Financial Engineering

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Exam Questions

June 2017

The exam consisted of 10 multiple choice questions (ABCD-kind), and 10 true-or-false questions. Correct multiple choice answers gave +3 points, incorrect ones -2 points. Correct true-or-false answers gave +1 point, incorrect ones -1 point. Leaving an answer blank was equivalent with 0 points. The exam did not say if the scores of the different kinds of question get reweighed or not. It was an open book exam. You could even use a laptop on flight mode (handy for ctrl+F or matlab programs). The duration of the exam was maximum 2 hours.

The questions below are not as elaborate written down as how they were asked at the exam, but complete enough (I hope) to give clear idea about what was asked.

  • Q1: Principle protected node. Determine the highest participation rate possible that a bank can offer and implement by taking a static position. There is no possibility of default. The maturity is 2 years. The stocks are non-dividend paying. S_0 = 50. Their are 3 European call options (maturity 2 years) given: (1) strike = 45, bid price = 8.05, ask price = 8.20, (2) strike = 50, bid price = 4.90, ask price = 5.00, (3) strike = 55, bid price = 2.60, ask price = 2.75. Putting €94 on a risk-free bank account results in €100 after 2 years.
    • A: 100-95%
    • B: 95-75%
    • C: 75-50%
    • D: < 50%
  • Q2: VIX. The VIX closed on a level of 10.38. This means that the implied probability of daily log-returns of SP500 of less than minus 2% is about
    • A: 33%
    • B: 5%
    • C: 0.13%
    • D: 0.005%
  • Q3: Calibration of a model leads to
    • A: estimation of parameters of risk-neutral measure
    • B: estimation of parameters of historical measure
    • C: A & B combined
    • D: None of the above, just a momentarily check to see if the simulation algorithm is correct.
  • Q4: In the Heston-Stochastic Volatility model
    • A: sample paths for stock prices are continuous, squared volatility process contains jumps
    • B: sample paths for stock prices are continuous, squared volatility process are continuous
    • C: sample paths for stock prices contain jumps, squared volatility process are continuous
    • D: sample paths for stock prices contain jumps, squared volatility process contains jump
  • Q5: Take a Brownian motion simulation with drift 5% and annualized volatility of 20% for a period of 1 year. Then the annualized realized volatility of the sample path is
    • A: always < 20%
    • B: always = 20%
    • C: = 5%
    • D: approximately 20%, but random
  • Q6: Implied volatilities for options on a stock
    • A: vary over different maturities and different strikes
    • B: are constant for different strikes, but vary for different maturities
    • C: are constant for different maturities, but vary for different strikes
    • D: are constant for different strikes and for different maturities
  • Q7: Carr & Madan's EC pricing formula uses as main input the characteristic function of
    • A: the stock price process
    • B: the logarithm of the stock price process
    • C: the volatility process
    • D: the EC price under BS
  • Q8: An euro denominated book consists of 1 type of stocks and plain vanilla options on this stock, modeled by BS. The portfolio has (1) delta = 0, (2) gamma = 50, (3) vega = 100, (4) theta = -25. If the implied volatility goes from 20% to 21%, then the P&L of the book is
    • A: < €10
    • B: €10-75
    • C: €75-150
    • D: > €150
  • Q9: Selling ATM put options with expiry in 1 year (non-dividend paying stock) implies
    • A: long vega, short gamma, long theta
    • B: short vega, short gamma, long theta
    • C: short vega, long gamma, short theta
    • D: long vega, short delta, long theta
  • Q10: EP, T = 1 year, implied volatility = 20%, non-dividend paying, S0 = 750. In BS (with no interest rates), the delta of an put option with strike K = 700 is around
    • A: 33%
    • B: -33%
    • C: 65%
    • D: -65%
  • Q11: Newton's method always needs only a single iteration step to find the minimum of a quadratic function f(x) = a*x²+bx+c, with a>0. (True/False)
  • Q12: The 3-year forward price of a dividend-paying stock can be replicated without any risk by taking positions in underlying stock and a risk-free bank account. (True/False)
  • Q13: A trader sells an European OTM call option, T = 1 year, no dividends, implied volatility = 20%. If he daily delta-hedges, he never makes a loss, if there are no transaction costs. (True/False)
  • Q14: Variance Gamma stock price model is an incomplete model, (i.e. there exists no unique equivalent martingale measure.) (True/False)
  • Q15: A trader sells an European OTM call option, T = 1 year, no dividends, implied volatility = 20%. By delta-hedging he always makes more money than selling the option uncovered. (True/False)
  • Q16: If 2 models produce identical vanilla option prices, they also produce the same prices for every exotic option. (True/False)
  • Q17: Call option, no dividends, no interest rates, ITM. If volatility drops from 20% to 18%, delta will increase. (True/False)
  • Q18: Call option, BS model, no dividends, r = 1%, S(0) = 100, T = 1 year, K =80, implied volatility = 20%. if stock price increases by 1%, delta will increase. (True/False)
  • Q19: A trader sells an EC at implied volatility of 20%, T = 1 year, no dividends, and strike is 10% below the current stock price. Then daily delta-hedging eliminates all his risk. (True/False)
  • Q20: Nelder-Mead generates in each iteration a new test position by extrapolating the behaviour of the objective function measured at each test point arranged as a simplex.