Quant Interview Questions 1

Below are some quant interview questions.

1. Difference between European, American and Bermudan options.

   Answer

   • European Options: Can only be exercised at expiration.
   • American Options: Can be exercised at any time before expiration.
   • Bermudan Options: Can be exercised on specific dates before expiration.

2. What are the assumptions of Black-Scholes model?

   Answer

   • European options.
   • Geometric Brownian motion.
   • No arbitrage opportunities.
   • Constant volatility.
   • Constant interest rate.
   • No dividends.
   • Frictionless Markets.
   • Perfectly hedged.
   • Continuous trading.
   • Infinite liquidity.

3. Derive the Black-Scholes formula.

   Answer

   Geometric Brownian motion of the stock price $`S_t`$:

   $$dS_t = \mu S_t dt + \sigma S_t dW_t$$

   where $`\mu`$ is the drift, $`\sigma`$ is the volatility, and $`W_t`$ is a Wiener process. It states that the infinitesimal rate of return on the stock has an expected value of $`\mu dt`$ and a variance of $`\sigma^2 dt`$.

   The value of the options depends on stock price $`S_t`$, time $`t`$, denoted as $`V(S_t, t)`$. Hence, ito's lemma gives:

   $$\begin{align*} d V(S_t, t) &= \left( \frac{\partial V}{\partial t} + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S_t^2 \right) dt + \frac{\partial V}{\partial S} d S \\ &= \left( \frac{\partial V}{\partial t} + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S_t^2 \right) dt + \frac{\partial V}{\partial S}\left( \mu S_t dt + \sigma S_t dW_t \right) \\ &= \left( \frac{\partial V}{\partial t} + \mu S \frac{\partial V}{\partial S} + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S_t^2 \right) dt + \sigma S \frac{\partial V}{\partial S}dW_t\\ \end{align*}$$

   Base on the behavior of the option price, we construct a portfolio that consists of the option and a short position in $`S_t`$:

   $$\Pi = - V(S_t, t) + \frac{\partial V}{\partial S} S_t$$

   It is a Delta ($\Delta = \frac{\partial V}{\partial S}$) hedging using the stock. Thus the net delta is zero.

   And assuming the portfolio is self-financing (no additional cash flow), we have (Assuming delta is constant):

   $$d\Pi = - dV + \frac{\partial V}{\partial S} dS_t$$

   By substituting the expression of $`dV`$ and $`dS_t`$ into the equation, we have:

   $$d\Pi = \left( - \frac{\partial V}{\partial t} - \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S_t^2 \right) dt$$

   Noted that the stochastic term is eliminated. Let's assume the portfolio have a risk-free return $`r`$, i.e.

   $$d\Pi = r \Pi dt$$

   Thus,

   $$\left( - \frac{\partial V}{\partial t} - \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S_t^2 \right) dt = r \left( - V + \frac{\partial V}{\partial S} S_t \right) dt$$

   Rearranging the equation gives:

   $$\frac{\partial V}{\partial t} + r S_t \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} - r V = 0$$

   This is the Black-Scholes PDE.

4. Sample 3 from [0, 1], what is the expectation of the maximum?

   Answer

   Let say $`X_1, X_2, X_3`$ are the three samples from uniform distribution $`U(0, 1)`$. The maximum is $`M = \max(X_1, X_2, X_3)`$. The cumulative distribution function (CDF) of $`M`$ is:

   $$P(M \leq x) = P(X_1 \leq x, X_2 \leq x, X_3 \leq x) = P(X_1 \leq x) P(X_2 \leq x) P(X_3 \leq x) = x^3$$

   The probability density function (PDF) is:

   $$f_M(x) = \frac{d}{dx} P(M \leq x) = 3x^2$$

   The expectation of $`M`$ is:

   $$E[M] = \int_0^1 x f_M(x) dx = \int_0^1 x \cdot 3x^2 dx = 3 \int_0^1 x^3 dx = 3 \cdot \frac{1}{4} = \frac{3}{4}$$

5. We have a stick with unit length which we cut randomly at two spots. What is the expected length of the tallest piece?

   Answer

   Let $`X_1`$ and $`X_2`$ be the two random cut points, uniformly distributed over the interval $[0, 1]$. Without loss of generality, assume $`X_1 \leq X_2`$. The lengths of the three pieces are:

   • Left piece: $`X_1`$
   • Middle piece: $`X_2 - X_1`$
   • Right piece: $`1 - X_2`$

   The maximum length is:

   $$M = \max(X_1, X_2 - X_1, 1 - X_2)$$

   To find the expected value of $`M`$, we can use the law of total expectation. The joint distribution of $`X_1`$ and $`X_2`$ is uniform over the unit square. We can compute the expected value by integrating over the region where $`X_1 \leq X_2`$.

   The expected value can be computed as:

   $$E[M] = \int_0^1 \int_{x_1}^1 \max(x_1, x_2 - x_1, 1 - x_2) dx_2 dx_1$$

   After evaluating this integral, we find that:

   $$E[M] = \frac{5}{6}$$

6. Moving dot and polygon relative position detection. Calculate the time complexity