3) This question concerns the use of the binomial model to approximate the price of options under the Black-Scholes model. Let the binomial tree be parameterized as follows: u=eh+hd=ehhh=1. A) Create a BinOptionPrice function in R that takes the variables as input: - S : the current price of the risky asset (St), - K : the exercise price of the option (K), - r : the risk-free rate ( r ), - T_t: the time in years until the option expires (Tt), - mu: the parameter in the binomial tree, - sigma: the parameter in the binomial tree, - n : the parameter n in the binomial tree, - isput: a Boolean variable such as TRUE indicates a put option and FALSE a call option, and which gives as output the value of a European vanilla option under the specified binomial model (either a call option or a put option). B) Evaluate your function for a call and put option with the following parameters: S0=100, K=105,r=2%,T=0.5,=20%,=r2/2,n=20. C) Using your BSOptionPrice and BinOptionPrice functions, produce numerical results which illustrate, for different values of , the convergence of the price of European vanilla options as a function of n towards the given price e by the Black-Scholes formula. Consider in particular the Cox-Ross-Rubinstein tree and the lognormal tree. Use the following assumptions: S0=100,K=105,r=2%,T=0.5,=20%. D) In the previous question, discuss the impact of the choice of the parameter . Is there a value of for which you observe faster convergence? E) Produce numerical results which illustrate that the random variable ST in the risk-neutral binomial tree is approximately lognormal LN(logS0+(r2/2)T,2T) when n is large . Use the following assumptions: S0=100,K=105,r=2%,T=0.5,=20%. Suggestion: Plot the density of ST in the binomial tree and that of a lognormal random variable LN(logS0+(r2/2)T,2T). F) Explain why using the binomial model to approximate the price of exotic options under the Black-Scholes model is not generally appropriate from a computer point of view