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Wiener Process Ito 's Lemma Derivation of Black-Scholes Solving Black-Scholes

Introduction to Financial Derivatives
Understanding the Stock Pricing Model

22M:303:002

Understanding the Stock Pricing Model

22M:303:002

Wiener Process Ito 's Lemma Derivation of Black-Scholes

Stock Pricing Model

Solving Black-Scholes

Recall our stochastic dierential equation to model stock prices:

dS = σ dX + µ dt S where µ is known as the asset 's drift , a measure of the average rate of growth of the asset price, σ is the volatility of the stock, it measures the standard deviation of an asset 's returns, and

dX is a random sample drawn from a normal distribution with mean zero.
Both µ and σ are measured on a 'per year ' …show more content…

Understanding the Stock Pricing Model 22M:303:002

Wiener Process Ito 's Lemma Derivation of Black-Scholes

Wiener Statistics

Solving Black-Scholes

Mean of dX , E[dX ] =

√

dt E[φ ] = 0

Variance of dX , Var[dX ] = E[(dX − 0)2 ] = E[φ 2 dt ] = dt E[φ 2 ] = dt · 1 = dt √ Standard deviation of dX = dt

Understanding the Stock Pricing Model

22M:303:002

Wiener Process Ito 's Lemma Derivation of Black-Scholes

The Pricing Model

Solving Black-Scholes

dS = σ dX + µ dt S
Since we chose

dX

such that

E[dX ] = 0

the mean of

dS

is:

E[dS ] = E[σ SdX + µ Sdt ] = µ Sdt
The variance of

dS

is:

Var [dS ] = E[dS 2 ] − E[dS ]2 = E[σ 2 S 2 dX 2 ] = σ 2 S 2 dt
Note that the standard deviation equals proportional to the asset 's volatility.

√ σ S dt ,

which is

Understanding the Stock Pricing Model

22M:303:002

Wiener Process Ito 's Lemma Derivation of Black-Scholes

Taylor 's +

Solving Black-Scholes

We need to determine how to calculate small changes in a function that is dependent on the values determined by the above stochastic dierential equation. Let f (S ) be the desired smooth function of S ; since f is suciently smooth we know that small changes in the asset 's price, dS , result in small changes to the function f . Recall that we approximated df with a Taylor series expansion, resulting in

df =

df 1 d 2f dS + dS 2 + · · · , dS 2 dS 2

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