Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. The key idea binary option payoff formula the model is to hedge the option by buying and selling the underlying asset in just the right way and, as a consequence, to eliminate risk. The model’s assumptions have been relaxed and generalized in many directions, leading to a plethora of models that are currently used in derivative pricing and risk management. Scholes formula, that are frequently used by market participants, as distinguished from the actual prices.
Scholes formula has only one parameter that cannot be directly observed in the market: the average future volatility of the underlying asset, though it can be found from the price of other options. Scholes model assumes that the market consists of at least one risky asset, usually called the stock, and one riskless asset, usually called the money market, cash, or bond. The rate of return on the riskless asset is constant and thus called the risk-free interest rate. The stock does not pay a dividend. It is possible to borrow and lend any amount, even fractional, of cash at the riskless rate. With these assumptions holding, suppose there is a derivative security also trading in this market.
It is a surprising fact that the derivative’s price is completely determined at the current time, even though we do not know what path the stock price will take in the future. Several of these assumptions of the original model have been removed in subsequent extensions of the model. Scholes equation is a partial differential equation, which describes the price of the option over time. The key financial insight behind the equation is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently “eliminate risk”. In this particular example, the strike price is set to 1.
Scholes formula calculates the price of European put and call options. In risk-neutral terms, these are the expected value of the asset and the expected value of the cash in the risk-neutral measure. Itō’s lemma applied to geometric Brownian motion. The equivalent martingale probability measure is also called the risk-neutral probability measure. Kac formula says that the solution to this type of PDE, when discounted appropriately, is actually a martingale.
Thus the option price is the expected value of the discounted payoff of the option. They are partial derivatives of the price with respect to the parameter values. The Greeks are important not only in the mathematical theory of finance, but also for those actively trading. Greeks that their traders must not exceed. Delta is the most important Greek since this usually confers the largest risk. Scholes are given in closed form below.