A Geometric Brownian motion (GBM) is used to model a stock price: dS(t) = S(t)(bdt + dWt).

 

A Geometric Brownian motion (GBM) is used to model a stock price: dS(t) = S(t)(bdt + σdWt). In the above, b and σ are termed as the drift and the volatility of the underlying stock.

Required

(1) Derive the explicit formula of S(t) in the above using Ito’s lemma. Also find a formula of t for the mean and variance of S(t).

(2) If a stock price follows S(t) = 100 exp{0.03t + 0.1Wt}, where Wt. is a BM. What is its drift and volatility?

(3) Write a Matlab code to do the following jobs: (1) Plot 100 paths; (2) compute its mean and variance of S(1).

Transcribed Image Text:

Consider a stock that pays no dividends, has a volatility of 30% per annum, and provides an expected return of 15% per annum with continuous compounding. In this case, μ = 0.15 and o = 0.30. The process for the stock price is ds S If S is the stock price at a particular time and AS is the increase in the stock price in the next small interval of time, the discrete approximation to the process is or AS AS S - S 0.15 dt +0.30 dz where has a standard normal distribution. Consider a time interval of 1 week, or 0.0192 year, so that At = 0.0192. Then the approximation gives = 0.15At+ 0.30€√At √0.0192 € = 0.15 x 0.0192 +0.30 x √ AS = 0.00288S+ 0.0416S€ Consider a stock that pays no dividends, has a volatility of 30% per annum, and provides an expected return of 15% per annum with continuous compounding. In this case, μ = 0.15 and o = 0.30. The process for the stock price is ds S If S is the stock price at a particular time and AS is the increase in the stock price in the next small interval of time, the discrete approximation to the process is or AS AS S - S 0.15 dt +0.30 dz where has a standard normal distribution. Consider a time interval of 1 week, or 0.0192 year, so that At = 0.0192. Then the approximation gives = 0.15At+ 0.30€√At √0.0192 € = 0.15 x 0.0192 +0.30 x √ AS = 0.00288S+ 0.0416S€

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Lets Given Stock pays no dividends Shashizak Volatility E of the stock is 30 per annum Expected vetw... View the full answer