Ito's lemma and Black-Scholes model(伊藤定理的简单推导与BS模型的推理)

Ito’s lemma and Black-Scholes model

Zhiming Liao (chiminhlyao@)

Ito’s lemma and Black-Scholes model are indispensable tools in financial applications. Ito lemma, named after its discoverer, Kishi Ito, is of great importance in finding the differential of a function of a particular type of stochastic process. When Fischer Black and Myron Scholes published their work-the pricing of options and corporate liabilities in the Journal of Political Economy in the 1970s, their prominent work immediately drew the interest of the Chicago option market, in 1997, they were awarded Nobel Prize in economics for this significant contribution to financial theory. Since then, Black-Scholes model has been playing a vital role in calculating options. This short paper will address Ito’s lemma and Black-Scholes model and their proofs.

Ito’s lemma

Suppose y(t) follows a diffusive stochastic process, that is to say

dy t=u y dt+σy dz t

Here, u y is the instantaneous expected rate of change in the y and σy is its instantaneous standard deviation.

And f y,t is a function differentiable twice in the first argument and once in the second. Then f also follows a diffusive process

d f y,t=ðf

+

ðf

u y+

1ð2f

2

σy2dt+

ðf

σy z t

Proof (informal derivation using Taylor series expansion formula) Taylor series expansion in two variables

f(x.y)=f(x0.y0)+x−x0

1!

f x x0.y0+

y−y0

1!

f y x0.y0+

x−x02

2!

f xx x0.y0

+x−x0y−y0

f xy x0.y0+

y−y02

f yy x0.y0+

x−x03

f xxx x0.y0

+x−x02

2!

y−y0

1!

f xxy x0.y0+

x−x0

1!

y−y02

2!

f xyy x0.y0

+y−y03

f yyy x0.y0+⋯

Assume that we are initially at some α,t and that a short interval of time ∆t passes. During this ephemeral period there will be some associated∆z. Using the Taylor expansion above,

f α+u y∆t+σy∆z,t+∆t

=fα,t+ u y∆t+σy∆z f y+∆tf t+1

u y∆t+σy∆z 2f yy+ u y∆t2+σy∆z∆t f yt

+1

∆t2f tt+t ird and ig er order terms

Therefore,

∆f=f α+u y∆t+σy∆z,t+∆t −fα,t

= u y∆t+σy∆z f y+∆tf t+1

2

u y∆t+σy∆z 2f yy+ u y∆t2+σy∆z∆t f yt+

1

2

∆t2f tt

+t ird and ig er order terms

Since z is following a standard Brownian motion, ∆z is normally distributed with expected value 0 and variance ∆t. It is easy to check that, E∆z2=∆t.

E∆f=1

σy2f yy+u y f y+f t∆t+second and ig er order terms in∆t

When ∆t tends to 0 or small enough, we can ignore higher order terms, so we can get the expected rate of change in f in accord with Ito’s lemma above.

Thus,

∆f−E∆f=σy∆zf y+second and ig er order terms in∆t and∆z

This e quation is the last part of the Ito’s lemma expression.

A formal correct proof can be fo und in Malliaris and Brock’s paper-Stochastic methods in economics and finance.

Example

In a world with a constant nominal interest rate r, a bond portfolio with value of $1 at time 0 and continuously reinvested coupon payments will be worth B(t)=e rt at time t. Suppose that the price level evolves randomly according to the stochastic process

dP=μPdt+σPdz

Where μ is the expected inflation rate and σ is its proportional standard deviation per unit time. The real value of the bond portfolio at time t will be

b t=

B t

What is the expected real return on the bonds?

Example 2 in the textbook- copula methods in finance-page 9

Notice that, given

ds t=μS t dt+σS t dz t

We can get f S,t=ln S t to obtain

d ln S t= μ−1

2

σ2dt+σdz t

If μ and σ are constant parameters, it is easy to obtain

ln Sτ∼N ln S t+ μ−1

σ2τ−t,σ2τ−t

Where N m.s is the normal distribution with mean m and variance s. Then, Pr Sτℑt is described by the lognormal distribution.

Black- Scholes model

In the simplest form, this model involves two underlying assets, a riskless asset like cash bond and a risky asset such as stock. The riskless asset appreciates at the short rate, or riskless rate of return r t, which is assumed to be nonrandom, although possibly time-varying. Thus, the price of the riskless at time t is assumed to satisfy the differential equation