The Stock Price SDE Let W bo a Brownian motion on a probability space (Ohm, F, P

Question

The Stock Price SDE Let W bo a Brownian motion on a probability space (Ohm, F, P). The price of a single share of S is assumed to satisfy the SDE dS/S = sigma dW + mu dt, where mu and sigma are constants called, respectively, the drift, and volatility of the stock. Equation (11.1) asserts that the relative change in the stock price has two components: a deterministic part mu dt, which accounts for the general trend of the stock, and a random component sigma dW, which reflects the unpredictable nature of S. The volatility Is a measure of the riskiness of the stock and its sensitivity to changes in the market. If sigma = 0, then (11.1) is an ODE with solution St = S_0e^mut. Equation (11.1) may be written in standard form as dS = digma S dW + mu S dt, Find dS^k in terms of dW and dt.

Explanation / Answer

10)

dS=SdW + Sdt (discrete returns dS/S are normally distributed with mean and volatility )

=>dS/S=dW + dt (divide by S both sides)

=>d(ln(S))=dW + dt

ln(St)-ln(S0)=( t)+(-2/2)( t) (integrating both sides from t=0 to t=t,continuous returns ln S are lognormally distributed with mean -2/2 and volatility )

kln(St)-kln(S0)=k( t)+k(-2/2)( t)

ln(Stk)-ln(S0k)=k( t)+k(-2/2)( t)

ln(Stk/S0k)=k( t)+k(-2/2)( t)

d(Sk)/Sk=kdW+k(-2/2)dt

d(Sk)=k*Sk(dW+(-2/2)dt)

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