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# Tutorial on Itˆo’s Formula

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### Tutorial on Itˆ o’s Formula

Zheng-Liang Lu

Department of Computer Science & Information Engineering National Taiwan University

November 18, 2019

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### Agenda

Wiener process.

Generalized Wiener process (Itˆo process).

Itˆo integral.

Martingale.

Itˆo’s formula.

Calculation examples.

Black-Scholes option pricing theory.

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### Wiener Process

A process W is called the Wiener process if the following conditions hold:

(1) W (0) = 0.

(2) The process W has independent increments: if r < s ≤ t < u, then W (u) − W (t) ⊥ W (s) − W (r ).

(3) For s < t, W (t) − W (s) ∼ N(0, t − s).

(4) W has continuous trajectories.

Note that W has a nowhere-differentiable trajectory (see the next page).

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### Itˆ o Process

A stochastic process X (t) is given by X (t) = a +

Z t 0

µ(s, X (s))ds + Z t

0

σ(s, X (s))dW (s), (1) where a is the initial condition of X (0), µ(t, X (t)) and

σ(t, X (t)) are two adapted1 processes, and W (t) is a Wiener process.

The third item at the right-hand side of Equation (1) is to be defined.

1Let X and Y be stochastic processes. Y is adapted to FtX-filtration if Y is FtX-measurable.

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### Itˆ o Integral

Let g be a process satisfying the following conditions:

g is square-integrable, that is, Z b

a

Eg2(s) ds < ∞.

g is adapted to the FtW-filtration.

We define the Itˆo integral as follows:

Z b a

g (s)dW (s) , lim

∆t→0 n−1

X

k=0

g (tk) [W (tk+1) − W (tk)] .

Why using theforward increments?

Because we cannot foresee the future.

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Then the following relations hold:

E

Z b a

g (s)dW (s)



= 0, (2)

E

"

Z b a

g (s)dW (s)

2#

= Z b

a

Eg2(s) ds, (3)

andRb

a g (s)dW (s) is FbW-measurable.2

2We could say that the integral is deterministic at time b.

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### Sketch of Proof for Equation (2)

E

Z b a

g (s)dW (s)



≈ E

"n−1 X

k=0

g (tk)∆W (tk)

#

=

n−1

X

k=0

E [g (tk)] E [∆W (tk)] (∵ W (tk) ⊥ ∆W (tk))

= 0.(∵ E[∆W (tk)] = 0)

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### Sketch of Proof for Equation (3)

For all i , j with i 6= j , we first calculate

E [∆W (ti)∆W (tj)] = E [∆W (ti)] E [∆W (tj)] = 0.

Note that the first equality results from the property of independent increments.

Then Equation (3) is proved as follows.

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E

"

Z b a

g (s)dW (s)

2#

≈ E

n−1

X

k=0

g (tk)∆W (tk)

!2

=

n−1

X

k=0

Eg2(tk) E ∆W2(tk) + X

i

X

j

E [g (ti)g (tj)] E [∆W (ti)∆W (tj)]

=

n−1

X

k=0

Eg2(tk) E ∆W2(tk)

=

n−1

X

k=0

Eg2(tk)

(tk+1− tk)

→ Z b

a

Eg2(s) ds.

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### Martingale

3

A stochastic process X is called an Ft-martingaleif the following condition hold.

For all t, E [ |X (t)| ] < ∞.

X is adapted to the filtration {Ft}t≥0.

For all s and t with s ≤ t, E [ X (t) | Fs] = X (s).

Now let X (t) =Rt

0 g (s)dW (s) with 0 ≤ t0 < t.

Then we have E

h

X (t) FtW0

i

= X (t0) +E

Z t

t0

g (s)dW (s) FtW0



= X (t0).

By Equation (2), every stochastic integral is a martingale.

3It is a notion of fair games.

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### Digression: Is the Market a Martingale?

For stock markets, the stock prices are not martingales.

Consider that you deposit S (0) in the bank with r ≥ 0.

Then S (t) = S (0)ert, which is riskless.

Or we rewrite the equation above like

E[S (t) | F0] = ertS (0) ≥ S (0).

This implies that the riskless asset is asubmartingale.

Because ofrisk aversion, one should expect a higher return for taking higher risk, that is,

E[S0(t) | F0] > ertS0(0), where S0(t) is a process of one risky asset.

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### Digression: Risk-Neutral Valuation & Martingale

Under a physical measure P, it is known that EPS0(t) F0 > ertS0(0).

Let Y (t) = e−rtS0(t).

Under the risk-neutralmeasure Q, the discounted asset price is a martingale because

EQ[Y (t) F0] = Y (0).

This result is used to price derivatives as follows:

p = EQh

e−rTΠ(S0(T )) F0i , where p is the derivative price and Π is a stochastic contingent claim for S0 with the time to maturity T .

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### Quadratic Variation of Wiener Process

Define ∆t = t − s and ∆W = W (t) − W (s) with s < t.

By definition, we have

E[∆W ] = 0,(∵ ∆W ∼ N(0, ∆t))

Var[(∆W )] = ∆t.

Now we are interested in the quadratic variation (∆W )2, which has:

E[(∆W )2] = ∆t,

Var[(∆W )2] = 2(∆t)2. (Why?)

This is because the trajectory of W isrough!

(dW )2= dt.

This identity will be used in the Itˆo’s formula.

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### Itˆ o Formula

For convenience, notations are simplified unless necessary.

For example, X (t) and µ(t, X (t)) are replaced by X and µ, respectively.

In a differential form, Equation (1) is equivalent to

dX = µdt + σdW . (4)

Let f be a C2-function.4

Define the process Z by Z = f (t, X ).

Then Z has a stochastic differential given by df = ∂f

∂t + µ∂f

∂x +1 2σ22f

∂x2



dt + σ∂f

∂xdW . (5)

4The function f is said to be of (differentiability) class Ckif the derivatives f0, f00, . . . , f(k)exist and are continuous.

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### Sketch of Proof for Itˆ o Formula

It is known that the second-order Taylor expansionfor f is

df = ∂f

∂tdt+∂f

∂xdX +1 2

2f

∂x2(dX )2 + ∂2f

∂t∂xdtdX + 1 2

2f

∂t2(dt)2 .

We then calculate (dX )2 with the identity (dW )2= dt so that (dX )2= µ2(dt)2+ 2µσdtdW + σ2(dW )2

∼ σ2dt.

Note that · · · is negligible compared to the dt-term.

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As a result,

df = ∂f

∂tdt +∂f

∂xdX +1 2

2f

∂x2(dX )2 +1 2

2f

∂t2(dt)2+ ∂2f

∂t∂xdtdX

= ∂f

∂tdt +∂f

∂xdX +1 2

2f

∂x2σ2dt

= ∂f

∂tdt +∂f

∂x(µdt + σdW )+1 2σ22f

∂x2dt

= ∂f

∂t + µ∂f

∂x +1 2σ22f

∂x2



dt + σ∂f

∂xdW .

Hence the proof is complete.

Note that · · · is used as the second form of Itˆo’s formula.

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4

### (t)] =?

Define Z by Z (t) = W4(t).

Then we have

∂Z

∂W = 4W3, and

2Z

∂W2 = 6W2.

By the Itˆo formula,

dZ = 6W2dt + 4W3dW with Z (0) = 0.

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Written in integral form, this reads Z (t) = 0 + 6

Z t 0

W2(s)ds + 4 Z t

0

W3(s)dW (s).

Taking the expected value on the equation above, the stochastic-integral term will vanish.

So we have

E [W4(t)] = 6 Z t

0

E [W2(s)]ds = 6 Z t

0

sds = 3t2.

Note that the exchange between doing an integration and taking an expected value works in most cases of financial math.5

This result could be used to prove Var[(∆W )2] = 2(∆t)2.

5See Fubini’s theorem.

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αW (t)

### ] =?

Define Z by Z (t) = eαW (t) with Z (0) = 1.

The Itˆo formula gives us dZ (t) = 1

2eαW (t)dt + αeαW (t)dW

= 1

2Z (t)dt + αZ (t)dW (t).

In integral form, this reads Z (t) = 1 +1

2 Z t

0

Z (s)ds + α Z t

0

Z (s)dW (s).

Why bother?6

6One can rewrite the stochastic process in form of · · · dt + · · · dW via the Itˆo formula. Starting from this form, it is easier to derive the expected values associated with the stochastic process. For most time, you cannot derive

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Now define m(t) = E [Z (t)] and differentiate the resulting equation as follows:

dm(t) = 1

2m(t)dt.

Using the ODE technique7, we have

m(t) = E [eαW (t)] = e12α2t.

Note that E [eαW (t)] is the moment-generating function (MGF)8 of W (t) so that you may follow the definition of MGF to produce the same result.

7To be more specific, you need the identity dxx = d ln x .

8See https://en.wikipedia.org/wiki/Moment-generating_function.

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t

0

### W (s)dW (s) =?

Define Z by Z (t) = W2(t).

By the Itˆo formula,

dZ (t) = dt + 2W (t)dW (t).

Z (t) = W2(t) = t + 2 Z t

0

W (s)dW (s).

So we have

Z t

0

W (s)dW (s) = W2(t) 2 −t

2.

The second term in the RHS differs from the ordinary calculus!

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### Example 4: Geometric Brownian Motion (GBM)

Let µ and σ be constant, and W be under the P measure.

A GBM is given by

dS = µSdt + σSdW .

Now take X = ln S with X (0) = ln S0.

It is easy to see that

∂X

∂S = ∂(ln S )

∂S = 1 S, and

2X

∂S2 = ∂2(ln S )

∂S2 = − 1 S2.

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By the Itˆo’s formula, dX = ∂(ln S )

∂t dt+∂(ln S )

∂S dS + 1 2

2(ln S )

∂S2 (dS )2

= 1

SdS+1 2(−1

S2)S2σ2dt

= 1

S(µSdt + σSdW )−1 2σ2dt.

= (µ −1

2)dt + σdW .

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In integral form, this reads ln S = ln S0+

Z t 0

(µ −1

2)dt + Z t

0

σdW

=ln S0+(µ −1

2)t+σW (t).

This gives us

ln S (t) ∼ N



ln S0+(µ −1

2)t,σ2t

 .

Note that the price volatility of one asset is σ√ t.

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In the end, we have

S (t) = S0e(µ−12σ2)t+σW (t),

which follows a so-called lognormaldistribution with E [S (t)] = S0eµt,

Var [S (t)] = S02

e(2µ+σ2)t− e2µt

.(Why?)

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### Exercise: Futures Price

Assume that S (t) follows a GBM.

It is known that the futures price F (t) is given by F (t) = S (t)er (T −t).

By the Itˆo’s formula,

dF = (µ − r )Fdt + σFdW .

If we shift to the Q measure (i.e., µ is replaced by r ), then dF = σFdW

with E [F (t)] = F0, which is a martingale.

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### Exercise: Product of GBM Processes

Let Y and Z be two GBM processes:

dY

Y = a dt + b dWY, dZ

Z = f dt + g dWZ, where dWY and dWZ has correlation ρ.

Consider the product of two GBM processes, U = YZ .

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By the Itˆo’s formula, dU = Z dY + Y dZ + dY dZ

= YZ (a dt + b dWY) + YZ (f dt + g dWZ)+

YZ (a dt+b dWY)(f dt+g dWZ)

= U [(a + f +bg ρ)dt+ b dWY + g dWZ] .

Rewrite the above equation as below:

dU

U = (a + f + bg ρ)dt + b dWY + g dWZ.

We show that the product of correlated GBM processes thus remains a GBM.

In particular, we can also show that Sn is also a GBM process for n ∈ N.

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### Exercise: Quotients of GBM Processes

Consider the quotient of two GBM processes, U = Y

Z,

where Y and Z are drawn from Example 6.

By the Itˆo formula, dU = 1

ZdY − Y

Z2dZ − 1

Z2dYdZ + Y Z3(dZ )2 ...

= U(a − f +g2− bg ρ)dt + b dWY − g dWZ .

This example reminds us to collect alldt-terms.

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### Example 5: Vasicek Model

9

X is a Vasicek process, defined by

dX = κ(θ − X )dt + σdW , with θ, κ, σ > 0.

Let Y = eκtX .

By the Itˆo’s formula, we then have dY = κeκtXdt + eκtdX

= κeκtXdt + eκt(κ(θ − X )dt + σdW )

= κθeκtdt + σeκtdW .

9Vasicek (1977). It is one of extension of the Ornstein-Uhlenbeck process, proposed by Ornstein and Uhlenbeck in 1930. Now the Vasicek model is out-of-date. The main focus aims at the LIBOR market model (LMM).

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eκtX = X0+ Z t

0

κθeκsds + Z t

0

σeκsdW .

Moreover, we could calculate

E [ X ] = X0e−κt+ θ(1 − e−κt), Var [ X ] = σ2

2κ 1 − e−2κt .

As t → 0, it is easy to see that E [ X ] = X0 and Var [ X ] = 0.

As t → ∞, E [ X ] = θ and Var [ X ] = σ2, which is finite due to the mean-reverting property!

Note that X is a process following a normal distribution.

(Why?)

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### Black-Scholes Option Pricing Theory

Assume that the stock price St follows a GBM (see p. 22).

For this stock, we now consider to sell a European call option which expires in time T and has the payoff function

Φ(ST) = (ST − K )+.

Insert a figure as an illustration of options.

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Define the call price Ct= f (t, St).

By the Itˆo’s formula, df = ∂f

∂tdt + ∂f

∂St

dSt+1 2

2f

∂St2(dSt)2

= ∂f

∂tdt + ∂f

∂St

(µStdt + σStdW ) + 1 2

2f

∂St2σ2St2dt

= ∂f

∂t + µSt

∂f

∂St2St2 2

2f

∂St2



dt + ∂f

∂StσStdW .

What is the fair priceof this call option?

The no-arbitrage principle comes into play.

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Construct a riskless portfolio as follows: buy ∆ = ∂S∂f

t shares of the stock and sell one European call.

The portfolio value V is V = ∆ × St− f .

For a small variation of St,

dV =∆ × dSt− df . (6)

If the market is free of arbitrage, then the risk-free asset must earn the risk-free rate, denoted by r > 0.

This gives us

dV = rVdt =r (∆ × St− f )dt. (7)

Now equate (6) and (7):

r (∆ × St− f )dt=− ∂f

∂t +σ2St2 2

2f

∂St2

 dt.

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Hence we derive the famous Black-Scholes PDE as follows:

∂f

∂t + rSt

∂f

∂St2St2 2

2f

∂St2 = rf . (8)

Define ∆ = ∂S∂f

t, Θ = ∂f∂t, and Γ = ∂S2f2 t.

Then we have another representation of BS-PDE as follows:

Θ + rSt∆ +σ2St2

2 Γ = rf .

If one considers the delta neutral (∆ = 0), then the previous equation becomes

Θ + σ2St2

2 Γ = rf .

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10

### Theorem

This discovery bridges two research domains (PDE andSDE)!

If f (t, x ) with t ∈ [0, T ] is a solution to

∂f

∂t + µ(x )∂f

∂x +1

2(x )∂2f

∂x2 = rf , f (T , x ) = Φ(x ), then f (t, x ) has a representation

f (t, X ) = e−r (T −t)EQ[ Φ(XT) | Xt = x ], where X follows a Itˆo process.

10Mark Kac (1914–1984), a Polish American mathematician.

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Now replace X by S .

Then the call price is

C = f (0, S0) = e−rTEQ[ (ST − X )+].

This is calledrisk-neutral valuation.

The price of European call options is

C = S0N(d1) − Ke−rTN(d2), (9) where N(·) is a cdf of a standard normal distribution,

d1 = log(SK0) + (r − σ22)T σ√

T ,

and d2 = d1− σ√ T .

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### References

John Hull, Options, Futures, and Other Derivatives, 10/e, 2018.

Tomas Bj¨ork, Arbitrage Theory in Continuous Time, 2009.

Steven Shreve, Stochastic Calculus for Finance II:

Continuous-Time Models, 2010.

Y.-D. Lyuu, lecture slides of Principles of Financial Computing, 2019.11

11See https://www.csie.ntu.edu.tw/~lyuu/finance1.html.

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