Ito’s lemma and Black-Scholes model
Zhiming Liao (chiminhlyao@gmail)
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
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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 de
viation.
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!
手机回收系统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
陶瓷颗粒+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
dB t
=r t B t Whose solution for the value B 0=1is B t =exp r s ds t
0 Proof
dB t dt =r t B t ⇔dB t B t =r t dt ⇒dln B t
dt =r t ⇔ln B t = r s t
ds +c
⟹B t =exp r s t
0 ds (As B 0=1, c=0)
The share price S t of the risky asset stock at time t is assumed to follow a stochastic differential equation of the form
dS t =μt S t dt +σS t dW t ,
Where W t is a standard Brownian motion, μt is a nonrandom function of t, and σ>0 is a constant called the volatility of the stock. Lemma 1
If the drift coefficient function μt is bounded, then the SDE(stochastic differential equation) has uniqu
e solution with initial condition S 0; and it is given by
S t =S 0exp σW t −σ2t
2+ μs ds t
Moreover, under the risk-neutral measure, it must be the case that
r t =μt .
Proof.
dW t =0+1dW t (u w =0,σw =1) Consider function S W t ,t =S 0exp σW t −σ2t 2
+ μs
ds t
果尔除草剂
0 Applying Ito’s Lemma,
S W =S t σ S WW =S t σ2
S t =S t μt −σ2
2
Then,
d S W t ,t = ðS ðt +ðS ðW u w +12ð2S ðW 2σw 2 dt +ðS ðW σW
dW t = S t μt −σ22 +1
2
S t σ2 dt +S t σdW t =μt S t dt +σS t dW t
So dS t =μt S t dt +σS t dW t , thus proved the first argument of this lemma.
Under risk-neutral assumption, the discounted share price of stock must be a martingale. The discount factor is B t , so the discounted share price of stock is
S t ∗
=S t t =S 0exp σW t −σ2t
2+ μs ds t 0 exp r s
ds t 0
=S 0exp σW t −σ2t + μS −r s ds t 0 Applying Ito’s lemma once more, we can have
dS t ∗=σS t ∗dW t +S t ∗ μS −r s dt
In order that S t∗ be a martingale, the dt term must be 0; this implies that r t=μt.
Lemma 2
Under the risk-neutral measure, the ln of the discounted stock price at time t is normally distributed
with mean lnS0−σ2t
2 and variance σ2t.
Proof,
Consider function f S t∗,t=lnS t∗,
From the proof of Lemma 1, we can say
dS t∗=σS t∗dW t+S t∗μS−r s dt=σS t∗dW t (Under risk-neutral assumption,μS−r s=0) That is dS t∗=σS t∗dW t
Applying Ito’s lemma, dlnS t∗=−σ2
2dt+σdW t
Analogous with example 2 in Ito’s lemma,
E(lnS t∗)=E lnS0−σ2
2
t+σdW t=lnS0−
σ2
2
t
Var(lnS t∗)=Var lnS0−σ2
2
t+σdW t=VarσdW t=σ2Var dW t=σ2t
The Black-Scholes formula for the price of a European Call option
European Call on the asset Stock with strike K and expiration date T is a contract that allows the owner to purchase one share of stock at price K at time T. Thus, the value of the Call at time T is S T−K+. According to the fundamental theorem of arbitrage pricing, the price of the asset Call at time t=0 must be the discounted expectation, under the risk-neutral measure, of the value at time t=T, which, by
Lemma 1, is
C S0,0;K,T =E S T∗−K
B T
+
Where S T∗ has the distribution specified in Lemma 2. A routine calculation, using integration by parts, shows that C x,0;K,T may be rewritten as
C x,0;K,T=xΦz−K
B T
Φ z−σT
Where z=ln xB T
K
+σ
2
2
T
σT
and Φis the cumulative normal distribution function.
Proof
As proved before,
S t∗=S t
B t
=S0exp σW t−
σ2t
2
+μS−r s ds
t
Under the risk-neutral measure, we have
S T∗=S0exp σW T−σ2T 2
Where W T follows standard Brownian motion, W T~N0,T Let W T=g T, so g~N0,1
Therefore
S T∗=S0exp σg T−σ2T
2
g~N0,1
C X,0;K,T=E S T∗−K
T+
S T∗−K
T
≥0 ⟺xexp σg T−
σ2T
≥
K
T
⟹σg T−
σ2T
≥ln
K
T
⟺g≥
σ2T
2−ln
xB T
K
σT
Denote z=ln xB T
K
+σ
2
2
T
σT
,
then S T∗−K
B T
≥0 ⟺g≥σT−Z
C X,0;K,T=E S T∗−K
B T
+
=E xexp σg T−
σ2T
硅片2
−
K
B T
+
= xexp σy T−σ2T
2
−
K
B T
e
−y2
2
2π
∞
σT−Z
= xexp σy T−σ2T
2
e
−y2
2
2π
∞
σT−Z −
K
B
e
−y2
2
2π∞
σT−Z
=x exp σy T−σ2T
2
e
−y2
2
2π
−
∞
σT−Z K
B T
e
−y2
2
2π
∞
σT−Z
=x exp −σy T−σ2T
2
e
−y2
2
2π
−
z−σT −∞K
B T
−y2
2
2π
z−σT
−∞
=x exp −σy T−σ2T e
−y2
2
2π
−
K
T
Φ z−σT
higgs粒子z−σT −∞
While x exp −σy T−σ2T
2
−y2
2
2π
z−σT −∞
=x
2π −σ s−σT T−σ2T
2
− s−σT
2
2
d s−σT
z
(Substitute y=s−σT)
=x
1
2π
−s2
2
ds z
−∞
=xΦz
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