Introduction to Martingale Pricing of options contracts. Introduction to Monte Carlo Simulation and to binomial and trinomial discretization of some stochastic processes used to price derivatives. A short introduction to Least Squares Monte Carlo. A short introduction to real options.

• Introduction to derivative securities: options
  • (a) payoff diagrams
  • (b) position diagrams
  • (c) put call parity
  • (d) composite positions:
    • i. spread;
    • ii. combination;
    • iii. hedge.
• An hands on introduction to stochastic processes most used in derivatives valuation modelling:
  • (a) time series modelling: additive shocks, multiplicative shocks.
    • i. MA(1), AR(1) representations
    • ii. from sufficient statistics of a normal distribution to those of the corresponding log normal;
  • (b) Wiener process as a limit case of a random walk
  • (c) Ito process as a generalization of a Wiener process
  • (d) Geometric Brownian Motion:
  • i. univariate GBM;
    • A. Ito's lemma application: log transform process parameters derivation, Arithmetic Brownian Motion
    • B. Monte Carlo simulation of Pde solution and of its Euler approximation
    • C. parameters empirical estimate;
    • D. binomial approximation through moment matching conditions: Cox, Ross, Rubinstein 1979
    • E. Brownian Bridge, concept and Monte Carlo Simulation;
  • ii. multivariate GBM with correlated Wiener Processes:
    • A. construction and simulation of a multivariate GBM with correlated Wiener processes;
    • B. bivariate case: analytic (Cholesky) transformation of two uncorrelated shocks into two correlated ones;
    • C. general case: Choleski decomposition of a correlogram;
    • D. parametric Monte Carlo study of the estimates of correlation between two time series generated by correlated GBMs. E. Boyle, Evnine, Gibbs 1989, multivariate binomial model: bivariate case programmed in Aptech Gauss;
  • (e) Ornstein Uhlenbeck:
    • i. original version with arithmetic shocks;
    • ii. Ito's lemma application: derivation of Schwartz 1997 version, geometric with spring effects on logarithms;
    • iii. OU process parameters estimates;
    • iv. Monte Carlo simulation of processes sub i. and ii.
    • v. Binomial approximation Sick 1995
  • (f) volatility estimate for univariate processes:
    • i. inverting Black e Scholes 1973 and deriving a volatility surface
    • ii. equally weighted estimates;
    • iii. ARCH(m);
    • iv. EWMA: exponentially weighted moving average;
    • v. GARCH(1,1):
      • A. volatility clustering detection;
      • B. leverage effect detection;
      • C. plain vanilla GARCH(1,1)
      • D. GARCH(1,1) as a discrete time counterpart of an Ornstein Uhlenbeck process;
      • E. I-GARCH
    • vi. review of some models that accomodate volatility leverage:
      • A. A-GARCH;
      • B. E-GARCH;
      • C. GRJ-GARCH;
      • D. NL-GARCH;
      • E. Smooth Transition GARCH;
      • F. Markov Switching GARCH;
    • vii. GARCH(1,1) estimation:
      • A. MLE methods in general;
      • B. MLE methods for GARCH(1,1) numerical examples on Excel:
        • 3 parameters estimation;
        • 2 parameters estimation variance targeting;
        • MLE estimate of EWMA;
        • fitness tests: Box Pierce, Ljung Box, autocorrelogram
    • viii. Use of GARCH() models to forecast volatility:
      • A. GARCH volatility term structure;
      • B. GARCH average volatility.
    • ix. GARCH models and Options Pricing:
      • A. local risk neutrality, Duan 1995;
      • B. numerical example: Monte Carlo simulation of a GBM with stochastic volatilty generated by a GARCH(1,1)
  • (g) Variance covariance matrix estimation for multivariate processes:
    • i. equally weighted estimates of covariances;
    • ii. EWMA with no cross terms.
    • iii. modelling of variance covariance matrix, review, with specification of the respective LL function:
    • iv. direct:
      • VEC GARCH,
      • BEKK GARCH,
    • v. indirect:
      • CCC GARCH,
      • DCC GARCH.
• Martingale Pricing for derivative securities:
  • (a) american options valuation: drift change and backward induction in the following models
    • i. Cox, Ross, Rubinstein 1979
    • ii. Sick 1995
    • iii. Boyle, Evnine, Gibbs 1989
  • (b) european options valuation, in addiction to the preceding sub (a):
    • i. derivation of Black e Scholes 1973 as a limit case for Cox, Ross, Rubinstein 1979;
    • ii. Stultz 1982, Johnson 1987 rainbow options valuation, bivariate case programming in Aptech Gauss
    • iii. Monte Carlo simulations for both univariate and multivariate cases;
• Real Options
  • (a) parallelism with decision tree analysis
  • (b) martingale pricing viability for an irregular uncertainty resolution, multiperiod securities markets di Harrison e Kreps 1979
  • (c) differences and analogies between real and financial options
  • (d) most frequent real options, Mickey mouse examples
    • i. option to wait;
    • ii. option to expand/contract
    • iii. option to mothball/restart
    • iv. option to switch use
    • v. option to abandon
    • vi. option to default
    • vii. operating default
    • viii. financial default
      • asset substitution moral hazard
      • underinvestment moral hazard
      • put call parity interpretation of bond holders equity holders wealth transfer
  • (e) different approaches to real options valuations:
  • (f) the general real options model of Kulatilaka-Trigeorgis:
    • i. mickey mouse example
    • ii. taxonomy of operating modes of an industrial plant Markov Chain states analogies;
    • iii. binomial lattice Cox, Ross, Rubinstein 1979, Mickey Mouse example.
• Least Squares Monte Carlo, Longstaff, Schwartz 2001 RFS:
  • (a) general introduction to the model and comparison with Tsitsiklis Van Roy model;
  • (b) american/bermudean put option valuation, example of Moreno Navas 2003 MF:
  • (c) LSMC for the Kulatilaka Trigeorgis general real options model, Gamba 2011 JMF

On successful completion of this course, the student should

Pre-Assessment Formal prerequisites are:

Models and Algorithms for Financial Management 1

Actual prerequisites are not assessed at the beginning of the course and they are considered as a given when tuning the teaching approach of finance topics. A good programming ability is required for the following applications: A) any spreadsheet, e.g. Excel, Calc; B) any matrix oriented language, e.g. MatLab, Gauss, Ox, Octave, Scilab. In the computer lab classes, Gauss will be used. Univariate and multivariate calculus is applied in most of the models. A solid background in probability theory is required. A good programming ability is required for the following applications: A) any spreadsheet, e.g. Excel, Calc; B) any matrix oriented language, e.g. MatLab, Gauss, Ox, Octave, Scilab. In the computer lab classes, Gauss will be used. Univariate and multivariate calculus is applied in most of the models. A solid background in probability theory is required.

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