Ito’s Lemma X(t) is a stochastic process

Ito’s Lemma

X(t) is a stochastic process if its value evolves stochastically over time. Start with discrete–time processes whose values can change only at discrete points in time. The change in X has some probability distribution.

then future stock price is log-normally distributed.

(2) _t is independent of _s for t s.

This specification implies that X(t+2) – X(t) = 2 + _t + _t+1
and that: X(t+2) – X(t) ~ (2, 2 ²),
and in general: X(t+n) – X(t) ~ (n, n ²)
Implications:

“drift” is proportional to n

variance is proportional to n; so standard deviation is proportional to

Digression:

Convert to daily: n = 1/250. Therefore,

Annual basis, odds about 23% market will fall if returns are normal distributed. For the market to fall, return must be below the mean by 12%, which is ¾ of a standard deviation: N() = 0.23.

However, on a daily basis, mean return is insignificant compared to . N() = .48. So the odds are essentially 50/50 market will rise or fall. Essentially a random walk.

A related paradox:
Suppose your portfolio is $1 million. Put it in a safe (real) annuity, and you can earn about $35,000 (real) annually, clearly not enough to retire on (at least in Boston).
Put it in the market index, and the daily standard deviation of your portfolio will be about .01  $1,000,000 = $10,000, which dominates your salary. [E.g., even with a salary of $250,000, you earn only $1,000 a day, relatively insignificant compared to your daily volatility.] So why bother working?
How do we resolve these two conclusions?

Given this analysis, we can write

X(t+n) – X(t) = n +   [where  ~ (0,1)]

Notice that variance = n²

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Innovation: _ _ _ • • •

Within any period, the random portion of the increment to X (the “innovation”) is _i ,

with variance ²h. Therefore, the change in X across any subperiod is :
X(t + ih) – X[t + (i – 1)h] = h +  _i
Notice that the mean and variance of X(t+1) – X(t) are unaffected by the number of subperiods:

X(t+1) – X(t) = (h +  _i) =  +  _i

Mean = 

Variance = ²h  var (_i) = ²h  n = ²

(b) “jump” in stock prices – this means stock prices are continuous.

Now take limit as h  dt:

dX = dt + dz; dz  

• independent increments

• normally distributed, dz ~ N(0, dt)
While I have written mean and std. deviation as constants,  and , we can be more general. An Ito process allows  and  to depend on X and t:
dX = (X, t) dt + (X, t) dz
This process is still Markov: transition probabilities depend only on current values of X and t.
Special cases:

1. 
   Arithmetic Brownian Motion
   

2. 
   Geometric Brownian Motion
   

dX = Xdt + Xdz

= dt + dz

Percentage increments to X have constant mean and variance. Implies X_t is log-normal [i.e., ln(X_t/X₀) is normal].

3. 
   Ornstein – Uhlenbeck process
   

This specification implies mean reversion to a “long–run” value of X*.

h0

= lim E[²h² + ²h² + 2h^3/2] = lim (²h² + ²h) = 0  MSE convergence

Differentiability:

If = X/t

converges in MSE to some value X'(t), this would be its time derivative.
We require E[ ()² ] to converge for the derivative to exist.
But lim E[ ()² ] = lim (² + ²/h) = 
Intuition: think about the stock market example:
= 

Example: The typical daily fluctuation is a very large rate of fluctuation: for example, a one standard deviation swing in return (e.g., 1% per day) is equivalent to a swing in the annualized % rate of change of

______________________________ time

Notice: E(dS/dt) does not exist, but E(dS)/dt does exist!

dX = dt + dz,

To make this example easy, suppose = 0.

Therefore, dX = dz  dX ~ N(0, ² dt), symmetric around 0.

S = exp(X)

 exp(X)

X₀ – X X₀ X₀ + X

Notice that although disturbance around X₀ is symmetric (± X), disturbance around e^X0 is not.
Convexity of function implies gain > loss, so there is upward drift in stock price even though rate of return is symmetric around zero.
E(S) = E(e^x) > e ^E(x) = e^X0 = S₀

In general, E[f(X)]  f[E(X)] This is “Jensen’s Inequality”

Therefore, while usual calculus states that

dS = e^XdX

we’ve just shown that

E(dS)  e^X E(dX) = 0

In this case E(dS) > 0
So usual calculus must be wrong for functions of stochastic processes.
Must consider effect of the curvature of the function




S  error from curvature



 Slope = f’(X₀)

____________________________________________ X

X₀
Obvious tool is a Taylor series expansion, which tells us how to correct for curvature:
f(X₀+ X) = f(X₀) + f^'(X₀) X + ½ f^’’(X₀) (X)² + 1/6 f^’’’(X₀) (X)³ + . . .

f^’ = initial slope

f^’’ adjusts for change in slope (convex f^’’ > 0 ; concave f^’’< 0)

f^’’’adjusts for change in rate of change in slope, etc.

In a loose sense, Ito’s lemma tells us that when we take a Taylor series, we can use the following “multiplication rules:”

(iii) (dz)² = dt

Intuition:

dz = 

(dz)² = ²dt

= E [(²dt – dt)²]

= E (⁴dt² – 2 dt² ² + dt²)

= dt²  E(⁴ – 2 ² + 1)

dX = (X,t) dt + (X,t) dz

then (dX)² = ²(X,t) dt² + 2 (X,t) dt (X,t) dz + ²(X,t) dz² = 0 + 0 + ²(X,t) dt
which is nonstochastic. When you square an Ito process, the only term that survives is variance.

Now consider a function of X, y = f(X,t ).

e.g., y may be the value of a call option and X the stock price on which the call is written.
Taylor series:

f(X + dX, t + dt) = f(X, t) + f_XdX + f_tdt + ½ f_XXdX² + ½ f_ttdt² + f_Xt dXdt

 df = f_X(X, t) dX + f_t(X, t) dt + ½ f_XX(X, t) dX² + 0 + 0

= f_X[(X, t) dt + (X, t)dz] + f_t dt + ½ f_XX ²(X, t) dt + 0 + 0

= [f_X (X, t) + f_t + ½ f_XX ²(X, t)] dt + f_X (X, t)dz

This is called Ito’s lemma.

____________________________________________ X

X₀
Since X is small for small t, may treat relationship as locally linear and correlation  1
What does this analysis imply about “portfolios” comprised of y and X? That they can be made riskless. More on this next week.
This graph also explains why the “extra” or second derivative term enters the drift for y. A Jensen’s inequality term that depends on both convexity and volatility.

Example 1: Log-normality and geometric brownian motion

If dX = dt + dz with ,  constant

[i.e., ln(S) has arithmetic brownian motion, equivalent to assuming ln(S_T) is normally dist.]

then

= (e^X + 0 + ½ e^X²) dt + e^Xdz

= e^X( + ½ ²) dt + e^Xdz



correction for Jensen's inequality

Define  =  + ½ ²

We will commonly write: dS/S =  dt + dz

S has G. B.M. which we now know is equivalent to: S_T /S₀ is log–normally distributed.

Suppose expected rate of return on stock = , but stock pays continuous proportional dividend at rate . Then

dS = ( – ) S dt + S dz [Notice that ²(S, t) = ²S²]

Spot-futures parity for futures maturing at time T implies that

F(S_t, t) = S_t e ^{(r – )(T–t)}
Ito's lemma 

dF = F_s dS + F_t dt + ½ F_ss(dS)²

= F_s( – )S dt + F_t dt + ½ F_ss²S² dt + F_sS dz
But F_S  S = F and F_t = – (r – ) F and F_ss = 0 

dF = ( – )F dt – (r – )Fdt + 0 + F dz

 dF/F = ( – r)dt + dz
Issues: 1) why is E(dF/F) less than expected rate of return on stock by r dt?

2) when will futures price be unbiased predictor of E(S_t)?

Consider a security paying a continuous dividend stream, i.e., a dividend in period dt of Xdt (equivalently, a rate of dividends X), forever.

  

Hypothesize a solution of form f(X) = kX

Then: f_XX = f

f_XX = 0

f + 0 + X = rf  kX + X = rkX 

k = 1/(r–)  f(X) = = growing perpetuity value (Gordon growth model)

Multivariate Ito’s lemma

Need one more multiplication rule. If dz₁ and dz₂ are two Weiner processes, with correlation of , then

dz₁  dz₂ = dt

Notice that dz₁  dz₂ = ₁ ₂= ₁₂dt .

E (₁ ₂) = cov (₁, ₂) = ₁₂ = 
Now let y = f (X₁, X₂, t)

where dX₁ = ₁ (X₁, X₂, t) dt + ₁(X₁, X₂, t) dz₁

dX₂ = ₂ (X₁, X₂, t) dt + ₂(X₁, X₂, t) dz₂
Then dy = f₁ dX₁ + f₂ dX₂ + f₃ dt + ½ f₁₁ (dX₁)² + ½ f₂₂ (dX₂)² + f₁₂ dX₁ dX₂

= [f₁ ₁ + f₂ ₂ + f₃ + ½ f₁₁₁²(X₁, X₂, t) + ½ f₂₂ ₂²(X₁, X₂, t) + f₁₂ ₁₂] dt

+ f₁₁ dz₁ + f₂₂ dz₂

The generalization to n sources of risk is obvious.

dL/L = _L dt + _L dz_L  dL = _LL dt + _LL dz_L

dK/K = _K dt + _K dz_K  dK = _KK dt + _KK dz_K
Notice that Q_L = Q/L = aL^a–1K^b and

Q_LL = aL^aK^b = aQ

Similarly: Q_KK = bQ

Q_KKK²= b(b–1)Q

Q_LLL² = a(a–1)Q

Q_LKLK = abQ

= Q_L (_L L dt + _L L dz_L) + Q_K (_KK dt + _KK dz_K) + ½ Q_KK(K² dt) +

½ Q_LL(L² dt) + Q_KL (KL _K _L dt)
= dt {aQ_L + bQ_K + ½ Q a(a–1)+ ½ Q b (b–1) + Q a b  _L _K }+

a Q _L dz_L + b Q _K dz_K

+ a _L dz_L + b _K dz_K

Performance of a perfectly successful timer as a function of market performance

Portfolio

return




r_f

Market index return

r_f

Notice: There is no meaning to "the beta" of the fund. Beta = 0 or 1, depending on the market forecast.

Can you guess the results of this strategy, if executed with perfect success?

Bills only: $1 $12.6 3.5%

Market index only: $1 $973 9.9%

Perfect timer (annual) $1 $41,022 15.7%