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In mathematical financethe 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 optionsor more generally, derivatives.
This hedge, in turn, implies that there is only one right price for the option, as returned by the Black—Scholes formula. 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 thetaand a term involving the second spatial derivative gammathe convexity of the derivative value with respect to the underlying value.
Black and Scholes' insight is that the portfolio represented by the right hand side is riskless: 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. 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.
Per the model assumptions above, the price of the underlying asset typically a stock follows a geometric Brownian motion. Note that Wand consequently its infinitesimal increment dWrepresents the only source of uncertainty in the price history of the stock.
Intuitively, W t is a process that "wiggles up and down" in such a random way that its expected change over any time interval is 0.
In addition, its variance over time T is equal to T ; see Wiener process: Basic properties ; a good discrete analogue for W is a simple random walk. The value of these holdings is. Thus uncertainty has been eliminated and the portfolio is effectively riskless. The rate of return on this portfolio must be equal to the rate of return on any other riskless instrument; otherwise, there would be opportunities for arbitrage.
Different pricing formulae for various options will arise from the choice of payoff function at expiry and appropriate boundary conditions. A subtlety obscured by the discretization approach above is that the infinitesimal change in the portfolio value was due to only the infinitesimal changes in the values of the assets being held, not changes in the positions in the assets.
In other words, the portfolio was assumed to be self-financing. This can be proven [ citation needed ] in the continuous setting and uses basic results in the theory of stochastic differential equations. Here is an alternate derivation that can be utilized in situations where it is initially unclear what the hedging portfolio should be. For a reference, see 6. In the Black—Scholes model, assuming we have picked the risk-neutral probability measure, the underlying stock price S t is assumed to evolve as a geometric Brownian motion:.
Since this stochastic differential equation SDE shows the stock price evolution is Markovian, any derivative on this underlying is a function of time t and the stock price at the current time, S t. This derivation is basically an application of the Feynman-Kac formula and can be attempted whenever the underlying asset s evolve according to given SDE s.
Once the Black—Scholes PDE, with boundary and terminal conditions, is derived for a derivative, the PDE can be solved numerically using standard methods of numerical analysis, such as a type of finite difference method. In certain cases, it is possible to solve for an exact formula, such as in the case of a European call, which was done by Black and Scholes.
To do this for a call option, recall the PDE above has boundary conditions. The last condition gives the value of the option at the time that the option matures. Other conditions are possible as S goes to 0 or infinity. For example, common conditions utilized in other situations are to choose delta to vanish as S goes to 0 and gamma to vanish as S goes to infinity; these will give the same formula as the conditions above in general, differing boundary conditions will give different solutions, so some financial insight should be utilized to pick suitable conditions for the situation at hand.
To solve the PDE we recognize that it is a Cauchy—Euler equation which can be transformed into a diffusion equation by introducing the change-of-variable transformation. Using the standard convolution method for solving a diffusion equation given an initial value function, u x, 0we have. From Wikipedia, the free encyclopedia. Options, Futures and Other Derivatives 7 ed. Retrieved from " https: All articles with unsourced statements Articles with unsourced statements from June Views Read Edit View history.