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

Credit ratings In the s&P rating system, AAA is the best rating. After that comes AA, A BBB. BB.B. and ccc The corresponding Moody's ratings are Aaa. Aa,a. Baa, Ba.b. and caa Bonds with ratings of BBB (or Baa) and above are considered to be investment grade

The Two-Factor Hul-white Model (equation 24.1, page 571) dx=0(0)+u-ax dt +o,dz1 du= -but +o,dz, where=f(r and the correlation between dz, and dz, is p The short rate reverts to a level dependent on u, and u itself is mean reverting

Why Interest rate Derivatives are Much more difficult to value Than Stock Options We are dealing with the whole term structure of interest rates; not a single variable The probabilistic behavior of an individual interest rate is more complicated than that of a stock price Options, Futures, and Other Derivatives, 5th edition C 2002 by John C. Hull

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

Binomial trees are frequently used to approximate the movements in the price of a stock or other asset In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d Options

What loss level is such that we are y%o confident it will not be exceeded in n business days?

A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock S=49,K=50,=5%,=20% T=20 weeks,μ=13%

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

A Simple Binomial model A stock price is currently $20 In three months it will be either $22 or