Questions and Answers of Introduces Quantitative Finance

Theprobability density function (PDF) for daily profits at Pyramid AssetManagement can be described by the following functions (see Figure 3.4),The density function is zero for all other values of
You estimate the standard deviation of daily returns for XYZ Corp.’s stock at 2.00%.XYZ does not pay a dividend. You own a put option and a call option on XYZ, both with one year to expiration and
You are asked to evaluate the risk of a portfolio containing two securities. The first security has an expected shortfall of −$400 (a loss of $400).The second security has an expected shortfall of
Which of the following distributions are associated with EVT:a) Poissonb) Weibullc) Fréchetd) Claytone) Gumbel
You have created a Monte Carlo simulation to calculate the one-day, 99% VaR of a portfolio containing a large number of options. In your simulation, you generate 1,000 sample one-day returns. The
ABC stock is currently trading at $100 and has a one-day 95% VaR of $4. What is the one-day 95% VaR of a portfolio that consists of $200 of cash and $800 of ABC?
The CEO of your firm wants to know what the worst possible monthly return could be over the coming year. You decide to use extreme value theory to answer this question.Specifically, you decide to use
Based on your analysis of Company ABC’s stock returns, rABC, you develop the following OLS regression model:where rA and rB are two uncorrelated indexes, and ε is a mean-zero disturbance term.If
You are asked to evaluate the risk of a portfolio that is long $100 of AAPL and short $100 of XOM. The standard deviation of returns for AAPL is 40%. The standard deviation of returns for XOM is 30%.
Calculate the population covariance and correlation of the following series: Series #1 Series #2 21% 20% 53% 32% 83% 80% 19% 40%
Given two random variables, XA and XB, with corresponding means μA and μB and standard deviations σA and σB, prove that the variance of XA plus XB is GA+B = 0 +ob +2PABGA OB
Prove Equation 5.29. That is, given the standard univariate regressionprove thatwhere σX and σY are the standard deviations of X and Y, respectively, and ρXY is the correlation between the two.
You perform a regression analysis of a hedge fund’s returns against an industry benchmark.You have 50 data points.The total sum of squares (TSS) is 13.50%.The residual sum of squares (RSS) is
Calculate the Cholesky decomposition for the following covariance matrix: 4 14 16 = |14 50 58 16 58 132
The following regression equation describes the daily returns of stock XYZ, rXYZ, in terms of an index return, rindex, and a mean-zero disturbance term, ε:where α and β are constants, ε is mean
Given the following annual return series for Fund A and Fund B, determine the coskewness, SAAB. Year Fund A Fund B 1 -3% -3% 2 - 1% 0% 3 0% 3% 4 1% 1% 5 3% - 1%
In addition to the assumption in the previous question, assume rindex has a mean of 0.05% and a standard deviation of 1.5%. What is the expected value of rXYZ? What is the standard deviation of rXYZ?
We will discuss bond defaults in greater detail in Chapter 8. In general, bond defaults tend to be correlated, but under certain circumstances, you may be able to assume that defaults are
You are evaluating the risk of two binary call options. The options have the same expiration date, but different underlying stocks.The following probability matrix gives the probabilities that either
Given the following joint probability density function, determine if X and Y are independent.where c is equal to 3/256. --- f(x, y) =. -2> X> 2,-2 > Y> 2 c(8-x2-) -2 x 2,-2 Y 2
Look at the two contour plots in Figure 6.12. For each plot, are the two variables positively or negatively correlated? Is the joint distribution elliptical or nonelliptical? FIGURE 6.12 Contour
Calculate Spearman’s rho for the FGM copula, Equation 6.20, in terms of α. C = uv[1 + a(1u)(1 v)] (6.20)
You have created a Monte Carlo simulation to model the risk of a stock index over 10,000 days. All of the daily log returns have a mean, standard deviation, skewness, and excess kurtosis of 0.10%,
You have been given the following VaR and iVaR statistics for a portfolio that contains three positions. What is the iVaR of Position B? Assume the iVaR and VaR are both one-day 95%, and have been
The total VaR of a portfolio is $10 million. The portfolio contains a position in XYZ worth $1 million, with an iVaR of $3 million. What would the approximate VaR of the portfolio be if the XYZ
You have developed the following regression analysis to describe the returns to a portfolio, rp, in terms of the returns to an index, rIndex, a mean-zero disturbance term, ε, and two constants, α
A portfolio, with a standard deviation of $30 million contains three positions.The three positions, in turn, have standard deviations of $10 million, $20 million, and$30 million. What is the
A portfolio contains three securities. You perform principal component analysis on the securities and find that the three principal components explain 60%, 30%, and 10%of the overall variance. What
You are interested in creating a levered version of an ETF. The ETF has a Sharpe ratio of 0.80 and an annual return standard deviation of 20%.The risk-free rate is 2%. Assuming you can borrow at the
A hedge fund is considering hiring two different portfoliomanagers, Alice and Bob.The fund’s Sharpe ratio is currently believed to be 1.00. Based on historical data, you believe that Alice has a
What is the present value of a two-year bond with an annual coupon of 10% and a$1,000 notional? Assume there is no probability of default and that the risk-free rate is 5%.
A one-year bond has a 4% coupon and a notional of $100. There is a 12% probability that the bond will default.The expected recovery rate is 20%.What is the present value of the bond? Assume a
A two-year bond with a 10% coupon and a notional of $100 is selling for $85. What is the yield of this bond?
A one-year zero-coupon bond with a 19% probability of defaulting currently has a market price of $72. The risk-free rate is 5%.What risk premium are investors demanding to accept the credit risk of
A company has $2,076 of debt outstanding and a market capitalization of $1,924. All of the debt matures in exactly one year. What is the probability that the company will default? Ignore drift and
The following table contains price and volume data over 15 days for shares of AAPL.What is the 15-day average trading volume in terms of shares? What is the 15-day average dollar volume? Calculate
A bond portfolio contains 40 identical bonds, each with a 5%probability of defaulting.Defaults are independent. What is the probability that exactly four bonds default?
You own 2 million shares of IBM. The current price of IBM is $80 and the average daily volume of IBM is 4 million shares. How many days’ volume of IBM do you own?
A trader has $1,000 to invest.The trader can either buy 1,000 shares of ABC at $1, or buy 1 share of XYZ, for $1,000. The bid-ask spread for ABC is $0.01, and for XYZ is $1. Both stocks have an
Assume the demand function for XYZ exhibits a constant elasticity of −4. The current demand in the market is 10 million shares, and the current price of XYZ is $64. If you try to sell an additional
The probability that gross domestic product (GDP) decreases is 20%.The probability that unemployment increases is 10%. The probability that unemployment increases given that GDP has decreased is
An analyst develops a model for forecasting bond defaults. The model is 90% accurate.In other words, of the bonds that actually default, the model identifies 90% of them;likewise, of the bonds that
As a risk analyst, you are asked to look at EB Corporation, which has issued both equity and bonds. The bonds can either be downgraded, be upgraded, or have no change in rating. The stock can either
Your firm is testing a new quantitative strategy. The analyst who developed the strategy claims that there is a 55% probability that the strategy will generate positive returns on any given day.
Your firm has created two equity baskets. One is procyclical, and the other is countercyclical. The procyclical basket has a 75% probability of being up in years when the economy is up, and a 25%
Calculate Kendall’s tau and Spearman’s rho for the following data set. A B C X 70% 40% 20% Y 5% 35% 10%
Calculate Kendall’s tau for the independent copula, C(u, v), given by C(u, v) = uv
You are an analyst at Astra Fund of Funds, but instead of believing that there are two or three types of portfolio managers, your latest model classifies managers into four categories. Managers can
You are monitoring a new strategy. Initially, you believed that the strategy was just as likely to be up as it was to be down or flat on any given day, and that the probability of being up was fairly
You have a model that classifies Federal Reserve statements as either bullish or bearish.When the Fed makes a bullish announcement, you expect the market to be up 75%of the time. The market is just
For the Bayesian network in Figure 10.7, each node can be in one of three states: up, down, or no change. How many possible states are there for the entire network?What is the minimum number of
Calculate the correlation matrix forNetwork 1, the network on the left, in Figure 10.6.Start by calculating the covariance matrix for the network. FIGURE 6.6 Joint Standard Normal Distribution with
Calculate the correlation matrix for Network 2, the network on the right, in Figure 10.6. Start by calculating the covariance matrix for the network. FIGURE 6.6 Joint Standard Normal Distribution
You are offered the opportunity to invest in an option with a 60% chance of gaining $100, and a 40% chance of losing $25. How much would you be willing to pay for this option if you were risk averse?
Assume that a consumer’s utility can be described by the following utility function:where α is a constant and C is the level of consumption. Currently the consumer’s consumption is uncertain,
You are the owner of a widget factory that is expected to generate $100 in profits over the next 12 months. Unfortunately, your factory is in a flood plain. There is a small chance that a flood will
Consider an investor who is subject to habit formation, where the investor’s habit level is a simple weighted average of consumption in the previous three periods,and the investors utility in any
You are the chief risk officer for a large pension fund. In a recent meeting, the manager of the fund’s equity portfolio, explained how she planned to shift the portfolio away from large retail
In the previous chapter, we calculated the VaR for Box Asset Management, which has profits that are described by the following PDF:What is the one-day 95% expected shortfall of Box Asset Management?
Calculate Halton sequences for bases 2, 3 and 4. Compare the results (you may wish to plot one base against another). What do you notice? How should this affect the bases you choose for a
Repeat the above using Halton sequences.Previous questionValue a European call option using a Monte Carlo simulation. Use the simulation to estimate the value of an integral, as opposed to simulating
Value a European call option using a Monte Carlo simulation. Use the simulation to estimate the value of an integral, as opposed to simulating the random walk for the asset.
Why is it difficult to use Monte Carlo simulations to value American options?
Modify the above spreadsheet to value various exotic options such as barriers, Asians and lookbacks.
Modify the spreadsheet in the above to ouput the option’s delta.
Simulate the risk-neutral random walk for an asset using a spreadsheet package, or otherwise. Use these data to calculate the value of a European call option.
Alter your compound option program to value a chooser option which allows you to buy a call or a put at expiry.
Write a program to value compound options of the following form:(a) Call on a call;(b) Call on a put;(c) Put on a call;(d) Put on a put.
Write a program to value a down-and-out call option, with barrier below the strike price.
Adjust your program to value call options with the forward price as underlying.
Write a program to value European call and put options by solving the Black–Scholes equation with suitable final and boundary conditions. Include a constant, continuous dividend yield on the
Sudden, large movements in a stock are usually accompanied by an increase in implied volatilities. Incorporate a vega term into the one-asset CrashMetrics model. What is the role of actual volatility
Extend the example CrashMetrics spreadsheet to incorporate many underlyings, all related via an index. How would interest rate products be incorporated?
We have seen how to apply the CreditMetrics methodology to a single risky bond, to apply the ideas to a portfolio of risky bonds is significantly harder since it requires the knowledge of any
Construct the intermediate steps in the derivation of the equation for the value of a risky bond (when default is governed by a Poisson process): av av + at + (u – Aw) - (r+p)V = 0. ar ar?
Repeat the analyses of the above two problems but with the extra assumption that on default their is a recovery rate of 10%, i.e. you will receive $10 at maturity if there has been default.
The same government and company as in the above question now issue five-year bonds with $100 principal and prices for government bond and corporate bond of $82.15 and $78.89 respectively. What does
A risk-free government, zero-coupon bond, with a principal of $100, maturing in two years has a value of $91.75. A risky corporate zero coupon bond with the same principal and maturity is worth
What criticisms of Value at Risk as described here can you think of? Consider distributions other than Normal, discontinuous paths and non linear instruments.
Assuming a Normal distribution, what percentage of returns are outside the negative two standard deviations from the mean? What is the mean of returns falling in this tail? (This is called the
What are the economic significances of α and β in the Capital Asset Pricing Model and how are they measured or estimated in practice?
Where should we be on the efficient frontier in question 1 if we wish to minimize the chance of a return less than 0.05?
Find the efficient frontier for the assets in the table above, when asset D is replaced by a risk-free asset, E, which has a mean of 0.10 over our time horizon of two years. Asset E is uncorrelated
Work out the efficient frontier for the following set of assets:The correlation coefficients between the four assets are given by Asset A 0.08 0.12 B 0.10 0.12 0.10 0.15 0.14 0.20 9.
Build a spreadsheet that simulates the above, in which you keep betting the same fraction of your accumulated wealth.
A die is weighted so that the probability of getting a 6 is 0.3, of getting a 1 is 0.1, and of getting 2, 3, 4 or 5 is 0.15 each. You bet on 6, receiving odds of 5 to 1 (fair odds for an unbiased
Using forward rate data, perform a principal component analysis. What are the three main components in the forward price movements and what are their weights?
Perform a simulation, using the method of section 19.7, to value an option on a zero-coupon bond. You will need to decide upon a suitable form for ν(t, T ), the forward rate volatility. Does your
Derive the equation for the evolution of the forward rate:fromand (3o²4, 7) – Hit. T) at -Tot, a (t, T) dX, dF(t; T) =
Find the approximate value of a cashflow for a floorlet on the one-month LIBOR, when we use the Vasicek model.
When an index amortizating rate swap has a lockout period for the first year, we must solvewith jump conditionwhereg(r, i) =1 if ti < 1,and with final conditionV(r, P, T ) = (r − rf )P.In this
How would a collar be valued practically? What is the explicit solution for a single payment?
Derive a relationship between a floorlet and a call option on a zero-coupon bond.
Write down the problem we must solve in order to value a puttable bond.
Use market data for zero-coupon bond prices to fit a Vasicek model for the interest rate.
Use market data for the price of zero-coupon bonds to fit a Ho & Lee model. Refit the model with data for a week later in time. Compare the two curves for η ∗(t).The second curve for η∗
Differentiate Equation (17.2) twice to solve for the value of η ∗(t). What is the value of a zero-coupon bond with a fitted Vasicek model for the interest rate?Equation (17.2) 1 3 :- n"(s)B(s;
Substitute the fitted function for A(t; T ), using the Ho & Lee model, back into the solution of the bond pricing equation for a zero-coupon bond,Z(r, t; T ) = eA(t;T )−r(T−t).What do you
What form does the bond pricing equation take when the interest rate satisfies the Vasicek modeldr = (η − γ r)dt + β1/2dX?Solve the resulting equations for A and B in this case, to findand 1 z(B
What final condition (payoff) should be applied to the bond pricing equation for a swap, cap, floor, zero-coupon bond, coupon bond and a bond option?

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