Think Value

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To describe the evolution of a derivative of stock price, we use Ito’s Lemma. For example, through it we can express the fair forward price of a stock as a derivative of its current spot price.

Recall that the Ito process is a generalized form of the Wiener process, where both the drift rate and the variance rate are functions of time and the underlying variable:

dx = a(x, t)dt + b(x, t)dz – where:

(a) — drift rate,
(b) — variance rate.

(Detailed: Ito’s Lemma & Wiener Process)

Ito proved that if (x) (the stock price) follows an Ito process, and (G) is a function of (x) and (t) (for example, an option price), then (G) also follows an Ito process.

In that case, the drift rate and variance rate of (G) are determined according to a specific formula, and (dz) represents the Wiener process.

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Partial Derivative Notation

The symbol ∂ denotes a partial derivative — meaning the effect of one factor while keeping others constant.

Symbol	Meaning	How to read it
∂G / ∂S	Change in (G) when (S) changes, keeping (t) constant	“Partial G with respect to S”
∂G / ∂t	Change in (G) when (t) changes, keeping (S) constant	“Partial G with respect to t”
∂²G / ∂S²	Second derivative of (G) with respect to (S); appears in Ito’s Lemma because of variance terms	“Second partial derivative of G with respect to S”

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We’ve also seen that a stock price process can be expressed as an Ito process, assuming that investors’ required and expected returns remain constant.
(Detailed: Geometric Brownian Motion of Stock Price)

dS = μS*dt +σS*dz, (Were – σ is Standard Deviation)

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Applying Ito’s Lemma

According to Ito’s Lemma, if (G(S, t)) is a function of the stock price and time, then the stochastic process for (G) can be written as:

Symbol	Meaning
(G(S, t))	Function depending on stock price (S) and time (t) (e.g., option price)
∂G / ∂t	Partial derivative with respect to time — measures time’s effect on (G)
∂G / ∂S	Partial derivative with respect to stock price — measures how (G) changes when (S) changes
∂²G / ∂S²	Second partial derivative — measures curvature, i.e., how sensitivity itself changes with (S)
( \mu )	Mean growth rate of the stock price (drift rate)
( \sigma )	Stock price volatility
( dz )	Wiener process — random change over time (dt)
( dt )	Infinitesimal time interval

Hence, if we know how the stock price (S) evolves over time, we can also describe the evolution of any derivative (G(S, t)).

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Application to Forward Prices

We know that the relationship between forward and spot prices is given by:

F₀​=S₀e^rT
(Detailed: Valuing Forward Contracts)

At any time (t < T), the forward price can be written as:

F​=Se^r(T-t)

To apply Ito’s Lemma, we calculate the partial derivatives of (F) with respect to (S) and (t), and substitute them into the lemma.

Put them in formula:

Since:

F​=Se^r(T-t)

we derive an interesting result:

the expected rate of change (drift) of the forward price equals (μ – r) —
that is, the stock’s excess return over the risk-free rate.

Source: Options, Futures & Other Derivatives, John C. Hull

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