Tutorial on Itˆo’s Formula

Tutorial on Itˆ o’s Formula

Zheng-Liang Lu

Department of Computer Science & Information Engineering National Taiwan University

November 18, 2019

Agenda

• Wiener process.

• Generalized Wiener process (Itˆo process).

• Itˆo integral.

• Martingale.

• Quadratic variation of Wiener process.

• Itˆo’s formula.

• Calculation examples.

• Black-Scholes option pricing theory.

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).

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 Ft^X-filtration if Y is Ft^X-measurable.

Itˆ o Integral

• Let g be a process satisfying the following conditions:

• g is square-integrable, that is, Z b

a

Eg²(s) ds < ∞.

• g is adapted to the F_t^W-filtration.

• We define the Itˆo integral as follows:

Z b a

g (s)dW (s) , lim

∆t→0 n−1

X

k=0

g (t_k) [W (t_k+1) − W (t_k)] .

• Why using theforward increments?

• Because we cannot foresee the future.

• 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

Eg²(s) ds, (3)

andRb

a g (s)dW (s) is F_b^W-measurable.²

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

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

= 0.(∵ E[∆W (tk)] = 0)

Sketch of Proof for Equation (3)

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

E [∆W (t_i)∆W (t_j)] = E [∆W (t_i)] E [∆W (t_j)] = 0.

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

• Then Equation (3) is proved as follows.

E

"

Z b a

g (s)dW (s)

²#

≈ E





n−1

X

k=0

g (t_k)∆W (t_k)

!2



=

n−1

X

k=0

Eg²(tk) E ∆W²(tk) + X

i

X

j

E [g (t_i)g (t_j)] E [∆W (t_i)∆W (t_j)]

=

n−1

X

k=0

Eg²(t_k) E ∆W²(t_k)

=

n−1

X

k=0

Eg²(tk)

(tk+1− t_k)

→ Z b

a

Eg²(s) ds.

Martingale

• A stochastic process X is called an F_t-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) F_t^W0

i

= X (t⁰) +E

Z t

t⁰

g (s)dW (s) F_t^W0



= X (t⁰).

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

3It is a notion of fair games.

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)e^rt, which is riskless.

• Or we rewrite the equation above like

E[S (t) | F₀] = e^rtS (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] > e^rtS⁰(0), where S⁰(t) is a process of one risky asset.

Digression: Risk-Neutral Valuation & Martingale

• Under a physical measure P, it is known that E^PS⁰(t) F0 > e^rtS⁰(0).

• Let Y (t) = e^−rtS⁰(t).

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

E^Q[Y (t) F₀] = Y (0).

• This result is used to price derivatives as follows:

p = E^Qh

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 .

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!

• In differential form, it reads

(dW )²= dt.

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

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 2σ²∂²f

∂x²



dt + σ∂f

∂xdW . (5)

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

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.

• 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 2σ²∂²f

∂x²dt

= ∂f

∂t + µ∂f

∂x +1 2σ²∂²f

∂x²



dt + σ∂f

∂xdW .

• Hence the proof is complete.

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

Example 1: E [W

(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.

• 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.

Example 2: E [e

] =?

• Define Z by Z (t) = e^{αW (t)} with Z (0) = 1.

• The Itˆo formula gives us dZ (t) = 1

2α²e^{αW (t)}dt + αe^{αW (t)}dW

= 1

2α²Z (t)dt + αZ (t)dW (t).

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

2α² 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

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

dm(t) = 1

2α²m(t)dt.

• Using the ODE technique⁷, we have

m(t) = E [e^{αW (t)}] = e^12α2t.

• Note that E [e^{αW (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 ^dx_x = d ln x .

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

Example 3: R

0

W (s)dW (s) =?

• Define Z by Z (t) = W²(t).

• By the Itˆo formula,

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

• In integral form this reads

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!

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.

• 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

2σ²)dt + σdW .

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

Z t 0

(µ −1

2σ²)dt + Z t

0

σdW

=ln S₀+(µ −1

2σ²)t+σW (t).

• This gives us

ln S (t) ∼ N



ln S0+(µ −1

2σ²)t,σ²t

 .

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

• In the end, we have

S (t) = S₀e^{(µ−12σ2)t+σW (t)},

which follows a so-called lognormaldistribution with E [S (t)] = S0e^µt,

Var [S (t)] = S₀²

e^(2µ+σ2)t− e^2µt

.(Why?)

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)e^{r (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.

Exercise: Product of GBM Processes

• Let Y and Z be two GBM processes:

dY

Y = a dt + b dW_Y, dZ

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

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

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

= YZ (a dt + b dW_Y) + YZ (f dt + g dW_Z)+

YZ (a dt+b dW_Y)(f dt+g dW_Z)

= U [(a + f +bg ρ)dt+ b dW_Y + g dW_Z] .

• Rewrite the above equation as below:

dU

U = (a + f + bg ρ)dt + b dW_Y + g dW_Z.

• 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.

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 dW_Y − g dW_Z .

• This example reminds us to collect alldt-terms.

Example 5: Vasicek Model

• X is a Vasicek process, defined by

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

• Let Y = e^κtX .

• By the Itˆo’s formula, we then have dY = κe^κtXdt + e^κtdX

= κe^κtXdt + e^κt(κ(θ − X )dt + σdW )

= κθe^κtdt + σe^κtdW .

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).

• So it reads

e^κtX = X₀+ Z t

0

κθe^κsds + Z t

0

σe^κsdW .

• 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 ] = ^σ_2κ², which is finite due to the mean-reverting property!

• Note that X is a process following a normal distribution.

(Why?)

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.

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

• By the Itˆo’s formula, df = ∂f

∂tdt + ∂f

∂St

dS_t+1 2

∂²f

∂S_t²(dS_t)²

= ∂f

∂tdt + ∂f

∂St

(µStdt + σStdW ) + 1 2

∂²f

∂S_t²σ²S_t²dt

= ∂f

∂t + µSt

∂f

∂S_t +σ²S_t² 2

∂²f

∂S_t²



dt + ∂f

∂S_tσStdW .

• What is the fair priceof this call option?

• The no-arbitrage principle comes into play.

• 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 S_t,

dV =∆ × dS_t− 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 (∆ × S_t− f )dt. (7)

• Now equate (6) and (7):

r (∆ × St− f )dt=− ∂f

∂t +σ²S_t² 2

∂²f

∂S_t²

 dt.

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

∂f

∂t + rSt

∂f

∂S_t +σ²S_t² 2

∂²f

∂S_t² = rf . (8)

• Define ∆ = _∂S^∂f

t, Θ = ^∂f_∂t, and Γ = _∂S^∂2f2 t.

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

Θ + rS_t∆ +σ²S_t²

2 Γ = rf .

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

Θ + σ²S_t²

2 Γ = rf .

Feynman-Kac

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

2σ²(x )∂²f

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

f (t, X ) = e^{−r (T −t)}E^Q[ Φ(X_T) | Xt = x ], where X follows a Itˆo process.

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

• Now replace X by S .

• Then the call price is

C = f (0, S₀) = e^−rTE^Q[ (S_T − 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(^S_K⁰) + (r − ^σ₂²)T σ√

T ,

and d₂ = d₁− σ√ T .

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.