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It's a very useful essay for those who are specialising in finance - Explain why it has proved impossible to derive an analytical formula for valuing American Puts, and outline the main techniques that are used to produce approximate valuations for such securities Investing in stock options is a way used by investors to hedge against risk. It is simply because all the investors could lose if the option is not exercised before the expiration …

…the American option price since v must satisfy the differential equation for both. They then show that when an approximation is made, the differential equation can be solved using standard methods. The techniques mentioned in this paper are those commonly used in practise. Although they are not perfect, they are still considered good enough for practical purposes. So far no one has managed to create a direct analytical valuation method for valuing American put options.