bond maths contd....

the term structure found can also be equa to the zero - coupon bond yield.
choos ean nterest rate model that is consistent with the yield curve obtained from market prices of bonds. To calclate the zero coupon yield curve implied by an interest rate.

Pricing interest Rate Sensitive Securities I

In case of pricing stocks we use a tree assuming the markovian model. This does not work n case of fixed income securities like bonds, since th amount to be paid at the end is fixed. What does determine the present value of a fixed income secuirty is an ineterst rate model. The interest rate model can be assumed to be markovian.
For veery interest rate model there is an equivalent term structure model that can be observed only in the market. The probability of rise and fall from a node in the interest rate tree is assumed to be (1) independent of time and state (2) dependent on state (3) dependent on time (4) dependent on time and state. Ingersoll and Ross methods capture the term structure the best but are computationaly difficult.
The main idea is that given the sequence of future discountings, the expected present value of asecurity can be calculated. In equity market the lognormal model is standard. In FIRC market there is no standard interest rate tree model. Each business house uses its own peoprietary MODEL.

Trend catching

I was reading this article in statistics and empirical finance, which talks of asymmetry in time horizons of investors. These things motivate many game theoretic strategies of automated trading. If we could generate a description of how different investor classes react to information on wall street then we can think of trading strategies which, looking at this and the current state of order books, can find a strategy of optimally placing requests to catch the majority of any trend.
It is clear that trends in price start after some external impulse directly related to the company. Hence understanding what information is relevant (broadly) is not difficult. There has to be a notion of automated/human corroborated interpretation of information. Often technical information, like moving averages etc are very easily discovered. Based on the different horizons and different investor class contribution to overall market movement, one can devise games to win in this model.

Garch as an estimator of volatility

Volatility estimaion may not give us a good clue on hedging bets, but we predict it simply as Prof. Steele says "because we can".
Volatility is defined as:
Given a return series {r1 to rT}, volatility observed till day-t is stdev of {r1 to rt}
There are many predictors of volatlity.
Implied volatility: Noting the sigma used for call price estimation, we get a good prediction of what he market perceives as volatility of the underlier.
Historical Volatility: Given an observation of returns for the previous 400 days say we can use garch models to predict the volatility of te next trading day. Why can't we do this for returns as well. Because we know that it is a stylized fact that returns show clumpiness in data and hence there is arch component in returns, or ar component in volatility and since arch does not explain things well and garch does a much better job, we can bet on that. On the other hand there is no stylized fact about returns that we can exploit so conclusively.

However, empirical research has shown that GARCH is not good a estimator of r_{t+1}^2 and that much improved volatility forecasts can be obtained if high frequency (intraday) returns data are taken into account.