Q.20 Finite difference methods a) Show how we can tie the methods of finite difference to the trinomial trees.

b) Define and explain how we can calculate the first and second derivatives (calculus) by numerical methods.

c) Explain the method of Crank-Nicholson from the definition of explicit and implicit methods.

Q.21 Using a variance reduction technique like the antithetic variance reduction method, redo Q.15 and compare your result. Discuss other variance reduction techniques. Q.22 Real options. Discuss real options and their use in business valuation (e.g., the valuation of high tech firms). Provide a numerical example using the trinomial tree (or the explicit finite difference method). Show how this approach can be used the value an option to expand. Hint : see Hull (2015), p. 801. Q.23 Volatility estimation methods. Discuss and compare the different methods of calculating volatility. -Explain the approach used in estimating a GARCH model. Donwload a time-series of prices (e.g., SP 500) and estimate an AR(1)-GARH(1,1) using a software of your choice (e.g., EViews, Excel, Matlab, etc.). Analyse your results.

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