Let's model the price of a stock (S) using the following stochastic differential equation (SDE):
dS = μ * S * dt + σ * S * dW
Where:
- S is the current price of the stock.
- μ is the drift coefficient (expected return).
- σ is the volatility coefficient.
- dt is the time step.
- dW is a Wiener process increment (Brownian motion) that follows a normal distribution with mean 0 and standard deviation √dt for each time step.
For this example, you have:
- S = $100 (current stock price)
- μ = 0.08 (annual expected return)
- σ = 0.2 (annual volatility)
-number of trading days in a year=252.
Now, to simplify the exercise, we'll provide a single hypothetical value for dW, even though in reality, it's a stochastic process. Let's assume dW = 0.02.
What is the stock price (S) after one time step?
💡 Please feel free to share your detailed step-by-step justification in the comment section.
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To solve for the stock price after one time step using the given SDE, we can follow these steps:
1. Decompose the SDE:
Change in Stock Price = Drift + Shock due to Volatility
dS = μ * S * dt + σ * S * dW
2. Substitute in given values:
We are given:
S = $100
μ = 0.08
σ = 0.2
dW = 0.02
number of trading days in a year = 252
Thus, dt = 1/252 (as we're considering a single trading day as our time step).
3. Plug these values into our SDE:
dS = 0.08 * 100 * (1/252) + 0.2 * 100 * 0.02
dS = 0.03175 + 0.4 = 0.43175
4. Add this change to our initial stock price to get the new price:
S_new = S + dS = 100 + 0.43175 = $100.43175
Thus, the stock price after one time step is $100.43175
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