Can someone briefly explain the definitions of each greek letter how they correspond to the markets. Thank you.

both refer to portfolio performance: beta is the correlation with "the market". alpha: correlation with "idiosyncratic factors" such as manager's skill. example: alpha positive: you outperform the market, alpha negative: you are a loser and everybody takes money from you. beta positive: you are correlated w/ the market (you are just a buy and holder of spy's msft etc) beta negative: you lose when the market goes up you win when the market goes down, beta zero: you always make/lose money regardless of market direction. trader's holy grail: maximize alpha with beta as close to zero as possible.

The delta, gamma, theta and vega are sensitivity measures of an option. They measures how much B will change when A changes: for delta: B = the option value. A = stock price (the underlying); for gamma: B = delta. A = stock price; for theta: B = the option value. A = days to expiration; for vega: B = the option value. A = volatility. Hope this helps.

those are for options or option portfolios, and refer to the "correlations" or the "mathematical derivatives" of the option/portfolio value with respect to the underlying: the stock the currency etc. for example if you're an option market maker you are like a bookie so you want to make money regardless of stock price so you want delta = zero. if you want to make money regardless of volatility you want vega=zero. if you think volatility will increase then you want to be long vega. the other greeks are for second derivatives curvatures expiration dates (time decay rate) etc... read the natenberg the cottle etc.

Beta is also applied to a given security, not just a portfolio (which can be done in aggregate) or a money manager perforamance. Each security's price movement is correlated to the wider market (usually the S&P 500) and has a beta coefficient that represents approximately how it behaves relative to the index, e.g., the SPY would have a beta of 1 because it moves in lock step with the SP500. Portfolio beta would be approximated by volume weighting the individual security betas in the portfolio.