Suppose that in Problem 10.15 the correlation among the S&P 500 index (mea- sured in dollars) and the FTSE a hundred index (measured in sterling) is zero.7, the cor- relation between the S&P 500 index (measured in greenbacks) and the dollar-sterling trade charge is 0.Three, and the daily volatility of the S&P 500 index is 1.6%.What is the correlation between the S&P 500 index (measured in bucks) and the FTSE one hundred index whilst it is translated to bucks? (Hint: For 3 variables X, Y, and Z, the covariance among X + Y and Z equals the covariance between X and Z plus the covariance among Y and Z.) 11.8 Suppose that variables V1 and V2 have uniform distributions wherein all values among 0 and 1 are equally possibly.Use a Gaussian copula to define the correlation shape among V1 and V2 with a copula correlation of zero.Three. Produce a table much like Table 11.Five considering values of 0.25, zero.50, and 0.Seventy five for V1 and V2. (A [12:31 PM, 11/9/2021] vii: spreadsheet for calculating the cumulative bivariate everyday distribution is at the author's internet site: www-2.Rotman.Utoronto.Ca/?hull/riskman.) 11.9 Assume that you have independent random samples z1, z2, and z3 from a trendy everyday distribution and need to convert them to samples ?1, ?2, and ?3 from a trivariate regular distribution the usage of the Cholesky decomposition. Derive 3 formulation expressing ?1, ?2, and ?3 in phrases of z1, z2, and z3 and the three corre- lations which might be needed to outline the trivariate regular distribution. 11.10 Explain what is supposed by tail dependence. How are you able to range tail dependence by way of the choice of copula? 11.11 Suppose that the marginal distributions of V1 and V2 are wellknown regular distribu- tions however that a Student's t-copula with four degrees of freedom and a correlation parameter of zero.Five is used to outline the correlation among the variables. How would you bought samples from the joint distribution? 11.12 In Table eleven.5, what's the chance density feature of V2 conditional on V1

1. Let a stock price S(t)be a geometric Brownian motion under the risk-neutral measure P", i.e., dS(t) = rS{t]dt + aS(t)dW(t), where r and o > 0 are constant and W is a Brownian motion under the risk- neutral measure . Consider a derivative security that pays 5'2 (T) at time T . Construct a portfolio that trades in the stock and a money market account with constant rate of interest rso that the nal value of the portfolio X(T)is S2 (T) almost surely. In particular. specify what X[[})should be, and how many shares of stock A(t] the portfolio should hold at each time t? 2. We have a stochastic interest rate given by the process R\") = 1' + 0W6), where a" and 1" are positive constants. Using this interest rate. determine the value of a zero-coupon bond at time t. {Hintz Assume an afne-yield model) Recall that our stock price process is modelled by S = So exploWt + (a - 10?)t} for te [0, T], where o E R, o > 0, and (Witco,r] be a Brownian motion. This process (Sthejo, is called a geometric Brownian motion. Let c be a real number. The process X = etc te at fedW, is called the Ornstein-Uhlenbeck process. Show that it is a solution to the SDE dXt = -ax dt + dW