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

1. 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

2. 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)

3. Same as in 2) but using multiple step Euler-scheme based simulation with N>500 and compare the results

4. Same as in 3) but applied to Asian options, i.e. path-dependent options with payoff = (1/T\int_0^T S_tdt - K)^+

5. (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)