Volatility as an Inverse Problem - A Very Simple Explanation
Feb 24, 2006
Inverse problems are a class of problems in mathematics where essentially causes - or input to the problem - are determined by observing the effects, i.e. the output. An inverse function (mathematical equivalent of an inverse problem) reverses the process of the original function
If is a function on domain Y then for every on this domain, the inverse function
is such that:
Consider the problem of option pricing: a call options price is dependent on five variables, spot, strike, time to maturity, risk free rate and the volatility. All except the volatility are observable. Volatility cannot be observed in the market but needs to be estimated. How do we do that? Let us express the call's price, C , as a function g such that:
All the above, including the call option's price C, are observable (call options trade in the markets and traders buy and sell them given the demand and supply of these options, which determines their bid and offer prices). But volatility, denoted by sigma in the function, is not observed. So how can we estimate the volatility? That is why we have expressed g as a function of only one variable (keeping others constant). In fact, in practice implied volatility is estimated as an inverse function of the above function g.
If we assume that a certain price of an option is caused (determined by) a certain volatility level and the process is given by the function (Black-Scholes equation) then that volatility level is found out by reversing that process.
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