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

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?

30.2 Big Losses by Financial Institutions Allied Irish Bank($700 million) Barings($1 billion) Chemical Bank($33 million) Daiwa ($1 billion) Kidder Peabody ($350 million) LTCM ($4 billion) Midland Bank ($500 million) National Bank ($130 million) Sumitomo ($2 billion)

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

European Options on Stocks 13.2 Providing a dividend yield We get the same probability distribution for the stock price at time T in each of the following cases 1. The stock starts at price so and provides a dividend yield =q 2. The stock starts at price Soe q l and provides no income

Long Short Hedges A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price A short futures hedge is appropriate when you know you will sell an asset in the future want to lock in the price

Company a buys default protection from b to protect against default on a reference bond issued by the reference entity, C A makes periodic payments to In the event of a default by c A has the right to sell the reference bond to B for its face value