Hi, Guys:I am confused by the following question which may be stupid to you, but I did not find the answer after searching the forum and other literature. In HULL & WHITE, it says that short rate models actually model the evolution of short rates in RISK neutral world. I have the following questions1. I know that risk neutral means that the expected return is the risk-free rate, that means we need to have a risk-free rate before we move to risk-neutral world, right?2. What is the practical impact of this claim? For example, suppose I am trying to model the LIBOR overnight rate, and I am taking it as the risk free rate in my application, do I need to do anything to transfer the distribution to a risk-neutral distribution?3. Suppose I have a non-risk-free rate, what is the general approach to model it in risk neutral world using short-rate models?It will be helpful also if you can give me link for explaining this issue. Thanks so much.scizj

I am interested too. My take on this is:Risk neutral means no risk preferrence, hence we are happy with the expectation being positive but don't care how it comes about - the bigger the better?Risk neutral measure: MM account as the numeraire.But for rates I thought one needs to use the forward measure as a numeraire.So I think that the forward measure for rates leads to risk neutral pricing?Cheers,Alk

You can model either the short rate or the forward rate as there exist forumlas for bond prices in terms of either rate. Moving to the risk-neutral world depends on what your are modelling in the first place. Say you have an SDE for either the short or forward rate. The basic idea (in easy terms) is to find the parameters of the SDE such discounted bond prices are martingales and then do all your pricing with the SDE with these parameters. This topic is in any standard mathematical finance book (for non-rigorous maths choose Hull) or google will have plenty of answers and lecture notes.

sorry, for might be simple question just trying to check myself understanding. We have SDE for S_mu_sigma on original probability space with measure P. What does one need to make to arrive at risk neutral world?

QuoteOriginally posted by: listsorry, for might be simple question just trying to check myself understanding. We have SDE for S_mu_sigma on original probability space with measure P. What does one need to make to arrive at risk neutral world?Just a couple words to clarify what i meant. It just related to the math concept and does not related to option pricing at all. We have an underlying, say a stock S ( mu, sigma). This random process is given on original probability space with measure P. The function that is called Black-Scholes call option price has its underlying a heuristic process S ( r, sigma). To highlight risk neutral concept we do not care whether the Black-Scholes call option price is defined correctly or not. Our intention to reveal a connection between real stock S ( mu, sigma) and ?risk-neutral? stock S ( r, sigma). I called the latter heuristic because it does not exist in reality though is used for option price formula. Note that both risk- neutral and real stocks are random processes that are defined on the original (real) probability space with respect to measure P. The first papers which developed risk-neutral concept intended to proof how from real stock one can legally arrive at risk neutral. On the same way as a change variable in an integral we can change the integrand the Girsanov transformation can replace drift mu on ?r?. As far as they were not professional in stochastic calculus they lost a fact that the change of the variable does not change the value of the integral. Later the French mathematics began to use the risk neutral concept. They Improve the explicit error by putting the original stock on probability space with respect to risk neutral measure Q. This measure is chosen such that the image of the original(real) stock will have drift ?r?. The drawback is that the real stock is defined with respect to the original measure P in contrast to French adjustment of the risk neutral concept. The conclusion: regardless how good or incorrect the BS option price is the risk neutral concept fails to connect heuristic underlying of the BS pricing formula and the real underling of the option. If one likes BS option very much he can use it but without its connection to change measure techniques.

basically I am interested in the concepts of risk neutral valuation for fixed income, I thinkit is easy to understand the issue in equity markets.I found a good reference for risk neutral valuation for interest rates:risk -neutral valuation, pricing and hedgiing of financial derivatives, second ed, springer, bingham, kiesel

I also have a very basic question. When constructing a short-rate binomial lattice, many books "assume" the risk-neutral probability is 0.5. I haven't been able to find justification for the assumption of this value. Can anyone clarify the justification for this assumption?

There are many (risk neutral/risk-adjusted) short rate models. They are usually expressed as SDE's of the formdr = b(r,t) dt + a(r,t) dB on some domain (r(min),r(max)), and possibly with boundary behavior specified.For quite general models of this sort, it is possible to construct approximating binomialtrees. In general, the transitions associated with these trees involve multi-step jumps and thetransition probabilities are NOT equal to 1/2. The main idea is that the jumps and the transitionprobabilities from the state (r,t) are constructed to match, at least in some limit, the drift and diffusion of the sde. See Nelson and Ramaswamy for details.