Quant Interview Questions (Equity Exotic Derivatives)

1) What is the Taylor expansion of exp(-1/x^2)?

Ans:
This function cannot be expanded at x = 0 because there is a singularity: the function is NOT analytic at x = 0.
http://en.wikipedia.org/wiki/Taylor_series#Analytic_functions

2) Suppose we have the prices of options with various strikes. How can we price an option with arbitrary payoff?

Ans:
Solve the Fokker-Planck Equation to obtain the risk-neutral probability density p(S). Then the price of an option with arbitrary payoff can be calculated as
exp(-r (T-t)) \int Payoff(S) p(S) dS
Note: In case of vanilla European call, p(S) can be found by the second partial derivative of the call price w.r.t. strike K

3) Suppose we have a volatility smile (same expiry, different strikes). What no-arbitrage condition can be imposed on the option implied volatilities (i.e. prices)?

Ans:
Recall the bull spread relation and the butterfly spread relation. In the continuous limit, they read
0 < -dC/dK < 1 [Note the negative sign]
d^2C/dK^2 > 0

4) Given two stock price time series of two stocks. After computing the correlation using the naive approach (rolling covariance divided by rolling standard deviations), the correlation matrix is NOT PSD. How can this be fixed?

Ans:
??

See here for more interview questions/brainteasers