مادة دراسية
Actu 468: Quantitative Methods in Finance
Chapter I. The Binomial Option Pricing Model
- Price options under a one-period binomial model on a nondividend-paying stock by:
- applying the principle of no-arbitrage, and identify arbitrage opportunities if any.
- applying the risk-neutral pricing formula.
- Extend the one-period binomial model on stocks in the following directions:
- to other underlying assets, including stock paying dividends continuously at a rate proportional to its price, currency, and futures contract.
- to a multi-period setting for pricing European and American options.
- Construct a binomial model from market stock price data using historical volatility and the following methods:
- Forward binomial tree, Cox-Ross-Rubinstein tree, lognormal tree
- Understand option pricing using real probabilities and calculate the appropriate risk adjusted interest rate for discounting.
Chapter II. The Black-Scholes Option Pricing Model
- Recognize the underlying assumptions behind the Black-Scholes model.
- Explain the properties of a lognormal distribution and calculate the following for future stock prices under the Black-Scholes model:
- probabilities and percentiles
- means and variances
- conditional expectations E[St | St > K] and E[St | St < K]
- Deduce the analytic pricing formulas for the following European options using risk-neutral pricing formulas:
- cash-or-nothing calls and puts
- asset-or-nothing calls and puts
- ordinary calls and puts (the Black-Scholes formula)
- gap calls and puts
- Explain the concepts underlying the risk-neutral approach to evaluate financial derivative.
- Generalize the Black-Scholes formula to price exchange options.
- Estimate a stock’s expected rate of appreciation and historical volatility from stock price data.
- Understand the concept of implied volatility.
Chapter III. Option Greeks and Risk Management
- Interpret and compute the following under the Black-Scholes model:
- Option Greeks (Delta, Gamma, Theta, Vega, Rho, and Psi)
- Option elasticity, Sharpe ratio and instantaneous risk premium for both an option and a portfolio of options and the underlying stock.
- Approximate option prices using delta, gamma and theta.
- Recognize the relationship among delta, gamma and theta (the Black-Scholes equation)
- Explain and demonstrate how to control stock price risk using the methods of delta-hedging and gamma-hedging.
Chapter IV. Interest Rate Derivatives
- Price interest rate derivatives under a binomial tree for interest rates.
- Recognize the features of a Black-Derman-Toy tree.
- Price interest rate caplets, floorlets and bond calls and puts by applying the Black formula.
- 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
https://ocw.mit.edu/courses/mathematics/18-s096-topics-in-mathematics-with-applications-in-finance-fall-2013/
https://ocw.mit.edu/courses/sloan-school-of-management/15-450-analytics-of-finance-fall-2010/