Monte Carlo Methods in Finance

Homework: Chapter 1

Please enter your answers in the homework unit at the end of the Chapter.The solutions to this homework will be posted in a separate unit after the duedate.

1. Consider the time series of daily closing prices of IBM, GOOGLE andSIEMENS shares, which are stored in the first, second, and third columnsof the data file

closingPrices_IBM_GOOG_SI_2007_07_01_2013_06_30.txt,

respectively (see the reference How do I obtain the time series of prices ofa financial asset? in Unit 1.0 for details on how to create this file). Opena session in either Octave or Matlab. Load these time series and storethem in a matrix S by entering the command

S = load(’closingPrices_IBM_GOOG_SI_2007_07_01_2013_06_30.txt’);

in the Octave/MATLAB command window. The series of IBM prices willbe stored in S(:,1), the first column of S, the GOOGLE prices in S(:,2)

and the SIEMENS prices in S(:,3).

From the time series of daily closing prices, compute the time series ofdaily log returns using the command

logRetS = log(S(2:end,:)./S(1:end-1,:));

Make a plot of the time series of daily closing prices and of the corre-sponding log returns for the three assets. Compute the average and thestandard deviation of both types of time series. Which of the followingstatements are true?

• For each of the three assets the average of the log returns is approx-imately zero.

• For each of the three assets the average of the log returns is approx-imately one.

• The average of GOOGLE prices is higher than the average of SIEMENSprices.

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• The historical volatility (i.e. the standard deviation of the log re-turns) of GOOGLE is higher than the historical volatility of SIEMENS.

• The time series of prices are approximately stationary in the weaksense.

Hint: Plots in Octave or MATLAB can be generated with the commandplot. Use the function mean to compute the means and the function std

to compute the standard deviations.

2. We continue studying the time series of asset prices from the previousexercise. What is the approximate value of the correlation between thelog returns of IBM and GOOGLE?

• 0.0

• 0.3

• 0.4

• 0.5

• 0.6

Hint: Correlations can be computed using the Octave/MATLAB functioncorr.

3. We continue with the analysis of the time series from the previous exer-cises. We will now compute the autocorrelations of the time series of logreturns, assuming that they are stationary. The autocovariance at lag nof a stationary time series {X1, X2, . . . , XN} is

γn = E [(Xm −E[Xm]) (Xm+n −E[Xm+n])] .

The lag-n autocorrelation is the autocovariance normalized by the stan-dard deviations

ρn =E [(Xm −E[Xm]) (Xm+n −E[Xm+n])]√

var[Xm] var[Xm+n].

The autocorrelations provide a measure of the linear predictability of thefuture values of the series in terms of the current values of the series. Whatis the closest value to the lag-1 autocorrelation of the IBM log returns?And of the the square of the IBM log returns?

• ρ1[log-ret] = 0.2, ρ1[log-ret2] = −0.2

• ρ1[log-ret] = 0.2, ρ1[log-ret2] = 0.0

• ρ1[log-ret] = 0.2, ρ1[log-ret2] = 0.2

• ρ1[log-ret] = 0.0, ρ1[log-ret2] = −0.2

• ρ1[log-ret] = 0.0, ρ1[log-ret2] = 0.0

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• ρ1[log-ret] = 0.0, ρ1[log-ret2] = 0.2

Hint: The lag-n autocorrelation of a time series can be obtained by com-puting the correlation between the values of the time series and the valuesof the same series displaced n time steps. Assuming that the time series isstored in vector X the Octave/MATLAB command to compute the lag-nautocorrelation is

autocorr = corr(X((1+n):end), X(1:(end-n)));

4. Consider an asset whose initial price at t0 (today) is S(t0) = S0. Theowner of a European digital call option on this asset with maturity t0 +Treceives an amount of money A if the value of S(t0 + T ) is above K (thestrike of the option) and 0 otherwise. Therefore, the payoff at maturity ofthis digital call option is

payoff (S(t0 + T )) =

{A if S(t0 + T ) ≥ K0 if S(t0 + T ) < K

Compute the price of a digital option whose parameters are S0 = 100,K = 90, r = 0.05, T = 2, σ = 0.4 and A = 90.

• 91.05

• 78.22

• 30.76

• 51.27

• 43.32

• 24.76

Hint: The price of this derivative product can be computed by meansof the function priceEuropeanOption. This function requires as input ahandle to another function that computes the payoff of the option. Forthe digital call, the handle is

digitalPayoff = @(ST)((ST >= K)*A);

Recall that in Octave/MATLAB the expression (ST >= K) evaluates to 1if ST ≥ K and to 0 otherwise. If either ST or K are vectors, the comparisonis made element by element.

5. Consider an asset whose initial price is S(t0) = S0. A European call optionis a derivative product that gives the owner the right to buy a specifiednumber of shares of the asset at some time in the future t0 +T (maturity),for a price K (the strike), which is set at t0, the time when the option iswritten. To acquire this right one needs to pay a fee, which is called thepremium or price of the option. At maturity, the seller (or writer) of the

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call option has the obligation to sell the asset at the agreed price (K) ifthe owner of the option decides to exercise her right to buy.

A European put option is a derivative product that gives the owner theright to sell a specified number of shares of the asset at some time in thefuture t0 + T (maturity), for a price K (the strike), which is set at t0, thetime when the option is written. At maturity, the seller (or writer) of theput option has the obligation to buy the asset at the agreed price (K) ifthe owner of the option decides to exercise her right to sell.

Let C be the price of the European call and P the price of the Europeanput. Derive the so-called Put-Call parity relation, which is a closed-formexpression for C - P , the difference between the prices of the call andthe put options. Use this expression to answer the following question:how does C − P behave when σ, the volatility of the underlying asset,increases?

• C − P decreases.

• C − P increases.

• C − P increases if S0 ≥ Ke−rT and decreases if S0 < Ke−rT .

• C − P decreases if S0 ≥ Ke−rT and increases if S0 < Ke−rT .

• C − P is independent of the volatility.

Hint: The price of a derivative product is the discounted expected valueof the corresponding payoff under risk-neutral probability

Price(t0) = e−rTE[Payoff(t0 + T )],

where r is the risk-free interest rate. The payoff at maturity for a calloption is max(S(t0 + T )−K, 0). Derive the expression of the payoff of aput option.

Consider an investor who purchases a call option and writes a put option.The payoff of this investor’s portfolio is the payoff of the call minus thepayoff of the put. Compute the value of this portfolio as the discountedexpected value of the payoff under risk-neutral probability. This value isC − P , the difference between the prices of the call and the put options.

In the derivation you will need to use the following results:

• The discounted expectation of the value of the asset at any time inthe future is its price today

e−r(t−t0)E [S(t)] = S0, t > t0.

• The expected value of a constant is the constant itself

E [K] = K

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• Taking expectations is a linear operation

E[a g1(X) + b g2(X)] = aE[g1(X)] + bE[g2(X)],

where a and b are arbitrary real constants.

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