We consider an investment in a European call option. The investment horizon is [0,7]. The option...

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We consider an investment in a European call option. The investment horizon is [0,7]. The option will mature at T > 7 with strike price K and it is written on a non-dividend-paying stock whose price follows a geometric Brownian motion: dS₁ = μSedt + o SidW₁ = rStdt+oSidWt, t> 0, where {W,:t> 0} and {W₁ : t≥ 0} are standard Brownian motions under the physical proba- bility measure P and risk-neutral probability measure Q respectively. From the two dynamics above, we have that, W₁ = W₁+"="t₁ t≥ 0. We are interested in the return and risk of this option position at the end of investment horizon. The rate of return R, can be calculated as follows: R where Co and C, are the option prices at time 0 and 7 respectively. For the following questions, we set the parameters as = 0.05, r =0.02, = 0.3, So K = 200, T = 0.5 and T = 1.0. Cr-e Co et Co 100, 1. Estimate the expected return EP[R] with standard Monte Carlo simulation. 2. Propose and implement an importance sampling estimator that improves your estimator in (1). 3. Based on (2), propose and implement a pathwise estimator and a likelihood ratio estimator for the sensitivity of expected return w.r.t. So, i.e., DE[Rr] aso We consider an investment in a European call option. The investment horizon is [0,7]. The option will mature at T > 7 with strike price K and it is written on a non-dividend-paying stock whose price follows a geometric Brownian motion: dS₁ = μSedt + o SidW₁ = rStdt+oSidWt, t> 0, where {W,:t> 0} and {W₁ : t≥ 0} are standard Brownian motions under the physical proba- bility measure P and risk-neutral probability measure Q respectively. From the two dynamics above, we have that, W₁ = W₁+"="t₁ t≥ 0. We are interested in the return and risk of this option position at the end of investment horizon. The rate of return R, can be calculated as follows: R where Co and C, are the option prices at time 0 and 7 respectively. For the following questions, we set the parameters as = 0.05, r =0.02, = 0.3, So K = 200, T = 0.5 and T = 1.0. Cr-e Co et Co 100, 1. Estimate the expected return EP[R] with standard Monte Carlo simulation. 2. Propose and implement an importance sampling estimator that improves your estimator in (1). 3. Based on (2), propose and implement a pathwise estimator and a likelihood ratio estimator for the sensitivity of expected return w.r.t. So, i.e., DE[Rr] aso

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