Module 3 - Equities & Currencies

• Computing the price of a derivative as an expectation
• Girsanov's theorem and change of measures
• The fundamental asset pricing formula
• The Black-Scholes Formula
• The Feynman-Kac formula
• Extensions to Black-Scholes: dividends and time-dependent parameters
• Black's formula for options on futures

• Show that in the 1-period binomial model, the risk-neutral measure and the equivalent martingale measure are the same
• Derive the Fundamental Asset Pricing Formula for the 1-period and multiperiod binomial model
• Derive the Black-Scholes PDE from the 1-period binomial model
• Establish the connection between the multiperiod binomial model yields the Black-Scholes formula from
• Define complete markets and incomplete markets
• Explain how the Black-Scholes model is a complete market
• Show that, in a complete market, the no-arbitrage approach and the martingale measure approach are strictly equivalent

• The justification for pricing by Monte Carlo simulation
• Grids and discretization of derivatives
• The explicit finite-difference method

• Characterisation of exotic options
• Time dependence (Bermudian options)
• Path dependence and embedded decisions
• Asian options

• The many types of volatility
• The market prices of options tells us about volatility
• The term structure of volatility
• Volatility skews and smiles
• Volatility arbitrage: Should you hedge using implied or actual volatility?

• Implicit finite-difference methods including Crank-Nicolson schemes
• Douglas schemes
• Richardson extrapolation
• American-style exercise
• Explicit finite-difference method for two-factor models
• ADI and Hopscotch methods

Simple ideas that can have profound importance
 

• Lessons from the mathematics of gambling
• "Happiness"
• Jensen's Inequality
• Discrete hedging
• Transaction costs
• Extreme markets
• Is volatility important?
   

A few of the things that quants get wrong:
 

• Sensitivity to parameters
• Correlation
• Lack of diversification/size of trades
• Reliance on dynamic hedging (arguments), risk neutral versus real
• Feedback
• Too much complexity
• Supply and demand

• The relationship between implied volatility and actual volatility in a deterministic world
• The difference between 'random' and 'uncertain'
• How to price contracts when volatility, interest rate and dividend are uncertain
• Non-linear pricing equations
• Optimal static hedging with traded options
• How non-linear equations make a mockery of calibration

• Size and importance of the FX and FX options market
• How FX has developed into the largest global market
• Current uses of FX options
• Volatility surface and out of money options
• Pricing of simple FX options and those replicated from vanilla calls and puts
• Path-dependent FX options American and Bermudan
• Risk management and basic delta hedging