Using Differentiation To Predict Future Stock Prices Simply Explained


Using Differentiation To Predict Future Stock Prices Simply Explained
Using Differentiation To Predict Future Stock Prices Simply Explained

Step 1 - Given Equation:


The equation \( \frac{dS}{dt} = \mu \cdot S_t \) 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 \( \mu \) (we call it the "drift") multiplied by the asset's current price \( S_t \).


Step 2 - Dividing Both Sides:


If we divide both sides of the equation by \( S_t \) (the asset's current price), and then multiply both sides by a tiny amount of time \( dt \), we get:


$$\frac{dS_t}{S_t} = \mu \cdot dt$$


This equation is important because it links how the percentage change in the asset's price relates to the drift \( \mu \) 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:


$$\ln(S_t) + c_1 = \mu \cdot t + c_2$$


Note: If you differentiate \( \ln(S_t) \) with respect to \( S_t \), you get \( \frac{1}{S_t} \).


Step 4 - Simplifying and Antilog:


If we subtract \( c_1 \) from both sides of the equation, we get:


$$\ln(S_t) = \mu \cdot t + (c_2 - c_1)$$


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 \( \ln(S_t) \) back into \( S_t \).


Step 5 - Getting the Final Equation:


By using the antilog on both sides, we get:


$$S_t = e^{\mu t + c_2 - c_1}$$


We can simplify \( c_2 - c_1 \) into just \( c \) (a single number).


Step 6 - Wrapping It Up:


When we have \( c = c_2 - c_1 \), and we set \( e^c \) equal to the initial price \( S_0 \), the equation becomes:


$$S_t = S_0 \cdot e^{\mu t}$$


This means that the future price of the asset is the initial price times \( e \) raised to the power of the drift \( \mu \) times \( t \).


In simpler terms, the equation helps us predict how the asset's price changes over time using a drift rate \( \mu \). 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.


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About the Author

 

 Florian Campuzan is a graduate of Sciences Po Paris (Economic and Financial section) with a degree in Economics (Money and Finance). A CFA charterholder, he began his career in private equity and venture capital as an investment manager at Natixis before transitioning to market finance as a proprietary trader.

 

In the early 2010s, Florian founded Finance Tutoring, a specialized firm offering training and consulting in market and corporate finance. With over 12 years of experience, he has led finance training programs, advised financial institutions and industrial groups on risk management, and prepared candidates for the CFA exams.

 

Passionate about quantitative finance and the application of mathematics, Florian is dedicated to making complex concepts intuitive and accessible. He believes that mastering any topic begins with understanding its core intuition, enabling professionals and students alike to build a strong foundation for success.