Using differentiation to determine future stock prices in layman’s terms….
Step 1 - Given Equation:
The equation dS/dt = μ * St tells us something important: it shows that during a tiny moment, the change in the price of an asset (like a stock) is equal to a certain number μ (we call it the "drift") multiplied by the asset's current price St.
Step 2 - Dividing Both Sides:
If we divide both sides of the equation by St (the asset's current price), and then multiply both sides by a tiny amount of time dt, we get dSt/St = μ * dt or (1/ St) dt This equation is important because it links how the percentage change in the asset's price relates to the drift μ and the tiny time step dt.
Step 3 - Solving the Differential Equation:
Now we're going to solve this equation. Think of it like solving a puzzle to find out how the asset's price changes over time. By integrating the right side of the equation (which means adding up all the small changes), we get:
ln(St) + c1 = μ * t + c2
NB: If you differentiate ln (St) with respect to St you get (1/St)
Step 4 - Simplifying and Antilog:
If we subtract c1 from both sides of the equation, we get ln(St) = μ * t + (c2- c1).
Now, let's use a special operation called "antilog" on both sides. It's like the opposite of taking the natural logarithm. This operation turns "ln(St)" back into "St."
Step 5 - Getting the Final Equation:
So, by using the antilog on both sides, we get St = e^(μt + c2 - c1).
Now, we see that c2 - c1 can be replaced with just "c" (a single number).
Step 6 - Wrapping It Up:
When we have c = c2 - c1, and we set e^(c) equal to the initial price (S₀), the equation becomes St = S₀ * e^(μt). This means that the future price of the asset is the initial price times "e" raised to the power of the drift μ times t.
In simpler terms, the equation helps us predict how the asset's price changes over time using a drift rate μ. It's like having a math formula to see how an asset's price will grow over time based on a constant rate of change.