuition behind the Black–Scholes option pricing model.

Part 1: Left-Skewed Returns

In the context of financial time series, describe the meaning of a 'left-skewed' return distribution. What does this skewness imply about the frequency and magnitude of positive versus negative returns, and why is this characteristic often perceived as a 'tail risk' for portfolio managers?

Part 2: Black–Scholes Intuition

The Black–Scholes formula for the price of a European call option is given by: C = S * N(d1) - K * e^(-r*T) * N(d2)

Explain the economic significance of the two main terms in this equation:

  1. S * N(d1)
  2. K * e^(-r*T) * N(d2)

Provide an intuitive explanation for what each term represents in terms of the risk-neutral probability of the option expiring in-the-money and the present value of the exercise cost.

Hints

Hint 1

For left-skewness, think about the position of the mean relative to the median and how the 'fatness' of the left tail affects the probability of extreme losses.

Hint 2

Consider that S * N(d1) represents the expected value of the asset at expiration given that the option ends in-the-money, scaled by the probability of that event occurring.

Hint 3

Recall that K * e^(-r*T) is the discounted strike price, and N(d2) acts as the risk-neutral probability that the option will be exercised.

Follow-up questions

  • How does high kurtosis (fat tails) in a return distribution affect the pricing of out-of-the-money options compared to a normal distribution?
  • If the underlying asset pays a continuous dividend yield, how does this modify the Black–Scholes formula, and why?
  • Explain the concept of 'Greeks' (Delta, Gamma, Vega) and how they relate to the sensitivity of the call price to changes in market parameters.