(a) Implement the right endpoint rule as a function in R. Use it to compute L /2 22 dz for several discretizations h. Compare your results to the exact answer and verify the order of convergence of the right endpoint method. (b) How might you use a numerical integrator to compute the integral L 212 dz? Implement a function that takes as input a and the number of subintervals n and numerically approximates the integral using a scheme of your choice. (c) The Black-Scholes formula for a European call option is: C()=Se(d+) - Ke*(-)(d) where (2)= 2T d+ log () + (r-q+) (Tt) oT-t dd-oT-1 = log () + (rq) (T-1) +(8-9-) Using your results from (b), write a function in R. that takes as input S, T, t, K, r, 8, and q and outputs the Black-Scholes call option price with all integrals computed numerically. (d) Use one of your finite difference schemes to approximate one of the Greeks using the function from (c) to compute the data points. Compare this to the exact answer, and comment on how the numerical integration affected your accuracy. In connection with the deduction of the Black & Scholes formula for the price of an option the following partial differential equation appears: u t =-pu+ax +x; t> 0, x R Ju | u(0, x) = (x K) + ; - 20u 12 x R, where p > 0, a, and K > 0 are constants and - (x K)+ = max(x K,0). - Use the Feynman-Kac formula to prove that the solution u of this equation is given by u(t, x) = e-pt (2t - 1 (x exp{(a } )t + y} K)+edy ; R t>0. Initials: 11 of 13 10. (10 pts) This question is about black holes, but you don't need to know anything about black holes to answer it. You just need thermodynamics. A black hole is an object whose gravitational escape velocity is the speed of light. From this definition and your mechanics class, you could easily calculate that the radius of a black hole of mass M is given by RBH = 2GM/c, where G is the gravitational constant and c is the speed of light. From your modern physics class, you could argue that the internal energy of a black of mass M is given by UBH = Mc. The entropy of a mass M black hole is given by the formula SBH (4kG/hc) M, which you should take as given (k is the Boltzmann constant). Actually, there a arguments (based on thermodynamics and a little relativity) that this formula is nobody knows how to get it by counting microstates---we don't know what the microstates of a black hole are! ) Please compute the temperature of a mass M black hole, TBH. More detail alculations show that black holes actually emit black-body radiation at this t