Q.13 Use the numerical method of the trinomial tree (or the explicit finite difference method) for computing the value an American put knowing the following information: S = 92; X = 96; RF = 0, 05; Sigma = 0.25; T = 5/12; b = 0.11; n = 900.
Q.14 Use Q.13 informations for the calculation of an American put using the method of Crank-Nicholson. Compare your results.
Q.15 a) Use the Monte Carlo method for the calculation of an Asian put knowing the following information: S = 80; K = 86; RF = 0.05; T = 1; Sigma = 0.20; nsim = 115; n = 100; b) Multi-assets options (exotic options). Find the price of an option that pays the maxium of 2 spreads (spread option) assuming: S1=120, S2=90, X=20, S3=95, rho12=0.6, rho13=0.6, rho23=0.7, T=3/12, rf=0.03, b1=0.02, b2=0.02, b3=0.02, v1=0.3,v2=0.3,v3=0.4, nsim=2000 (Hint: see Racicot, Notes on Monte Carlo, or Haug, 2007).
Q.16 Redo the calculations of Q.15 with Sobol method (optional).
Q.17 Use implicit trinomial tree to perform the calculation of a put knowing the following information: S = 95; X = 106; RF = 0, 05; Sigma = 025; T = 5/12; b = 0.12; n = 200.
Q.19 Using the Monte Carlo Least Squares approach, evaluate the price of an Amerian put assuming an Heston (1993, 2000) stochastic volaitiliy model having the following caracteristics: K=10, r=0.05, T=4/12, k=5, theta=0.16, sigma=0.9, rho=0.1, v0=0.0625, S = 8 to 12, and values of American puts : 2.000, 1.107064, 0.520030, 0.213668. Define and explain each concepts. Hint : see Rouah (2015).
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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