# Financial Engineering

## 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 â‚¬94 on a risk-free bank account results in â‚¬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: < â‚¬10
- B: â‚¬10-75
- C: â‚¬75-150
- D: > â‚¬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*xÂ²+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. (True/False)