The Black–Scholes Model
黑-舒尔斯模型模型

In this lecture.
在这堂讲座。 . .

the assumptions that go into the Black–Scholes model
进入Black–Scholes模型的假设。

foundations of options theory: delta hedging and no arbitrage
期权理论的基础:对冲和无套利

the Black–Scholes partial differential equation
Black–Scholes偏微分方程

the Black–Scholes formulæ for calls, puts and simple digitals
calls,puts和简单数字的Black–Scholes公式

the meaning and importance of the ‘greeks,’ delta, gamma, theta, vega and rho
‘希腊字母’delta,gamma, theta,vega和rho的意义和重要性

By the end of this lecture you will be able to
通过这节讲座的结束，您将能够⽣成完成⼀标准的期权

derive the Black–Scholes partial differential equation
推导Black–Scholes偏微分方程

quote formulæ for simple contracts
货币对规则为简单的合约

understand the meaning of the common greeks
理解意义⼀个常见的值希腊字母

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Introduction
Black-Scholes 方程是期权定价中的最大突破

The Black–Scholes equation was the biggest breakthrough in the pricing of options.
Black–Scholes⽅程是期权定价最重要的突破之⼀。

The theory is quite straightforward, using just the ideas from stochastic calculus that we have already seen.
该理论相当直接，仅使用我们已经看到的随机微积分的思想。

The end result is a diffusion-type partial differential equation which can be used for the pricing of many different derivatives.
最终结果是一种扩散型偏微分方程，可用于许多不同衍生品的定价。

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What determines the value of an option?
是什么决定期权的价值?

The value of an option is a function of the stock price S and time t
期权的价值是股价 S 和时间 t 的函数.

The value of the option is also a function of parameters in the contract, such as the strike price E and the time to expiry T − t, T is the date of expiry.
期权的价值也是合同中参数的函数，如行权价E和到期时间T ，其中，T是到期日，t是当前时间。

The value will also depend on properties of the asset, such as its drift and its volatility, as well as the risk-free rate of interest:
价值还取决于资产的属性，比如其漂移和波动性，以及无风险利率。

V (S, t; σ, µ; E, T ; r).
V(S,t;σ,µ;E,T;r).。

Semi-colons separate different types of variables and parame- ters.
分号分开不同类型的变量和参数。

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S and t are variables;
S和t是变量；

σ and µ are parameters associated with the asset price;
σ和µ是与资产价格相关的参数。

E and T are parameters associated with the particular con- tract;
E和T是与具体合同相关的参数;

r is a parameter associated with the currency.
r是与货币相关的参数.

For the moment just use V (S, t) to denote the option value.
暂时只使用V(S,t)表示期权价值.

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The Black–Scholes assumptions
布莱克-舒尔斯假设

The risk-free interest rate is a known function of time
风险免费利率是时间的已知函数

There are no dividends on the underlying
基础上没有派息

Delta hedging is done continuously
Δ对冲是连续进行

There are no transaction costs on the underlying
基础上没有交易成本

There are no arbitrage opportunities
基础上没有套利机会

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And more.
更多内容。 . .

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A very special portfolio
一个非常特别的投资组合

We assume that the asset evolves according to
我们假设资产按照发展演变根据到

dS = µS dt + σS dX.

Then we imagine constructing a special portfolio.
然后我们想象构建一个特殊配置

Use Π to denote the value of a portfolio of one long option position and a short position in some quantity ∆, delta, of the underlying:
使用 Π 表示 一个 价值 的 一个 投资组合 中 一个 长期 期权头寸 和 空头头寸 中 的 一些 数量 ∆， 德尔塔 的 的 基础：

Π = V (S, t) − ∆S. (1)

Intuition: Think of moves in S and accompanying move in V , for a call.
直觉:思考在S中的动作和相关的V中的动作，为通话。

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How does the value of the portfolio change?
如何改变组合的价值？

The change in the portfolio value is due partly to the change in the option value and partly to the change in the underlying:
资产组合价值变化部分是由于期权价值的变化以及基础资产价值的变化：

dΠ = dV − ∆ dS.

Notice that ∆ has not changed during the time step.
注意∆在时间步长期间没有发生改变。

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From Itˆo we have
从Itˆo我们有

dV =

∂V

∂t

∂V

∂S

2 2 ∂2V

Thus the portfolio changes by

dΠ =

∂V

∂t

∂V

∂S

2 2 ∂2V

The right-hand side of (2) contains two types of terms, the deterministic and the random.
右侧包含两种项，确定性项和随机项。

The deterministic terms are those with the dt.
确定性项是带有dt 的项。

The random terms are those with the dS. These random terms are the risk in our portfolio.
随机项是带有dS 的项。这些随机项是我们投资组合中的风险。

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Elimination of risk: Delta hedging
风险的消除：Delta对冲

Is there any way to reduce or even eliminate this risk? This can be done in theory by carefully choosing ∆.
是否有任何方法减少或甚至消除这种风险？在理论上可以通过谨慎选择∆.

If we choose
如果我们选择

∆ = ∂V

(3)

then the randomness is reduced to zero.
那么随机性被降低到零。

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Any reduction in randomness is generally termed hedging.
任何随机性的减少通常被称为套期保值。

The perfect elimination of risk, by exploiting correlation be- tween two instruments (in this case an option and its under- lying) is generally called Delta hedging
风险的完美消除，通过利用两个工具之间的相关性（在这种情况下是期权和其基础工具），通常被称为 Delta 对冲.

Delta hedging is an example of a dynamic hedging strategy.
Deltahedging是动态对冲策略的一个示例

changes, since
变化，因为

it is, like V a function of the ever-changing variables S and t
它，像 V 一样是一个关于不断变化的变量 S 和 t 的函数.

This means that the perfect hedge must be continually rebal- anced.
这意味着完美对冲必须不断重新平衡。

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No arbitrage
无套利

After choosing the quantity ∆ as suggested above, we hold a portfolio whose value changes by the amount
之后选择数量 ∆如上述，我们持有一个按照上述所建议的价值变动的投资组合。

dΠ =
dΠ=

∂V + 1
∂V+1

2 2 ∂2V
22∂2V

∂S2
∂S2

dt. (4)
dt。(4)

This change is completely riskless.
这个改变是完全无风险的。

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If we have a completely risk-free change dΠ in the value Π then it must be the same as the growth we would get if we put the equivalent amount of cash in a risk-free interest-bearing account:
如果我们完全无风险地改变Π的价值ΠdΠ一致则我们放入等价金额的现金在无风险的利息计息帐户

dΠ = rΠ dt. (5)

This is an example of the no arbitrage principle.
这是一个例子的无套汇原则。

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The Black–Scholes equation
Black–Scholes方程式

Substituting (1), (3) and (4) into (5) we find that
替换(1),(3)和(4)到(5)我们发现这

∂t 2

2 2 ∂2V

∂S2

dt = r

V S∂V

dt.

On dividing by dt and rearranging we get
在除以dt并重新排列我们得到

∂V 1 2 2 ∂2V ∂V

This is the Black–Scholes equation.
这是Black–Scholes方程。

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Observations:

The Black–Scholes equation equation is a linear parabolic
Black–Scholes方程方程是一个线性抛物线

partial differential equation
偏导数偏微分方程

The Black–Scholes equation contains all the obvious vari- ables and parameters such as the underlying, time, and volatil-
Black–Scholes方程包含所有明显的变量和参数，如基础、时间和波动率

ity, but there is no mention of the drift rate µ.

This means that if two people agree on the volatility of an asset they will agree on the value of its derivatives even if
这意味着如果两个人就一个资产的波动性达成一致，甚至如果他们

they have differing estimates of the drift.
他们有不同的估计漂移。

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Replication
复制

Another way of looking at the hedging argument is to ask what happens if we hold a portfolio consisting of just the stock, in a quantity ∆, and cash.
看问题的另一种方式是对冲论点问发生什么如果我们持有一个仅包含股票的组合以及现金。

If ∆ is the partial derivative of some option value then such a portfolio will yield an amount at expiry that is simply that option’s payoff.
如果∆是一些期权价值的偏导数，那么这样的投资组合将在到期时产生某个数额，即该期权的支付。

In other words, we can use the same Black–Scholes argument to replicate an option just by buying and selling the underlying asset.
换句话说，我们可以使用相同的Black–Scholes论点来复制一个期权，只需买入和卖出这个标的资产。

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Final conditions
最终条件

The Black–Scholes equation knows nothing about what kind of option we are valuing.
Black–Scholes公式并不了解我们正在评估什么种类的期权。

This is dealt with by the final condition. We must specify the option value V as a function of the underlying at the expiry date
这是由最终条件处理的我们必须指定期权价值V作为底层在到期日的函数

T . That is, we must prescribe V (S, T ), the payoff.
T这意味着我们必须规定V(S,T)支付。

For example, if we have a call option then we know that
例如，如果我们具有看涨期权，那么我们知道那

V (S, T ) = max(S − E, 0).

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Options on dividend-paying equities
股息支付股权上的期权

Assume that the asset receives a continuous and constant divi- dend yield, D
假设资产获得连续且恒定的股息收益 D.

Thus in a time dt each asset receives an amount DS dt.
因此在时间dt内每个资产收到一定数量DSdt。

This must be built into the derivation of the Black–Scholes equa- tion.
这必须被纳入到Black–Scholes 方程的推导中。

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Take up the Black–Scholes argument at the point where we are looking at the change in the value of the portfolio:
接受Black–Scholes论据的论点，我们将观察投资组合价值的变化：

dΠ =

∂V

∂t

∂V

∂S

2 2 ∂2V

The last term on the right-hand side is the amount of the div- idend per asset, DS dt, multiplied by the number of the asset
右侧的最后一个术语是资-产每股分红 DS*dt 乘以资产的数量

held, −∆.
持有的数量， -∆。

The ∆ must still be the rate of change of the option value with respect to the underlying for the elimination of risk.
∆必须仍然是对基础资产的期权价值变化率以消除风险。

End result:
结果：

∂V 1 2 2 ∂2V ∂V

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Currency options
货币期权

Options on currencies are handled in exactly the same way.
货币期权是以完全相同的方式处理。

In holding the foreign currency we receive interest at the foreign rate of interest rf .
在持有外币时，我们按照外汇利率获得利息。

This is just like receiving a continuous dividend:
这就像是持续分红一样：

∂V 1 2 2 ∂2V ∂V

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Commodity options
大宗商品期权

The relevant feature of commodities requiring that we adjust
大宗商品需要调整的相关特性我们必须调整

the Black–Scholes equation is that they have a cost of carry. That is, the storage of commodities is not without cost.
Black-Scholes方程式是他们的资金持有成本。 换言之，商品的存储并非无成本。

Let us introduce q as the fraction of the value of a commodity that goes towards paying the cost of carry.
让我们将q引入作为用于支付持有成本的商品价值的一部分，这部分用于支付持有成本。

To be precise, for each unit of the commodity held an amount
为了使准确地每个持有的商品单位的数量

qS dt will be required during short time dt to finance the holding:
qSdt将在短时间内需要dt来资助持有：

∂V 1 2 2 ∂2V ∂V

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Solving the equation and the greeks

The Black–Scholes equation has simple solutions for calls, puts and some other contracts. Now we are going to go quickly through the derivation of these formulæ.
Black–Scholes方程具有简单的解决方案呼叫，看跌期权和某些其他合同。现在我们将快速浏览推导这些公式。

The ‘delta,’ the first derivative of the option value with respect to the underlying, occurs as an important quantity in the deriva- tion of the Black–Scholes equation. In this lecture we see the importance of other derivatives of the option price, with respect to the variables and with respect to some of the parameters.
‘δ’，波行的幽第一导数，是期权价值在导数派扣-赛克尔斯方程中的重要量。在本节课中，我们将看到期权价格的其他导数对变量以及对某些参数的重要性。

These derivatives are important in the hedging of an option position, playing key roles in risk management.
这些衍生品在期权头寸的对冲中起着重要作用，在风险管理中扮演着关键角色。

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Derivation of the formulæ for calls, puts and simple digitals
推导公式的方法，对看涨期权、看跌期权和简单数字期权

The Black–Scholes equation is
布莱克 - 夏尔斯方程是

∂V 1 2 2 ∂2V ∂V

This equation must be solved with final condition depending on the payoff: each contract will have a different functional form prescribed at expiry t = T , depending on whether it is a call, a put or something else.
这个方程必须根据给定的最终条件进行求解：每个合同在到期时将有一个不同的功能形式，该形式由取决于是否为看涨期权，看跌期权或其他情况。

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The first step in the manipulation is to change from present value to future value terms.
Manipulation的第一步是从现值变为未来价值。

Recalling that the payoff is received at time T but that we are valuing the option at time t this suggests that we write
回想收到报酬的是在时刻T，但我们是在几个时间价值的期权在t this建议我们写

V (S, t) = e−r(T −t)U (S, t).

∂t ∂t

This takes our differential equation to

∂U 1 2 2 ∂2U ∂U

(Remember this result for later, present valuing means that one of the terms disappears.)

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The second step is really trivial. Because we are solving a back- ward equation we’ll write

τ = T − t.

This now takes our equation to
这现在带到我们的

∂U 1 2 2 ∂2U ∂U

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When we first started modeling equity prices we used intuition about the asset price return to build up the stochastic differential equation model. Let’s go back to examine the return and write
当我们开始建立股票价格模型时，我们使用了关于资产价格回报的直觉来建立随机微分方程模型。让我们回头来检查回报并写下

∂ −ξ ∂

∂2 −2ξ ∂2

−2ξ ∂

∂S ∂ξ

∂S2 = e

∂ξ2 − e ∂ξ.

Now the Black–Scholes equation becomes
现在Black–Scholes方程变成

2

+ （r − 1σ2） ∂U.

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The last step is simple, but the motivation is not so obvious. Write

This is just a ‘translation’ of the co-ordinate system. It’s a bit like using the forward price of the asset instead of the spot price as a variable.
这只是坐标系的‘翻译’。这有点像使用资产的远期价格而不是现货价格作为一个变量。

After this change of variables the Black–Scholes becomes the simpler
在这个变量改变后，Black–Scholes变得简单

∂τ 2

2 ∂2W

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And you’ve seen this equation before!

You’ve even solved it to find a special solution—out of the infinite number of possible solutions—and exactly the same solution will be needed here. (Lucky!)
You’ve even solved it to find a special solution—out of the infinite number of possible solutions—and exactly the same solution will be needed here.(幸运!)

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To summarize,
总之

x−（r−1σ2）τ

） −rτ

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We are going to derive an expression for the value of any option whose payoff is a known function of the asset price at expiry.
我们将推导出任何期权价值的表达式，其回报是已知函数的资产价格在到期时的。

This includes calls, puts and digitals. This expression will be in the form of an integral.
这包括认购期权，认沽期权和数字期权。这个表达式将会以积分的形式给出。

For special cases, we’ll see how to rewrite this integral in terms of the cumulative distribution function for the Normal distribution. This is particularly useful since the function can be found on spreadsheets, calculators and in the backs of books.
对于特殊情况，我们将看到如何以标准正态分布的累积分布函数重写这个积分。这特别有用，因为可以在电子表格、计算器和书籍背面找到这个分布函数。

But there are two steps before we can write down this integral.
但是在我们写下这个积分之前有两个步骤。

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The first step is to find a special solution of (8), called the
第一步是找到一个特殊的方程解，即（8）的解，称为基本

fundamental solution. This solution has useful properties.
解。这个解具有有用的性质。

The second step is to use the linearity of the equation and the
第二步是利用方程的线性性和方程的性质，这个方程的性质

useful properties of the special solution to find the general solution of the equation.
有用的特殊解法的性质可以帮助我们找到方程的通解。

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The first step is easy, just recall solving the equation from the earlier lecture. The solution we want is
第一步很简单，只需回想从前面讲座中解出方程的解法。我们需要的解是

1 −(x−x叮)2

√2πτ σe

2σ2τ .
2σ2τ。

And this also strongly hints at a relationship between option values and probabilities. More anon!
和这也强烈地暗示在选项值和概率之间的关系。更多匿名！

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4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

-2 -1 0 1 2 3 4

Above is plotted Wf as a function of x叮 for several values of τ
上面是绘制Wf作为一个函数的x叮为几个值的τ .

At x叮 = x the function grows unboundedly, and away from this point the function decays to zero, as τ → 0.
在x叮= x函数不断增长且远离此点处函数衰减为零,随着τ趋于0.

Although the function is increasingly confined to a narrower and narrower region its area remains fixed at one.
虽然函数越来越局限在一个越来越狭窄的区域内,但其面积保持不变为1.

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These properties of decay away from one point, unbounded
这些性质从一个点衰减,不受限制地向外扩散

growth at that point and constant area, result in a Dirac delta function δ(x叮 − x) as τ → 0.
生长在那个点和恒定区域，导致一个Dirac delta函数δ（x叮− x）作为τ → 0。

The delta function has one important property, namely
Delta函数具有一个重要属性，即

Z δ(x叮 − x)g(x叮) dx叮 = g(x)

where the integration is from any point below x to any point above x
在 x 下面的任何点到 x 上面的任何点积分的地方.

Thus the delta function ‘picks out’ the value of g at the point where the delta function is singular i.e. at x叮 = x
因此，δ函数在δ函数是奇异的点处“挑选出”g 的值，即在 x 叮 = x 处.

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In the limit as τ → 0 the function W becomes a delta function at x = x叮. This means that
当τ趋近于0 时，函数W变成一个位置为x的δ函数。这意味着该函数会在x处取值叮。这意味着

 1  r ∞

−(x叮−x)2

τ →0 σ
τ趋近于0 时σ

  e

2σ2τ g(x叮) dx叮 = g(x).
2σ2τg(x叮)dx叮=g(x)。

Whoa! This is tricky!
哇！Thisistricky！

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I am going to ‘cut to the chase’ and quote the solution:
我在去‘cuttothechase’和引用解决方案：

V (S, t) =

e−r(T −t) σv2π(T − t)

dS叮

This ‘formula’ works for any European, non path-dependent, option on a single lognormal underlying asset, all you need to know is the payoff function.
这个‘公式’适用于任何欧洲，非路径依赖的期权上的单一的对数正态基础资产，你需要知道的就是报酬函数。