Change of Numeraire 

Thanks Brian! Indeed it was their paper,www.jstor.org/stable/3215299, that generalised the approach, though the technique was known before then!

It was Helyette German who amplified something that was practically known, and made the expressions rigorous. But notice: all these “substitutions of Numéraire” are basically reducible to a Mragrabe option, “exchange one asset for another.” As most of Quantpie lessons point out: all these problems reduce to the volatilities, the volatility measures, whether the vol is stochastic, and then the “correlation” of the volatilities, in a stable relationship it is a ratio, but in duel stochastic it introduces a third probability density function.

@@GeorgeSmileyOBE how do you really understand these things, particularly stoch vol, intuitively and relate them to pricing and use for hedging?

@@kxm73 “The Nobel Prize?! Gee, thanks fellas!” Army Man, America’s Only Magazine, George Meyer, editor. Let me further complicate things: when you have stochastic volatility, you must have a time barrier (expiration) or the probability space goes to infinity for any variable. Look at it this way: Asset-Liability management. If I have an asset of a dollar, and deposit it in a bank, it is still an asset for me, but a liability for the bank. When it was a dollar with me, there was zero (0) probability of default, or transaction costs, for the asset. Once I deposit it in the bank, I begin to bear those risks, and must be compensated for them, and the time value of money as well. Asset-Liability management therefore is broken into four (4) types: Type 1, you know the amount, and the time (a CD). Type Two, You know the amount, but you don’t know the time ( a 1m life insurance policy), type 3, you you do not know the amount, but you do know the time (some repos, most court judgements, contracts for difference, etc.), and last Type 4, you know neither the amount, nor the time. Type 4 is easiest, it is pure random. Type 1 is easy in stable interest rate environment, but has some variance but only rare events (acts of God, etc.). Type 3 is pretty easy, because it is based on 1) initial conditions of the variable in question (ex: price now of a stock) and the time horizon over which a Brownian-Bachelier process could evolve. Type 2 is really the toughest, and that is why an insurance company employés actuaries and pools risks and hopes the law of large numbers and good pricing provides enough profit (margin) amongst the most un correlated life/car/house insurance (you would not insure houses only made of wood, car insurance only for 18 year old males, or life insurance only for 60+ folks….risk is too concentrated and not ‘randomized’ enough. With that intuition out of the way, we come to the numéraire. What is ‘a dollar?’ It is simply a unit measure. What is a ‘stock?’ Same thing, but it is simply a one-step (derivative) of something else (proportional claim on cash flows). What is a ( ‘risk free’) bond? It is simply the unit, in terms of its growth of units, over (a bounded period) of time. Expressing the unit in terms of a dollar, a share of stock, or a (portion) of a bond or even another derivative is simply a matter of expressing the volatility possible over a period of time. Could the value of a dollar jump to infinity in a nano second? Sure, but the *probability* of that is extremely low. Once we have standard statistics, some sort of stable measurable or model able volatility, every numéraire can be discounted to NPV or compounded to FV.