Black Scholes

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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' basis.
Understanding the Stock Pricing Model 22M:303:002

Wiener Process Ito's Lemma Derivation of Black-Scholes

Ecient Market Hypothesis

Solving Black-Scholes

Past history is fully reected in the present price, however this does not hold any further information. (Past performance is not indicative of future returns) Markets respond immediately to any new information about an asset.

Understanding the Stock Pricing Model

22M:303:002

Wiener Process Ito's Lemma Derivation of Black-Scholes

Markov Process

Solving Black-Scholes

Denition A stochastic process where only the present value of a variable is relevant for predicting the future. This implies that knowledge of the past history of a Markov variable is irrelevant for determining future outcomes. Markov Process⇔Ecient Market Hypothesis

Understanding the Stock Pricing Model

22M:303:002

Wiener Process Ito's Lemma Derivation of Black-Scholes

Investigating the Random Variable
Consider a random variable, X , that follows a Markov stochastic process. Further assume that the variable's change (over a one-year time span), dX , can be characterized by a standard normal distribution (a probability distribution with mean zero and standard deviation one, φ = ϕ(0, 1)). What is the probability distribution of the change in the value of the variable (dX ) over two years?

Solving Black-Scholes

Understanding the Stock Pricing Model

22M:303:002

Wiener Process Ito's Lemma Derivation of Black-Scholes

Investigating the Random Variable
Since X follows a Markov process, the two probability distributions are independent. Thus, we can sum the distributions. The two year mean is the sum of the two one-year means. Similarily, the two year variance is the sum of the two one-year variances. However, the change is best represented by the standard deviation, so the probability distribution that describes dX over two years is: √ ϕ(0, 2).

Solving Black-Scholes

Understanding the Stock Pricing Model

22M:303:002

Wiener Process Ito's Lemma Derivation of Black-Scholes

Investigating the Random Variable
Assumption Changes in variance are equal for all identical time intervals. For a six month period, the variance of change is 0.5 and the √ standard deviation of the change is 0.5. The probability distribution for the√ change in the value of the variable during six months is ϕ(0, 0.5). √ Similarily, dX over a three month period is ϕ(0, 0.25). The change in the √ value of the variable during any time √ period, dt , is ϕ(0, dt ) ⇔ φ dt . This is because the variance of the changes in successive time periods are additive, while the standard deviations are not. Understanding the Stock Pricing Model 22M:303:002

Solving Black-Scholes

Wiener Process Ito's Lemma Derivation of Black-Scholes

Wiener Process

Solving Black-Scholes

The process followed by the variable we have been considering is known as a Wiener process; A particular type of Markov stochastic process with a mean change of zero and a variance rate of 1 per year. The change, dX during a small period of time, dt , is √ dX = φ dt where φ = ϕ(0, 1) as dened above. The values of dX for any two dierent short intervals of time, dt , are...
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