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In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives. For a European call or put on an underlying stock paying no dividends, the equation is: : where ''V'' is the price of the option as a function of stock price ''S'' and time ''t'', ''r'' is the risk-free interest rate, and is the volatility of the stock. 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". This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula. ==Financial interpretation== The equation has a concrete interpretation that is often used by practitioners and is the basis for the common derivation given in the next subsection. The equation can be rewritten in the form: : The left hand side consists of a "time decay" term, the change in derivative value due to time increasing called ''theta'', and a term involving the second spatial derivative ''gamma'', the convexity of the derivative value with respect to the underlying value. The right hand side is the riskless return from a long position in the derivative and a short position consisting of shares of the underlying. Black and Scholes' insight is that the portfolio represented by the right hand side is riskless: thus the equation says that the riskless return over any infinitesimal time interval, can be expressed as the sum of theta and a term incorporating gamma. For an option, theta is typically negative, reflecting the loss in value due to having less time for exercising the option (for a European call on an underlying without dividends, it is always negative). Gamma is typically positive and so the gamma term reflects the gains in holding the option. The equation states that over any infinitesimal time interval the loss from theta and the gain from the gamma term offset each other, so that the result is a return at the riskless rate. From the viewpoint of the option issuer, e.g. an investment bank, the gamma term is the cost of hedging the option. (Since gamma is the greatest when the spot price of the underlying is near the strike price of the option, the seller's hedging costs are the greatest in that circumstance.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Black–Scholes equation」の詳細全文を読む スポンサード リンク
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