1. (30 points) Consider a Term Life insurance contract with:- Policyholder is 45 years old at the start of the contract.- Single premium P, so the policyholder only pays at the start of the contract (t = 0).- Benefit of $100,000, which is paid at the end of the year of death.- Maturity of the contract is 5 years.- Condition to receive the benefit is that the policyholder has to die before the maturitydate.- r = 0.03 is the interest rate the company is expecting to earn each year. We use continuouscompounding, which means that v = er⇥t.The actuarial value is used to price the product, meaning to find a premium at t = 0 for thecontract. The premium is equal to:P = 100, 000 A 145:5= 100, 000 X4k=0vk+1k|q45= 100, 000 (v q45 + v21|q45 + v32|q45 + v43|q45 + v54|q45)= 100, 000 (v q45 + v2 p45 q46 + v32p45 q47 + v43p45 q48 + v54p45 q49)where npx is the probability that somebody aged x will survive for n years and qx is the probabilitythat somebody aged x will die in the next year.For the calculation of the premium, you are given the following vector:q = ⇥q45 ; 1|q45 ; 2|q45 ; 3|q45 ; 4|q45⇤q = [0.080780387 ; 0.087867314 ; 0.095821596 ; 0.104749008 ; 0.11477648]This means, there is a 0.095821596 probability that the policyholder dies in the third year.Page 2Questions:a. (5 points) Calculate the single premium the policyholder needs to pay for this contract.b. (10 points) Set up a Risk Management strategy such that the company is able to pay thepolicyholders until time T = 5. Do this first for 1 policyholder.- At time t = 0 the policyholder is alive and signs the contract. You want to test in100 000 scenarios if the policyholder survives or dies in the upcoming 5 years, untilthe contract ends.- Liabilities: Consider the random variable Vt, which denotes the value of the liabilities.For example between t = 0 and t = 1 there are two possibilities: or he diesand there is a payout at the end of the year, or he doesn’t die and the contract staysin force without payment.- Assets: Consider a random variable At, whereAt = P eYt where Yt ⇠ N(µY t, 2Y t)We assume that µY = r2Y2 . If for example Y= 0 we have thatAt = ert- Look at all the different time-steps of the contract until you reach year 5 (maturity).t = 0 t = 1 t = 2 t = 3 t = 4 t = 5You will have for each time-step a variable V , A and L. You can do this by using aloop.- At t = 5, you make a histogram of the loss random variable you have at that time.- Explain what you see.c. (5 points) Do the same for when you have 100 policyholders and 100 000 policyholders.Explain what happens.d. (10 points) Calculate the premium for which the insurer only has a 30% loss. Don’t dothis by trial and error but create a function that helps you in calculating this.Page 32. (50 points) We need the following information for the this question:Brownian motion (Wiener process):Definition:A stochastic process X = {Xt, t0} is called a standard Brownian motion if1. X0 = 02. X has independent increments: for 0 s1 Xs1 and Xt2Xs2 areindependent random variables.3. X has stationary increments: The statistical distribution of Xt+sXs is independent ofs (and so identical in distribution to Xt).4. Xt+sXt ⇠ N(0, s)5. X is a continuous function of t.The standard brownian motion is denoted by W = {Wt, t0}, where W stands for Wiener.The geometric Brownian motion for the stock price:It can be shown that the stock price St has a lognormal distribution:Consider a trader who has a position of 100 000 written call options on a non-dividend-payingstock. The portfolio is hedged weekly and the time to maturity of the option is one year. Wehave the following information:- Stock price is S0 = $100- Strike price is K = $100- N = 52 weeks and T = 1 year- r = 3% (bank)- µ = 7%-= 20%Do the following steps in R:a. (5 points) Find the price of 1 call option.b. (5 points) Plot the option price (y-axis) in function of the underlying stock prices (xaxis).How can we relate delta with this graph.c. (5 points) Find the delta of this option at time t = 0.Plot delta (y-axis) in function of different stock prices (x-axis). Explain the graph.d. (10 points) Create a dynamic hedging strategy by simulating the stocks with the abovegeometric Brownian motion process. You should have something as followsStock Shares Cost of CumulativeWeek Price Delta Purchased Shares Purchased Cash Outflow Interest Cost0 100 0.5987 59,870.6326 5,987,063.26 5,045,723 2,911.8341 101.63551 0.6291 3,040.0916 308,981.26 5,357,616 3,091.824– Explain what代做Risk Management作业、R程序语言作业调试、R课程设计作业代写 帮做Haskell程序|代做R语言程序 happens in week 0, 1 and 2.– Explain what happens in week 52.– Calculate the total cost of hedging.e. (10 points) By running this program multiple times (e.g. 100 000), we will get eachtime a new stock price process and thus also the corresponding hedging cost. Plot thesedifferent hedging costs into a histogram and explain what you observe.f. (5 points) Instead of rebalancing once every week, we can also rebalance two times aweek or daily. Create a histogram of the hedging costs for both cases (twice per weekand daily) and explain what you see.g. (5 points) Do the same as in steps (d. and e.), but now with volatility= 30%,= 40%,= 50%,= 60%. Explain what you see in the different histograms of the hedgingcosts.h. (5 points) What happens if we change µ?Page 53. (5 points) Suppose that the expected return on a stock in the real world is 7%. The stock priceis currently $25 per share and its volatility is 20%. Assume you own 10,000 shares. Howmuch could you expect to lose during the next 6 months with a 99% level of confidence.4. (10 points) A financial institution has just sold 1,000 7-month European call options on theJapanese yen. Suppose that the spot exchange rate is 0.80 cent per yen, the exercise price is0.81 cent per yen, the risk-free interest rate in the United States is 8% per annum, the risk-freeinterest rate in Japan is 5% per annum, the volatility of the yen is 15% per annum.Assume that• Delta = 0.5250• Gamma = 4.206• Vega = 0.2355• Theta = 0.0399• Rho = 0.2231Question: Interpret each number (Delta, Gamma, Vega, Theta and Rho).Page 65. (15 points) A financial institution has the following portfolio of over-the-counter options oneuros:Type Position Delta of Option Gamma of Option Vega of OptionCall -1,500 0.7 2.5 1.5Call -400 0.9 0.55 0.45Put -1,850 -0.40 1.6 0.65Put -1,200 -0.70 1.75 1.2A traded option is available with a delta of 0.3, a gamma of 1.2, and a vega of 0.5.Questions:a. (5 points) What position in the traded option and in euros would make the portfolio bothgamma neutral and delta neutral?b. (5 points) What position in the traded option and in euros would make the portfolio bothvega neutral and delta neutral?c. (5 points) Suppose that a second traded option, with a delta of 0.1, a gamma of 0.5, anda vega of 0.6, is available. How could the portfolio be made delta, gamma and veganeutral?Page 76. (15 points) Figure 1 below shows the distribution of daily revenues of the company K in2018. In total there are 250 observations and the average daily revenue is about $5.56 millionwith a standard deviation of $9.25 million.Figure 1: The distribution of the company K’s daily revenues in millions of $a. (5 points) What is the 95% 1-day normal VaR?b. (5 points) What is the 95% 1-day historically simulated VaR? Assume an equal weightingscheme.c. (5 points) Suppose that the 95% VaR is breached on 20 trading days. Is the VaR measurein line with the confidence level set forth?Page 87. (15 points) Let us consider the market risk measurement of JPMorgan Chase & Co. during2007-2008. According to its 10-K financial reports, the average 1-day market risk VaR constituted$107 million in 2007, $196 million in 2008, while reaching $266 million on December31, 2008 and $100 million on December 31, 2007. For simplicity, assume that the averageVaR over 60 days ending on December 31, 2007 and December 31, 2008 coincide with theaverage VaR observed over 2007 and 2008, respectively. The number of VaR violations thatJPMorgan reports during 2007-2008 is summarized in Table 1 below:Year-Quarter Number of VaR violations (per quarter) Number of VaR violations (per year)2007Q1 0 0*2007Q2 0 0*2007Q3 5 5*2007Q4 3 82008Q1 2 102008Q2 0 102008Q3 1 62008Q4 0 3Table 1: The number of VaR violations reported by JPMorgan Chase & Co. during 2007-2008.Source: Quarterly (10-Q) and annual (10-K) filings of JPMorgan Chase & Co., available online*There were no VaR violations during 2006.a. (5 points) Based on Table 1 above, determine times during which the JPMorgan’s riskmodel passed the regulatory backtest and during which it was requested to be revised (orpossibly rejected).b. (5 points) What would be the 1-day market risk capital requirements of JPMorgan as atJanuary 2, 2008 and January 2, 2009 under Basel I and II? (assume there is no specificrisk charge)c. (5 points) What would the 1-day market risk capital requirements of JPMorgan as at January2, 2008 and January 2, 2009 become under Basel II.5? Since these are all backwardcomputations, let us consider 2008 as a year of the stressed market conditions in bothcases.Page 9转自：http://www.daixie0.com/contents/18/4503.html
