Computational Finance: Midterm Examination

April 23, 2001
Philip H. Dybvig

This is a closed-book examination. Answer all questions as directed. Mark your answers directly on the examination. On the valuation question, make sure your answer is clearly indicated. There are no trick questions on the exam. Good luck!
A. General Concepts Multiple choice: 20 points Circle the one best answer.

  1. Which of the following is an interest derivative?

    A. ARCH
    B. collar <----
    C. warrant
    D. PDA

  2. In risk-neutral probabilities, the mean change in futures price is which of the following?

    A. riskfree rate
    B. mean return on the underlying security
    C. risk premium on the underlying security
    D. zero <----

  3. Which is most likely to be a useful tool for valuing mortgage derivatives?

    A. Black-Scholes model
    B. binomial model
    C. simulation <----
    D. duration

  4. Suppose a bond manager overlays an active bond portfolio with S&P index futures to convert bond market profits into a portfolio that is to be judged against an S&P 500 benchmark. What is the best description of this situation?

    A. portable alpha <----
    B. asset-liability management
    C. portfolio insurance
    D. hedge fund

  5. Which of the following is the main concept underlying the binomial option pricing model?

    A. absence of arbitrage <----
    B. time value of money
    C. Modigliani-Miller irrelevancy propositions
    D. value-at-risk

B. Financial Computation 40 points

A Simple Option Pricing Problem in Two Periods

Riskless bond (riskfree interest rate is 25%):

16 --- 20 -- 25

Stock price:

            |- 225
    |- 120 -|
64 -|       |-  75
    |-  40 -|
            |-  25

Derivative security (binary option paying 100 when the stock price is between 50 and 100 and paying zero otherwise):

            |-   0
    |-  ?  -|
?  -|       |- 100
    |-  ?  -|
            |-   0

The actual probabilities are 2/3 for up and 1/3 for down at each node.

  1. What are the risk-neutral probabilities of up and down? (The risk-neutral probabilities are the same at every node: compute them at the first node. Hint: check to make sure the stock price is consistent with your answer.)

                 120 - 64               40 - 64
    r = 25%  u = -------- = 87.5%   d = ------- = -37.5%
                    64                     64
      *   r - d    25% - (-37.5%)    62.5%   1
    pi  = ----- = ---------------- = ----- = -
      u   u - d   87.5% - (-37.5%)    125%   2
      *         *   1
    pi  = 1 - pi  = -
      d         u   2
    check with the stock:
     120/2 + 40/2    80
     ------------ = ---- = 64
         1.25       1.25
  2. What is the value of the binary option at each node?

                                      |-   0
                   0/2 + 100/2        |
               |-  -----------  = 40 -|
     40        |       1.25           |
    ---- = 32 -|                      |- 100
    1.25       |   100/2 + 0/2        |
               |-  -----------  = 40 -|
                       1.25           |
                                      |-   0
  3. In the portfolio strategy that replicates the binary option, what are the holdings in the stock and the bond at the initial node?

        |- 40
    32 -|
        |- 40
    This is riskless, so just invest $32 in the bond and
    $0 in the stock.
  4. (thought question -- answer in a sentence or two of ordinary length) Suppose you have a view that volatility will be smaller than most people expect. Your investment banker has proposed the purchase of these binary options as a way of profiting if your view is correct. Discuss.

    This is a good way of betting that the stock will not
    go up or down too much, but only if your investment
    banker does not charge too much for the option!
C. Computer Implementation Short Answer: 40 points

The questions in this section are based on the Stochastic Volatility Option Pricing Applet in Homework 5. You should not need to use the complete program listings, but they are included at the end of the exam in case you would like to have a look at them.

Here is the method eurCall in SVPriceEngine, which computes the price of a European call option.
public double eurCall(double stock,double strike,long nsimu) {
  long i,n;
  double x;
  for(n=0;n<nsimu;n++) {
    stockP = stock;
    sigma = sigma0;
    for(i=0;i<npers;i++) {
      stockP *= stocktotret();
    x += Math.max(stockP-strike,0);}
  return(x/((double) nsimu * Math.pow(r1per,(double) npers)));}
  1. How would eurCall be modified to price a European put option?
     Change the line
        x += Math.max(stockP-strike,0);}
        x += Math.max(strike-stockP,0);}
  2. What is the variable x in eurCall? Discuss its role in the option valuation.
     The variable x contains the running sum of the option
     values.  After the for-loop completes, x contains the
     sum of all of them.  When it is divided by nsimu, we
     have the average of the values, which is discounted
     to compute the option price.
  3. Explain the role of Math.pow(r1per,(double) npers) in the denominator of the quantity returned by eurCall.
     Math.pow(r1per,(double) npers) is the factor used to
     discount the average option price.
  4. (thought question -- answer in a sentence or two of ordinary length) It is quite difficult to modify a simulation program such as the one used in eurCall to price an American option. Why is this?
     It is hard because we do not know when to exercise.
     We know the value of an exercised option at each node,
     but we do not know the continuation value.  By contrast,
     the binomial computes the continuation value at each
     point as it steps through the tree.
D. Advanced Questions and the Speakers Short Answer: up to 20 make-up points

  1. Sometimes interest-only obligations (i.o.'s) increase in value in response to rising interest rates, and sometimes they respond in the opposite direction. Explain briefly why the response can go either way.

     If expected repayments do not change much (as when market
     rates are much higher than the issue rate), the i.o. is
     like a bond and declines in value when the rate increases.
     If expected repayments fall significantly with the rate
     rise (as when the rates are at a new low in the life of
     the mortgages), the i.o. gains a longer life and increases
     in value.
  2. Describe briefly an example given by one of the class speakers of a practical problem solved in practice using the tools of this course.

     Dan Scholz of NISA described how simulation was used to
     analyze the trade-off between tracking error and transaction
     costs to determine a cost-smart rebalancing strategy.  (Many
     other answers are possible.