Overview

You will use R and Excel with the R package openxlsx to compute the value of European Call and Put options using the Black-Scholes model and the closed-form solution, a binomial lattice, Monte-Carlo simulation, and the explicit and implicit finite difference methods. You will be asked to describe each of the modeling methods to include providing mathematical equations using Words Equation Editor. For each of the binomial, Monte-Carlo simulation, and explicit and implicit finite difference methods, you will be asked to examine the convergence of the option prices to those obtained via the Black-Scholes closed-form solution as you shrink the length of the time-steps with each of the numerical methods. You will include your R code in an appendix with your submission.

Black-Scholes Closed Form Solution

Provide a brief description of the Black-Scholes closed form solutions for European Call and Put options to include the relevant formulae and assumptions of the model. Use Words Equation Editor for all formulaic expressions. Briefly discuss limitations of the model (e.g., can be used to value only European style options) and issues associated with the model (e.g., assumption of constant volatility). You will then value European Call and Put options for the following inputs:

Using the above inputs, plot the value of the European Call and Put options (two separate graphs) for the range of stock prices and times to expiration. Briefly discuss what you see and how the values comport with the Black-Scholes model and your knowledge of option values.

Binomial Lattice

Choose a lattice specification (e.g., Cox, Rubinstein, and Ross, or Jarrow and Rudd) to implement a binomial model to value European Call and Put options. Provide the relevant lattice values including the risk-neutral probability and up-and-down moves. Use Words Equation Editor for all formulaic expressions. Briefly discuss how one obtains the values for the risk-neutral probability and up-and-down moves so that the lattice produces option values which converge to that of the Black-Scholes closed form solution. You will then value European Call and Put options for the following inputs:

Using the above inputs, plot the value of the European Call and Put options (two separate graphs) for stock prices of 75, 100, and 125 and the number of time-steps to illustrate convergence of the lattice to the Black-Scholes closed for solution. That is, you will have 6 graphs (stock prices of 75, 100, and 125, European Call, European Put) where you can see how the value obtained from the lattice model converges to that of the Black-Scholes model for the same set of inputs. Briefly discuss what you see and how the values comport with the Black-Scholes model and your knowledge of option values.

Monte-Carlo Simulation

Implement Monte-Carlo simulation to value European Call and Put options. Provide any relevant mathematical expressions and describe how one uses Monte-Carlo simulation to value options. Use Words Equation Editor for all formulaic expressions. You will then value European Call and Put options for the following inputs:

Using the above inputs, plot the value of the European Call and Put options (two separate graphs) for stock prices of 75, 100, and 125 and the number of simulations to illustrate convergence of the simulated option value to the Black-Scholes closed for solution. That is, you will have 6 graphs (stock prices of 75, 100, and 125, European Call, European Put) where you can see how the value obtained from the simulation converges to that of the Black-Scholes model for the same set of inputs. Briefly discuss what you see and how the values comport with the Black-Scholes model and your knowledge of option values.

Finite Difference Methods

Implement the explicit and implicit finite difference methods to value European Call and Put options. Provide any relevant mathematical expressions and describe how one uses the finite difference approach to value options. Use Words Equation Editor for all formulaic expressions. You will then value European Call and Put options for the following inputs:

Using the above inputs, plot the value of the European Call and Put options (two separate graphs) for stock prices of 75, 100, and 125 and the number of changes in the stock-price dimension to illustrate convergence of the finite difference option value to the Black-Scholes closed for solution. That is, you will have 12 graphs (stock prices of 75, 100, and 125, European Call, European Put, Explicit Method, Implicit Method) where you can see how the value obtained from the finite difference method converges to that of the Black-Scholes model for the same set of inputs. Briefly discuss what you see and how the values comport with the Black-Scholes model and your knowledge of option values.