北美精算师Course8V——SOA真题5
 
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　　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
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　　(a) Evaluate potential problems in using this method to model the asset returns.
　　(b) Explain and *uate 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 *uate 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 *uate 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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