The purpose of these reports is to learn the basic tools and methodologies for solving financial problems.
The main topics of each Report are:
Report 1: Provide the price of a Call by using the (static) binomial model and compare it with the market quotes.
Report 2: Learn how to use the Put-Call Parity and Box-Spread equations.
Report 3: Compare the convergence of the Binomial Model and the Leisen-Reimer method.
Report 4: Write a VBA code to compute Greeks.
Report 5: Estimate the Value at Risk of an equibalanced Portfolio.
Report 6: Evaluate the price of an option using several alternatives of the Monte Carlo simulation.
● Go in Yahoo finance and select an asset (Nokia in this case)
● Go in "Options" and select a Call with maturity around 3 months and Strike AtTheMoney
● Consider the Mid price: this market quote is the target price for our model
● Go in "Historical Data" and download in EXCEL the last 3 months of daily data (take Adj Close that take into account for dividends splits etc)
● From daily prices compute daily returns
● Compute sigma_day = the standard deviation of the daily returns
● Compute sigma_year = the annual volatility = sigma_day*sqrt(252)
● Compute the parameters u,d of the binomial model that are related to sigma and T= 3 months
● Go in global-rates.com and select the Libor rate corresponding to T=3 months
● Compute the capitalisation factor using simple compounding = 1+LiborT and simple discounting = 1/(1+LiborT)
● Compute the risk neutral probability weight q and apply the risk neutral pricing formula for the price of a Call -
● Compare your price with the market quote at point 4)
● Repeat the procedure for T= 6 months.
● Choose a dividend paying asset ( Pepesico in this case)
● Fix a maturity T and consider Call a Put with K1 and K2 (K1<<K2)
● Find the corresponding discount factor for maturity T
● Find the implicit dividend for maturity T using the put-call parity and using At The Money options (that is new Call and Put with strikes ATM)
● Repeat the procedure for T=1m, 3m, 6m, 1year and deduce the term structure of implied dividends
● Set S=100; r=1%; vol=20%, T=1,K=100
● Write a VBA script that prices a Call with a binomial model where n (number of steps) is an input parameter
● Compare with the B&S formula
● Investigate the rate of convergence and compare with the Leisen and Reimer method
(theoretical)
● Set K=100; r=1%; vol=20%
● Provide a VBA script that computes the Greeks for a B&S model for S=60,65,70,..,130, 135, 140 and time to maturity tau=0.1,0.2,…,1, 2, 3, 4, 5 years
● Same as in 1 with a shock of the volatility of +-50%
● Display in a 3D graph the corresponding surfaces for all greeks
(empirical)
● Choose in Yahoo an asset which does not pay dividends and on which there is a book of European options
● Visualize in a 3D picture the implied volatility surface (as a function of the strikes and time to maturities) using the Call prices
● Check the presence of the implied volatility smile/skew
● Check the smoothness of the Greeks as time to maturity increases
● Compute average and variance of an equibalanced portfolio of 2 assets (6 months time window of daily returns)
● Compute the parametric single and joint Normal VaR at (95%, 99% , 99,5%) with T=1,..,100 days horizon (estimation of sigma flat)
● Same as in 2 with sigma estimanted following Riskmetrics EWMA with lambda=0,94
● Compute the MonteCarlo VaR with N (free input) simulations with T (free input) at the levels (99%,99,5%,95%)
● Compute the historical VaR (according to the historical value of the returns
● Compute the VaR with the method of historical simulation (not mandatory)
● Check in all cases the additivity (non additivity) of the VaR
● Fix some generic parameters for the BS market model
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simulate N trajectories (N free input of the script) for the GBM through a VBA code and visualise the paths in a figure for N=100
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Build up a pricer of vanillas (call/put) through MC by 1 step simulation (that is by simulating N>500 values of the random variable S_T not the entire path)
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Same as in 2) but using multiple step Euler-scheme based simulation with N>500 and compare the results
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Same as in 3) but applied to Asian options, i.e. path-dependent options with payoff = (1/T\int_0^T S_tdt - K)^+
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(OPTIONAL) Modify the existing codes in order to deal with other path dependent options, like e.g. lookback options with payoff = S_T - Min(S_t, t<=T)