Computational Finance: Midterm Examination Answers

For the Exam given April 12, 1999
Philip H. Dybvig

A. General Concepts Very Short Answer: 20 points

1. Which of the following is the conceptual basis for option pricing: capital asset pricing model (CAPM), absence of arbitrage, or the weighted-average cost-of capital (WACC)?

   absence of arbitrage
   

2. Name three examples of derivative securities that can be priced by a binomial model.

    
   call option, put option, cap (other
   answers are possible) 
   

3. Suppose you have a simple option pricing task. Assuming the problem is simple enough so that either can be used, which would be faster, a single-variable binomial model or a simulation?

    
   binomial model
   

4. Suppose you have a complex option pricing task that involves many sources of noise and stochastic volatility. Which approach is probably more appropriate, Black-Scholes, binomial, or simulation?

    
   simulation 
   

5. Name two common uses of simulations in finance.

    
   simulating investment strategies, pricing options  (other
   answers are possible)
   

B. Financial Computation 40 points

A Simple Option Pricing Problem in One Period

Riskless bond (interest rate is 50%):

100 --- 150

Stock:

    |- 125
50 -|
    |-  50

Derivative security (call option with a strike of 80):

    |- 45
 ? -|
    |-  0

1. What is the portfolio of the stock and the bond that replicates the option?

       
   Suppose the replicating portfolio has S dollars
   worth of stock and B dollars worth of bonds,
   replicating in both states implies that:
   
   (125/50) S + (150/100) B = 45 (up state)
   
   and
   
   (50/50) S + (150/100) B = 0.
   
   Solving for S and B, we have that 1.5 S = 45 or
   S = 30 (which is purchase of 3/5 of a share) and
   30 + 1.5 B = 0 or B = -20 (which shorting 1/5 of a
   bond).
   

2. What is the price of the portfolio?

       
   The price of the replicating portfolio is
   30 - 20 = 10.  
   

3. What are the risk-neutral probabilities of the two states? (I recommend that you verify that your answer gives the correct price for the stock, the bond, and the option.)

       
   Let p be the risk-neutral probability of the up
   state.  Then 1-p is the risk-neutral probability
   of the down state.  To price the stock correctly,
   we must have that
   
   50 = (1/1.5) x (125 p + 50 (1-p)).
   
   Solving for p we have that
   
   p = 1/3 and 1-p = 2/3.
   

4. (thought question) Suppose we have inherited many shares of the stock and would like to trade them to diversify but we are precluded from trading the stock due to terms of an inheritence. If the inheritence does not preclude trading the option, how can this help us? (Note: do not perform any computations in this answer.)

       
   If we short a portfolio of options and
   bonds that replicates the stock,
   we can undo the risk exposure without
   violating at least the letter of the terms
   of our inheritence.
   

C. Computer Implementation Short Answer: 40 points

The questions in this section are based on the asset-allocation simulation in Homework 4. 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 [note: not included in this answer sheet]. All of the references in this section are to classes defined in the file AssetAlloc.java.

1. The TextField inrisky appears in the following statements in the class ValuePlotFrame: near the start of the class definition and not in any method:

         TextField r,mu,sigma,inrisky;
   

   in the class constructor ValuePlotFrame():

         inputs.add(inrisky = new TextField("70",10));
   

   in the "start simulation" method startSimu():

         ir = text2double(inrisky)/100.0;
   

   in the method reset()

         inrisky.setText("70");
   

   What is the TextField inrisky? Briefly, how is it used?

   The TextField inrisky is the input
   cell where the user puts the proportion
   to be put in the risky asset.  This
   is put in the inputs Panel of the
   ValuePlotFrame and the text is set
   as a default to 70 both initially
   and by reset().  When needed for
   computations, it is converted to the
   double ir using text2double() and
   dividing by 100 to convert from
   a percentage to a decimal.  (Note:
   this answer is more comprehensive
   than is necessary.)
   

2. The method stockTotRet in AssetAllocEngine is defined by

     double stockTotRet() {
       return mean1per + std1persqrt12 * (Math.random()-0.5);}
   

   where the constandt std1persqrt12 would be set previously by the method newPars in the line

       std1persqrt12 = sigma * Math.sqrt(12.0 * tinc);
   

   of code. What is the point of the 0.5 and the square root of 12.0 in these lines?

   The 0.5 adjusts the mean of the
   Math.random() to 0.0 and
   the Math.sqrt(12.0) adjusts the std
   deviation to 1.0 to make it easy to
   convert to the mean and std deviation
   we want.
   

3. In the method fixProps of the class AssetAllocEngine, the following lines are within a for loop that steps through the periods of the investment problem:

         wealth = stock + cash;
         stock = wealth * inrisky;
         cash = wealth - stock;
   

   What do these lines do?

   The first line computes the wealth,
   the second line then gives the
   amount to hold (after trading) in
   the stock, and the third line
   computes the cash as a residual.
   

D. Advanced Questions Short Answer: up to 20 make-up points

These questions are intended to be more challenging conceptual questions. Answering these questions correctly can give you up to 20 points to substitute for points missed in Parts A-C.

1. In the binomial stock option pricing program (Crr in Homework 1), the expressions for up and down returns are based on a mean total stock return of one plus the interest rate, while in the corresponding expression in the binomial futures option pricing program (FutOp in Homework 3) the mean is one. Why are the two different?

   The stock is an investment (a store of
   value) and we need to get interest on
   the value we put in, while we do not
   put any money into a futures contract.
   

2. In the asset allocation simulation (AssetAllocEngine in Homework 4), the mean return of the stock is a parameter supplied to the program, while in the stochastic volatility option pricing simulation (SVPriceEngine in Homework 5), the mean is taken equal to the interest rate. Why are the two different?

   For the asset allocation simulation we
   care about returns in the actual
   probabilities investors will face, while
   for option pricing we want to use the
   risk-neutral probabilities that price
   derivative securities correctly.