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北美精算师Course8V——SOA真题5

来源：高顿网校 2014-06-27

　　北美精算师Course8V——SOA真题5

　　高顿网校小编提醒您，以下北美精算师的试题是由高顿网校小编为您“量身定制”的。

　　COURSE 8: Fall 2003 - 17 - GO ON TO NEXT PAGE

　　Investment

　　Afternoon Session

　　21. (6 points) A model of the type GARCH(1,1) is being used to set volatility for integer

　　values of strike price for an option pricing model that is based on a lognormal

　　distribution. You have the following information on the model for a specific underlying

　　asset:

　　? A weight of 10% is given to a variance of 4% that is reached

　　asymptotically as the strike price increases toward infinity.

　　? The model assumes returns are a constant 20%.

　　? A weight of 80% is given to the variance determined for the prior strike

　　price.

　　? The variance is 9% for a strike price of 0.

　　(a) Calculate the change in volatility for each of strike prices of 1 and 2 assuming the

　　variance for a strike price of 0 decreases from 9% to 8%.

　　(b) Assess whether the model with the given parameters is more appropriate for an

　　equity option or a currency option. Support your answer by interpreting the

　　resulting implications for the volatility smile.

　　(c) Describe how to determine a maximum likelihood estimate for the above model,

　　assuming the distribution of returns conditioned on the variance is normal.

　　COURSE 8: Fall 2003 - 18 - GO ON TO NEXT PAGE

　　Investment

　　Afternoon Session

　　22. (7 points) You would like to calibrate a new Monte Carlo pricing model by comparing

　　the model price to the Black-Scholes price of a European call option on a non-dividend

　　paying stock. The European call option and the underlying stock have the following

　　characteristics:

　　Current stock value = 100

　　Expected growth rate of the stock price = 10%

　　Volatility = 25%

　　Exercise price = 95

　　Time to maturity = 6 months

　　Risk-free interest rate = 6%

　　Black-Scholes price of the option = 11.37

　　Assume that the percentage change in the stock price is normally distributed.

　　(a) Calculate the control variate technique adjustment using stratified sampling on 3

　　intervals.

　　(b) Describe additional techniques that can be used to improve the efficiency of the

　　Monte Carlo method.

　　(c) Describe how the Monte Carlo method can be adapted to calculate the price of

　　American options.

　　COURSE 8: Fall 2003 - 19 - GO ON TO NEXT PAGE

　　Investment

　　Afternoon Session

　　23. (6 points) You are given the following information about an interest rate collar with

　　quarterly resets. Assume quarterly compounding unless otherwise indicated.

　　Notional amount = 100,000,000

　　Strike interest rate for cap = 7.0%

　　Strike interest rate for floor = 4.5%

　　Time to expiration = 2 years

　　Time t Forward 3-month interest

　　rate starting at time t

　　Zero-coupon interest rate

　　to time t (continuous

　　compounding)

　　9 months 5.0% 4.49%

　　12 months 5.2% 4.64%

　　15 months 5.3% 4.77%

　　(a) Compare the collar to a portfolio of bond options with equivalent payoffs.

　　(b) Calculate, using Black’s model, the price of the floorlet that prevents the interest

　　rate from being lower than the strike price for 3 months starting in one year.

　　Assume that the volatility of the underlying three-month rate is 20%.

　　(c) Assess the suitability of Black’s model for valuing options embedded in a

　　mortgage backed security.

　　COURSE 8: Fall 2003 -20- STOP

　　Investment

　　Morning Session

　　24. (3 points) You have been asked to review the trading costs of the following stock:

　　Time Price Number of

　　shares traded

　　(Previous close is 44 1/8)

　　1:40 pm 44 1/8 1000

　　1:42 44 1/4 2000

　　1:45 44 500

　　1:46 43 7/8 5000

　　1:50 43 3/4 2000

　　1:51 43 7/8 400

　　1:54 44 1/4 6000

　　(a) Explain why the measurement of trading costs is important.

　　(b) Calculate the price impact of trading for the above stock using the “money flow”

　　system as adapted by Birinyi.

　　***END OF EXAMINATION***

　　COURSE 8: Fall 2003 - 15 - GO ON TO NEXT PAGE

　　Investment

　　Afternoon Session

　　19. (5 points) The current value of the assets and liabilities are respectively A0 and L0 . The

　　liability in five years is designated L5 . Senior management is concerned that the asset

　　portfolio could drop in value below the liability value in five years.

　　One method to model the asset returns would be using a multivariate normal model

　　and modified Cholesky decomposition with the following assumptions:

　　m = [0.12, 0.06, 0.085]=[equities, fixed income, real estate]

　　covariance matrix =

　　0.04 0.003 0.014

　　0.003 0.0025 0.0015

　　0.014 0.0015 0.01

　　é ù

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　　(a) Evaluate potential problems in using this method to model the asset returns.

　　(b) Explain and evaluate the use of an option-based portfolio hedging strategy for a

　　portfolio of assets, A0 , and liabilities, L0 , where A0L0 , using

　　(i) a put, and

　　(ii) a call.

　　(c) Demonstrate that the strategies in (b) for the put and call options are equivalent,

　　assuming that cash flows are reinvested.

　　(d) Explain and evaluate the use of dynamic hedging by replicating the put option

　　using a risky portfolio and a duration-matched Treasury portfolio.

　　COURSE 8: Fall 2003 - 21 - GO ON TO NEXT PAGE

　　Investment

　　Afternoon Session

　　20. (6 points) A financial institution has sold short three European call options on a stock

　　with the following characteristics:

　　Stock price: 100

　　Strike price: 110

　　Volatility: 20%

　　Expiry: 2 years

　　Contract size: 1,000 shares

　　Dividends: none

　　Risk-free rate: 5%

　　(a) (4 Points) Describe and evaluate strategies the financial institution can use to

　　manage the risk of this portfolio.

　　(b) (1 Point) Calculate delta, theta and gamma for the portfolio.

　　(c) (1 Point) Verify the calculations of delta, theta and gamma by showing that the

　　results for the portfolio satisfy the Black-Scholes-Merton differential equation.

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