Actu 468: Quantitative Methods in Finance

Chapter I. The Binomial Option Pricing Model

  1. Price options under a one-period binomial model on a nondividend-paying stock by:
    1. applying the principle of no-arbitrage, and identify arbitrage opportunities if any. 
    2. applying the risk-neutral pricing formula.
  2. Extend the one-period binomial model on stocks in the following directions:
    1. to other underlying assets, including stock paying dividends continuously at a rate proportional to its price, currency, and futures contract.
    2. to a multi-period setting for pricing European and American options.
  3. Construct a binomial model from market stock price data using historical volatility and the following methods:
    1. Forward binomial tree, Cox-Ross-Rubinstein tree, lognormal tree
  4. Understand option pricing using real probabilities and calculate the appropriate risk adjusted interest rate for discounting.

Chapter II. The Black-Scholes Option Pricing Model

  1. Recognize the underlying assumptions behind the Black-Scholes model.
  2. Explain the properties of a lognormal distribution and calculate the following for future stock prices under the Black-Scholes model:
    1. probabilities and percentiles
    2. means and variances 
    3. conditional expectations E[St | St > K] and E[St | St < K]  
  3. Deduce the analytic pricing formulas for the following European options using risk-neutral pricing formulas: 
    1. cash-or-nothing calls and puts
    2. asset-or-nothing calls and puts 
    3. ordinary calls and puts (the Black-Scholes formula)
    4. gap calls and puts
  4. Explain the concepts underlying the risk-neutral approach to evaluate financial derivative.
  5. Generalize the Black-Scholes formula to price exchange options.  
  6. Estimate a stock’s expected rate of appreciation and historical volatility from stock price data.
  7. Understand the concept of implied volatility.

Chapter III. Option Greeks and Risk Management

  1. Interpret and compute the following under the Black-Scholes model:
    1. Option Greeks (Delta, Gamma, Theta, Vega, Rho, and Psi) 
    2. Option elasticity, Sharpe ratio and instantaneous risk premium for both an option and a portfolio of options and the underlying stock.  
  2.   Approximate option prices using delta, gamma and theta. 
  3.   Recognize the relationship among delta, gamma and theta (the Black-Scholes equation)
  4.   Explain and demonstrate how to control stock price risk using the methods of delta-hedging and gamma-hedging.

Chapter IV. Interest Rate Derivatives

  1. Price interest rate derivatives under a binomial tree for interest rates.
  2. Recognize the features of a Black-Derman-Toy tree. 
  3. Price interest rate caplets, floorlets and bond calls and puts by applying the Black formula.
  4. Apply put-call parity to European options on zero-coupon bonds.

Reference textbook: Derivatives Markets (Third Edition), 2013, by McDonald, R.L., Pearson Education, ISBN: 978-0-32154-308-0

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