Black-Scholes equation

The N di) term in the Black-Scholes equation represents an option s delta. Delta indicates how much the contract s value, or premium, changes as the underlying asset s price changes. An option with a delta of zero does not move at all as the price of the underlying changes one with a delta of 1 behaves the same as the underlying. The value of an option with a delta of 0.6, or 60 percent, increases 60 for each 100 increase in the value of the underlying. The relationship is expressed formally in (9-1). 

The current value (fair price) V of the option should depend upon the current asset price S and the time remaining to expiry, T — t. The Black-Scholes equation (derived in Chapter 7) states that V(S, t) satisfies... 

Stochastic calculus is used heavily in quantitative finance, a significant employer of numerate engineers. In Problem 6.B.5, we solved the Black-Scholes equation for the fair value of an option. Here, we show how this equation is obtained, through stochastic calculus. 

Swaptions are typically priced using the Black-Scholes or the Black pricing model. With a European swaption, the appropriate swap rate on the expiry date is assumed to be lognormal. The swaption payoff is given by equation (7.19). 

Most option pricing models use one of two methodologies, both of which are based on essentially identical assumptions. The first method, used in the Black-Scholes model, resolves the asset-price model s partial differential equation corresponding to the expected payoff of the option. The second is the martingale method, first introduced in Harrison and Kreps (1979) and Harrison and Pliska (1981). This derives the price of an asset at time 0 from its discounted expected future payoffs assuming risk-neutral probability. A third methodology assumes lognormal distribution of asset returns but follows the two-step binomial process described in chapter 11. 

Equation (8.21) can be simplified as (8.22), the well-known Black-Scholes option pricing model for a European call option. It states that the fair value of a call option is the expected present value of the option on its expiry date, assuming that prices follow a lognormal distribution. 

Let S t) be the variable representing our concerned data at time t. S(t) is supposed to satisfy a Markov, continuous-time, geometric Stochastic Differential Equation (SDE). A classical model of stochastic process is the BS (Black Scholes) lognormal diffusion process. The BS model follows a basic stochastic differential equation given by ... 

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