简述 BS 公式怎么来

两条经典路线：

• 风险中性定价：在 $\mathbb{Q}$ 下 $dS_t=(r-q)S_tdt+\sigma S_tdW_t$，计算贴现 payoff 的期望得到 closed-form。
• PDE 路线：delta 对冲构造无风险组合 → BS PDE；再配上到期条件 $V(S,T)=\text{payoff}$ 解 PDE 得 closed-form。

数值上也可由二叉树/有限差分在细分极限下收敛到 BS。

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英文解析

The Black- Scholes formulas give the values of plain vanilla European put and call options on an underlying asset with lognormal distribution. Several methods for deriving the Black- Scholes formulas are:

• Risk neutral pricing: the expected value of the payoff of the option at maturity computed under the assumption that the price of the underlying asset has a lognormal distribution with drift equal to the risk free rate gives the Black-Scholes value of the option.- Black-Scholes PDE solution: the Black-Scholes value of the option satisfies the Black-Scholes PDE with boundary conditions given by the payoff of the option at maturity. The Black-Scholes PDE is transformed into the heat PDE using a lognormal change of variables, and the closed form solution of the heat PDE is then used to derive the closed form solution of the Black-Scholes PDE, which is

the Black- Scholes value of the option.

• Binomial tree model pricing: the evolution of the underlying asset is modeled using a binomial tree calibrated to converge in the limit to a lognormal distribution with drift equal to the risk free rate. For every tree, an approximate option value is obtained from the binomial tree model. The limit of these binomial tree option values as the number of time intervals in the tree goes to infinity is the Black-Scholes value of the option.

Note that twelve different ways to derive the BlackScholes formula can be found in Wilmott [4].