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Pricing credit derivatives under a jump-to-default extended Black-Scholes mode We consider a jump-to-default extended Black-Scholes (JDBS) model. The pre-default stock price process is a geometric Brownian motion, dSt=Stdt+StdWt, where and >0 are constants, and {Wt:t0} is a standard Brownian motion under the physical probability measure P. Upon default, the stock price drops to zero. The default intensity process {t:t0} is a square-root diffusion, dt=(t)dt+tdBt, where ,,>0 are constants and {Bt:t0} is a standard Brownian motion under the physical probability measure P. The default time is specified as =inf{t0:0tsdsE}, for an independent exponential random variable EExp(1). Then the defaultable stock price St,t0 is set as St={S~t,tT]) (II) Proportional recovery which pays the holder S at the default time if default occurs before T. Its price is given by Ppr=E[erS1{T}]. You are required to answer the following questions regarding the pricing of these two credit derivatives. Set the parameters as r=0.02,q=0.01,T=1.0,=0.20=0.8,=0.3, =0.8,=0.2,=0.5,=0.5 and S0=100. The simulations of St and t are not necessary to be exact. 1. Show that Q[>T]=E[e0Ttdt] 2. Based on (1), propose a Monte Carlo simulation estimator for Pfi. 3. Propose variance reduction techniques for the estimator in (2) with appropriate control variate and antithetic variate respectively. 4. Implement the estimators in (2) and (3) and compare their performance (variance) by numerical experiments. 5. Show that St=S0e(rq+t2/2)t+Wt. 6. Show that E[erS1{rT}]=0TertE[e0tsdstSt]dt=TE[erTU0TUsdsTUSTU], for an independent random variable UU(0,1). 7. Based on (5) and (6), propose a Monte Carlo simulation estimator for Ppr and implement it