| We consider the problem of reconstruction of unknown coefficients in the parabolic partial differential equation from measurements of a solution at a final moment of time.; The linearized inverse source problem for a general evolution operator is investigated. A new method of weighted energy type estimates is applied, and the global uniqueness is established. As an application of this result, we obtain uniqueness theorems for various inverse coefficient problems. A counterexample is given showing that it is impossible to reconstruct two unknown coefficients by measuring a solution at a single moment of time.; We show that an important inverse problem that arises in financial markets can be understood as the problem with the final observation. Valuation of options and other derivative securities critically depends on the stochastic process assumed for the underlying asset. The inverse problem of option pricing is to recover the nature of this stochastic process, namely, time-independent variance of expected asset returns known as local volatility from market prices of options with different strikes.; We give a rigorous mathematical formulation of this inverse problem, establish one uniqueness result, and suggest an efficient numerical solution. We apply the method to S&P 500 Index and conclude that the index is negatively skewed with the higher probability of the sudden decline of the US stock market. The implied local volatility function can be used for various purposes such as risk management and hedging, arbitrage and volatility trading, and pricing exotic and more complicated over-the-counter options.; New analytic formulas for the option premium and implied volatility are obtained. The formula for the option premium shows explicitly effects of various factors that determine the option price. A formula for the implied at-the-money volatility is very simple, and can be easily programmed on pocket calculators and used by market participants. |
