What are the differences of local volatility, implied volatility and historical volatility?Thanks

1. Historical volatility is the volatility that a given asset has displayed over some period in the past. For example the long-term historical volatility of the S&P index is somewhere around 14% - 17% depending on how far back you go. It is a measure of how volatile a given asset has been in the past.2. Implied volatility is a bit different in that it is not actually representative of what an asset has actually done in the past. In basic European option pricing, there are a few different values that have to be plugged in to a standard pricing formula (Black-Scholes) in order to get a price. Taking an option on a non-dividend paying stock, for example, one would need to know the current stock price, the strike price of the option, the time to expiration of the option, the spot rate of interest that applies from now to maturity, and an estimate of the volatility of the asset price. Notice that the only one that is an estimate is the volatility; all of the others are either specified by the exact option contract (strike, expiration) or they can be observed in the market (stock price, interest rate). Therefore the volatility parameter will uniquely determine the value of the option ... i.e. there is a one-to-one relationship between volatility and the option value (the higher the volatility, the higher the value). So one can either quote a volatility, and then calculate a price, or equivalently, one can get a quoted price and "backsolve" for the volatility. This volatility, the one that when plugged into the Black-Scholes formula returns the correct option price, is called the implied volatility (since it can be implied directly from the price of the option).3. Local volatility is more complicated still. The problem with implied volatility is that it is only a meaningful calculation for a single option. Different options in the market have different implied volatilities, even though in a philosophical context it is hard to imagine that one asset can have two volatilities over a single time period. So, let's say you have to price a non-standard option with some weird features, for which a simple pricing formula like Black-Scholes does not exist. You are now faced with the problem of determining what volatility to use, since there are many different implied volatilities out there in the market. You can pick one, but then you run into the problem that your pricing will be "consistent" with this one European option but not with the rest of the market. Ideally you would like to have a special type of model which can correctly price all of these European options; and then you would feel good about using it to price your exotic option since it would be "consistent" with the market. So, the concept of local volatility takes a bit of a philosophical leap; if you are willing to assume that stock price volatility is a deterministic function of stock price and time, so that given any (S,t) there is a volatility that corresponds to the asset, this task of finding a "consistent" model is possible. This is called local volatility. You calculate a volatility function of S and t that when applied to your universe of European options (all with different implied volatilities), they are all priced correctly. Now you have a more general model that you can use to price your exotics.Sorry for the long-windedness, I hope this helps.Rgds

There's an interesting interpretation of the local vol^2 (i.e. local variance) as the risk-neutral expectation of the instantaneous variance conditioned on S_T=K (i.e. the final stock price being equal to the strike price). Check out "Stochastic Implied Trees: Arbitrage pricing with stochastic term and strike structure of volatility" by Derman and Kani in the International Journal of Theoretical and Implied Finance, Volume 1, or Jim Gatheral's lecture notes, available at <http://www.math.nyu.edu/fellows_fin_mat ... html>.Also, Alex Lipton provides an interesting relation between local and implied vol in section 10.6 of his book "Mathematical Methods for the Foreign Exchange".And, if you want a good discussion of the "smile problem" in general, along with a healthy dose of sardonic French humor, check out "Can anyone solve the smile problem?" available at <www.ito33.com>.

My attempt at more concise definitions:1. Historical vol: The standard deviation of observed past returns, with a few variations using log returns, zero-mean, standard error, and GARCH/EWMA results.2. Implied vol: The value that when plugged into the Black-Scholes formula recovers the observed price of an option. In theory, this is the market expectation of gamma-weighted future implied volatility, which corresponds to the expected gamma profits of delta-hedging the options. The same underlying often has different implied vols for different strikes and tenors of options.3. Local volatility: A deterministic function s(X,t) calibrated such that if an Ito process is simulated using the vol value for that time and spot, recovers option prices as close as possible to all or many observed option prices simultaneously.