这是用户在 2024-9-3 11:40 为 https://app.immersivetranslate.com/word/ 保存的双语快照页面,由 沉浸式翻译 提供双语支持。了解如何保存?

The Black–Scholes Model
黑-舒尔斯模型模型

In this lecture.
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简单数字Black–Scholes公式

the meaning and importance of the ‘greeks,’ delta, gamma, theta, vega and rho
‘希腊字母’delta,gamma, theta,vegarho意义重要性

Certificate in Quantitative Finance
证书量化金融

10

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

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

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

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

Certificate in Quantitative Finance

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.
理论相当直接,使用我们已经看到的随机微积分思想。

The end result is a diffusion-type partial differential equation which can be used for the pricing of many different derivatives.
最终结果一种扩散型偏微分方程,可用于许多不同衍生品定价。

Certificate in Quantitative Finance

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

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.
期权价值合同参数函数行权E到期时间T 其中T到期t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漂移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.
分号分开不同类型变量参数

Certificate in Quantitative Finance

S and t are variables;
St变量

σ and µ are parameters associated with the asset price;
σµ资产价格相关参数

E and T are parameters associated with the particular con- tract;
ET具体合同相关参数;

r is a parameter associated with the currency.
r货币相关参数.

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

Certificate in Quantitative Finance

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

The underlying follows a lognormal random walk with known volatility
基础遵循对数正态随机游走,其已知波动率

The risk-free interest rate is a known function of time
风险免费利率时间已知函数

There are no dividends on the underlying
基础没有派息

Delta hedging is done continuously
Δ对冲连续进行

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

There are no arbitrage opportunities
基础没有套利机会

Certificate in Quantitative Finance

And more.
更多内容。
. .

Certificate in Quantitative Finance

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

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 一个 投资组合 一个 长期 期权头寸 空头头寸 一些 数量 ∆, 德尔塔 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S中的动作相关的V中的动作,为通话。

Certificate in Quantitative Finance

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

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.
注意时间步长期间没有发生改变

Certificate in Quantitative Finance

From Itˆo we have
Itˆo我们

dV =

∂V

dt +

∂t

∂V

1

2

dS +

∂S

22 2V

σ

S∂S2 dt.

Thus the portfolio changes by

dΠ =

∂V

dt +

∂t

∂V

1

2

dS +

∂S

22 2V

σ

S∂S2 dt dS.(2)

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

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

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

Certificate in Quantitative Finance

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 ∆.
是否任何方法减少甚至消除这种风险?理论上可以通过谨慎选择.

If we choose
如果我们选择

= ∂V

∂S

(3)

then the randomness is reduced to zero.
那么随机性降低零。

Certificate in Quantitative Finance

Any reduction in randomness is generally termed hedging.
任何随机性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动态对冲策略一个示例

∂S

From one time step to the next the quantity ∂V
一个时间步到下一个,

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。

Certificate in Quantitative Finance

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动的投资组合

dΠ =
dΠ=

∂V + 1
∂V+1

σ

S

∂t2
∂t2

22 2V
222V

∂S2
∂S2

dt.(4)
dt。(4)

This change is completely riskless.
这个改变完全无风险的

Certificate in Quantitative Finance

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一致我们放入等价金额的现金无风险利息计息帐户

dΠ = rΠ dt.(5)

This is an example of the no arbitrage principle.
一个例子套汇原则。

Certificate in Quantitative Finance

The Black–Scholes equation
Black–Scholes方程式

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

σ

S

∂V + 1

∂t2

22 2V

∂S2

dt = r

VS∂V

∂S

dt.

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

∂V122 2V∂V

∂t + 2σ S∂S2 + rS ∂S rV = 0.(6)

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

Certificate in Quantitative Finance

Observations:

The Black–Scholes equation equation is a linear parabolic
Black–Scholes方程方程一个线性抛物线

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.
他们不同的估计漂移

Certificate in Quantitative Finance

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对冲论点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.
如果一些期权价值的偏导数,那么这样的投资组合在到期时产生某个数额该期权的支付。

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,只需买入卖出这个标的资产。

This leads to the idea of a complete market. In a complete market an option can be replicated with the underlying, thus making options redundant.
这导致了完全市场的概念,在完全市场中,期权可以通过标的资产复制,从而使期权变得多余。

Certificate in Quantitative Finance

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
最终条件处理我们必须指定期权价值V作为底层到期日的函数

T . That is, we must prescribe V (S, T ), the payoff.
T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

V (S, T ) = max(S E, 0).

Certificate in Quantitative Finance

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.
因此时间dt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。

Certificate in Quantitative Finance

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

dΠ =

∂V

dt +

∂t

∂V

1

2

dS +

∂S

22 2V

σ

S∂S2 dt dS DS dt.
S∂S2dtΔdSDΔSdt.

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:

∂V122 2V∂V

∂t + 2σ S∂S2 + (r D)S ∂S rV = 0.

Certificate in Quantitative Finance

Currency options
货币期权

Options on currencies are handled in exactly the same way.
货币期权完全相同方式处理

In holding the foreign currency we receive interest at the foreign rate of interest rf .
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:
持续分红一样

∂V122 2V∂V

∂t + 2σ S∂S2 + (r rf )S ∂S rV = 0.
∂ t+2σS∂ S2+(rrf)S∂ SrV=0.

Certificate in Quantitative Finance

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方程式他们资金持有成本。 换言之,商品存储并非无成本。

Let us introduce q as the fraction of the value of a commodity that goes towards paying the cost of carry.
我们q引入作为用于支付持有成本商品价值一部分,这部分用于支付持有成本。

To be precise, for each unit of the commodity held an amount
为了使准确持有商品单位数量

qS dt will be required during short time dt to finance the holding:
qSdt时间需要dt资助持有:

∂V122 2V∂V

∂t + 2σ S∂S2 + (r + q)S ∂S rV = 0.

Certificate in Quantitative Finance

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.
这些衍生品期权头寸的对冲起着重要作用,风险管理扮演关键角色。

Certificate in Quantitative Finance

Derivation of the formulæ for calls, puts and simple digitals
推导公式方法看涨期权、看跌期权简单数字期权

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

∂V122 2V∂V

∂t + 2σ S∂S2 + rS ∂S rV = 0.(7)

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

Certificate in Quantitative Finance

The first step in the manipulation is to change from present value to future value terms.
Manipulation第一现值变为未来价值

Recalling that the payoff is received at time T but that we are valuing the option at time t this suggests that we write
回想收到报酬时刻T我们几个时间价值的期权t this建议我们

V (S, t) = er(T t)U (S, t).

∂V = rer(T t)U + er(T t)∂U .

∂t∂t

This takes our differential equation to

∂U122 2U∂U

∂t + 2σ S ∂S2 + rS ∂S = 0.

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

Certificate in Quantitative Finance

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
现在我们

∂U122 2U∂U

∂τ = 2σ S ∂S2 + rS ∂S .
∂τ=2σS∂S2+rS∂S

Certificate in Quantitative Finance

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 = log S. With this as the new variable, we find that
ξ = 对数 S。通过将此作为新变量,我们发现 that

ξ

22ξ 2

2ξ

= eand

∂S∂ξ

∂S2 = e

∂ξ2 e∂ξ.

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

2

∂τ

2

∂ξ2

2

∂ξ

∂U = 1σ2 U

+ r 1σ2 ∂U.

Certificate in Quantitative Finance

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

2

x = ξ + r 1σ2 τandU = W (x, τ ).

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.
只是坐标系的‘翻译’有点像使用资产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
这个变量改变后,Black–Scholes变得简单

∂W = 1

∂τ2

2 2W

σ

∂x2 .(8)

Certificate in Quantitative Finance

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!)

Certificate in Quantitative Finance

To summarize,
总之

e

2

,T τ

= e

W (x, τ ).

V (S, t) = er(T t)U (S, t) = e U (S, T τ ) = e U (eξ,T τ )

= e

U

xr1σ2τ

Certificate in Quantitative Finance

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,其回报已知函数资产价格到期时的

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

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.
但是我们写下这个积分之前两个步骤。

Certificate in Quantitative Finance

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

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

The second step is to use the linearity of the equation and the
第二步利用方程线性性方程的性质,这个方程性质

useful properties of the special solution to find the general solution of the equation.
有用特殊解法性质可以帮助我们找到方程通解

Certificate in Quantitative Finance

The first step is easy, just recall solving the equation from the earlier lecture. The solution we want is
第一简单,只需回想前面讲座解出方程解法。我们需要的

1(xx)2

Wf (x, τ ; x) =
Wf(x,τ; x=

2πτ σe

2σ2τ.
2σ2τ

As you know, this is the probability density function for a Normal random variable x having mean of x and standard deviation στ
正如您所知,这是正态随机变量 x 的概率密度函数,其均值为 x,标准偏差为σ√τ
.

And this also strongly hints at a relationship between option values and probabilities. More anon!
强烈地暗示选项值概率之间关系。更多匿名!

Certificate in Quantitative Finance

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

-2-101234

Above is plotted Wf as a function of x for several values of τ
上面绘制Wf作为一个函数x几个τ
.

At x = x the function grows unboundedly, and away from this point the function decays to zero, as τ 0.
x= x函数不断增长远离此点函数衰减为,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,其面积保持不变1.

Certificate in Quantitative Finance

These properties of decay away from one point, unbounded
这些性质一个点衰减,不受限制地向外扩散

growth at that point and constant area, result in a Dirac delta function δ(x x) as τ 0.
生长那个恒定区域,导致一个Dirac delta函数δxx作为τ 0。

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

Z δ(x x)g(x) dx = 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 = 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
.

Certificate in Quantitative Finance

In the limit as τ 0 the function W becomes a delta function at x = x. This means that
τ趋近于0 时,函数W变成一个位置为xδ函数这意味着函数会在x处取值叮。这意味着

1 r

(xx)2

lim

τ 0 σ
τ趋近于0 时σ

e

2πτ−∞

2σ2τg(x) dx = g(x).
2σ2τg(x)dx=g(x)

Whoa! This is tricky!
哇!Thisistricky!

Certificate in Quantitative Finance

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

V (S, t) =

er(T t) σv2π(T t)

e

2

Payoff(S)

.(9)

r log(S/S)+r1σ2(T t)2I2σ2(T t)
rlog(S/S)+r1σ2(Tt2I2σ2Tt

0

dS

S

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,非路径依赖的期权单一的对数正态基础资产,你需要知道的就是报酬函数。

Certificate in Quantitative Finance

Observations
观察

This is a general formula (see above conditions on type of option)
一个通用公式(参见上述期权类型的条件)

It is of the form of a) a discounting term multiplied by b) the integral of the payoff multiplied by c) another function
形式一个折现乘以另一函数

This other ‘function’ is known as a Green’s function
“函数”称为Green函数

This function can be interpreted as a probability
函数可以解释概率

The whole expression can be interpreted as the present value of the expected payoff
整个表达式可以被解释为预期支付的现值现值期望收益

Certificate in Quantitative Finance

Let’s look at special cases.
让我们看看特殊案例

Certificate in Quantitative Finance

Formula for a call
看涨

The call option has the payoff function
看涨期权支付功能

Payoff(S) = max(S E, 0).
支付(S)=max(SE,0)

Expression (9) can then be written as

er(T t)

r log(S/S)+r1σ2(T t)2/2σ2(T t)

dS

e

2

σv2π(T t)E

(S E) S .
(SE) S

Certificate in Quantitative Finance

2

2

Return to the variable x = log S, to write this as
返回变量x= logS作为

er(T t) σv2π(T t)

r

log E
log E

e

x+log S+
x+log S +

r1σ

2(T t)2

/2σ

(T t)(exE) dx

r(T t)r

12

22

=

e

σv2π(T t)

e

log E

x +log S+

r2σ

(T t)
(Tt) =>

/2σ
/2σ => σ

(T t)ex dx

er(T t)r
er(Tt)r => erTtr

x+log S+r1σ2(T t)2/2σ2(T t)
x+logS+r1σ2(Tt2/2σ2(Tt => x+logS+r1σ2Tt2/2σ2Tt

E

σv2π(T t)

e2

log E

dx.
dx

Certificate in Quantitative Finance

Both integrals in this expression can be written in the form

d

e

2

dx

r 1x2

for some d (the second is just about in this form already, and the first just needs a completion of the square).

Certificate in Quantitative Finance

Thus the option price can be written as two separate terms involving the cumulative distribution function for a Normal dis- tribution:
因此期权价格可以被写成两个涉及正态分布的累积分布函数的独立术语:

Call option value= SN (d1) Eer(T t)N (d2)
看涨期权价值=SN(d1)Eer(Tt)N(d2)

where
其中

log(S/E)+ (r + 1