The Stock Price Assumption Consider a stock whose price is s In a short period of time of length 8t, the return on the stock is normally distributed

Put-call parity p +spe-ql=c+Xer holds regardless of the assumptions made about the stock price distribution It follows that

Standard approach to Estimating Volatility Define on as the volatility per day between day n-1 and day n, as estimated at end of day Define S: as the value of market variable at end of day i

Binary options Nonstandard American Lookback options options Shout options Forward start options Asian options Compound options Options to exchange Chooser options one asset for another Barrier options Options involving

Derivatives Dependent on a single Underlying Variable Consider a variable, 0, (not necessarily the price of a traded security) that follows the process d e S Imagine two derivative s dependent on e with prices f, and f2. Suppose

Term Structure models Blacks model is concerned with describing the probability distribution of a single variable at a single point in time a term structure model describes the evolution of the whole yield curve

Valuation of Swaps The standard approach is to assume that forward rates will be realized This works for plain vanilla interest rate and plain vanilla currency swaps, but does not necessarily work for non- standard swaps

Models to be Considered Constant elasticity of variance (CEV) Jump diffusion Stochastic volatility Implied volatility function (IVF) Options, Futures, and Other Derivatives

Weather derivatives: Definitions Heating degree days (HDD): For each day this is max(o, 65-4)where a is the average of the highest and lowest temperature in°F Cooling Degree Days(CDD): For each day this is max( o, A-65)

An alternative to the NPV Rule for Capital Investments Define stochastic processes for the key underlying variables and use risk- neutral valuation This approach (known as the real options approach) is likely to do a better job at valuing growth options, abandonment options, etc than NPV