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期权定价方法之B-S模型

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如何理解 Black-Scholes 期权定价模型?

C_t f_t for 期权定价方法之B-S模型 期权定价方法之B-S模型 f_t is the price of any European type derivative at time t.

如何从Greeks 来理解 BS differential Equation ?

这里的 f是欧式(European Style)期权的价格。

If gamma and 期权定价方法之B-S模型 delta neutral. \Gamma = 0\\~\\ \Delta = 0

f here is the risk - free bond,gain income with the time flows。

记得我们在二叉树里面,我们只需要delta-hedging即可就可以risk-free,但是当我们到了连续的情况我们需要delta + gamma hedging。

If 期权定价方法之B-S模型 期权定价方法之B-S模型 gamma and theta neutral. \Gamma = 0\\~\\ \theta= 0

f here is c shares of stock,that's saying if you don't like theta(time decay) and gamma (curvature accelerating), you only need to trade stock。

curvature accelerating什么含义? 就是说你的股票价格涨1%,组合涨1%,再涨1%,组合涨的更多比如2%。

time decay呢? 就是说如果你只买股票,仅仅是时间变动,你的股票价值是不会变化的,股票的theta是0。

记得我们在二叉树里面,我们只需要delta-hedging即可就可以risk-free,但是当我们到了连续的情况我们需要delta + gamma hedging。

如何理解 (dt)(dB_t) = 0, (dB_t)^2=dt Stochastic Calculus rules

我们知道 dB_t 是服从 正态分布的,那么为什么他的平方会趋近于一个常数?(而不是卡方?)

dB_t = B_-B_t \sim N(0,dt)

V(dB^2_t ) ->0 rapidly.

我们知道 dtdB_t 是正态分布的,为什么会等价于0?

V(dtdB_t ) ->0 rapidly.

E(dtdB_t ) =dtE(dB_t) = dtE(B_-B_t)= dt(E(B_)-E(期权定价方法之B-S模型 B_t))=0\\ \forall t B_t \sim N(0,t)

Var(dtdB_t ) =dt^2Var(期权定价方法之B-S模型 dB_t)=dt^3 -> 0

E((dB_t)^2 ) =E((B_-B_t)^2) \backsimeq E((B_-B_0)^2) = E(B_^2) = dt

我们同时要证明, Var((dB_t)^2 -dt) = Var((dB_t)^2) -> 0

Then E((dB_t)期权定价方法之B-S模型 ^4) = dt^4E(Y^4)

Then Var((dB_t)^2) = dt^4E(Y^4) 期权定价方法之B-S模型 - dt^2

也会按照至少 dt^2 的速度收敛到0

如何理解 Ito's Lemma?

s - > ds_t = s_trdt+s_t\sigma dB_t

为什么这里 (ds)^2 没有被忽略掉?

因为正常的全微分下,我们的dt 和 ds都是默认的极小量,其高阶极小量忽略,而这里:

(ds)^2 = (s_t\sigma dB_t)^2=\sigma^2s^2(dB_t)^2=\sigma^2s^2dt

相关面试/练习题解析

股票价格$500. 波动率40%, the risk-free rate 期权定价方法之B-S模型 is at 5%. 投资组合delta 0.1, gamma 0.06 ,theta -800. 在股票无红利派发的情况下给出投资组合的价格。

先给一些背景知识在这里:

什么是dollar to dollar exposure?

我们如何用期权构造一个dollar to dollar exposure呢?

首先根据上面的公式我们需要抹除gamma的影响,所以我们可以用Deep in the money call option(看payoff 图,你的strike price离股票现在价格太远(太小),使得变动delta几乎是1)。

但是买deep in the money call 和 股票还是有不同,这个不同就在Theta上,他是time decay的。

假设你通过购买Google股票获得了dollar to dollar 的Google股票exposure。 θ是多少?

\theta + 0.03*500*1 + 0 = 0.期权定价方法之B-S模型 03*250\\~\\ \theta = -7.5

注意,our position is losing value at the rate of 7.5$ a year.

你能用Google价格的1/4进行构造相同的dollar to dollar exposure吗? 如何实现?

you could do deep in the money call of the call option.

call^3 -> 期权定价方法之B-S模型 call^2->call->stock\\~\\ price:62.5->125->250->500\\~\\ theta:-15->-10->-7.5->0\\~\\

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