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Stochastic calculus is a way to conduct regular calculus when there is a random element.<\/p>\n\n\n\n
Regular calculus is the study of how things change and the rate at which they change.<\/em><\/p>\n\n\n\n Description<\/strong><\/p>\n\n\n\n Think of stochastic calculus as the analysis of regular calculus + randomness.<\/p>\n\n\n\n Regular Calculus<\/strong><\/p>\n\n\n\n Regular calculus studies the rate at which things changes.<\/p>\n\n\n\n The red lines indicate the rate of increase at the black dots.<\/p>\n\n\n\n As the red lines become steeper, the increase in values is going up at a faster pace.<\/p>\n\n\n\n Imagine that we are climbing up a ladder, but now we are climbing up at a faster pace.<\/p>\n\n\n\n Randomness<\/strong><\/p>\n\n\n\n Let’s talk about randomness before combining this with the earlier section on regular calculus.<\/p>\n\n\n\n This is what a bunch of random charts look like:<\/p>\n\n\n\n This behavior is described as Brownian motion. <\/p>\n\n\n\n This means that their behaviour is random, but over the long run and with enough samples, their overall movement resembles a bell shape. In other words, they are normally distributed.<\/p>\n\n\n\n This randomness is not so random after all. The end result of all these random movements is a bell shaped output. (See the bell shape by tilting your head to the right.)<\/p>\n\n\n\n That means that most of the data points end in the middle while the rest are spread out across the sides.<\/p>\n\n\n\n More info on Normal Distribution: <\/em>Normal Distribution – MathIsFun<\/em><\/a><\/p>\n\n\n\n In Brownian motion, the values can be negative. However, stock prices can’t be negative.<\/p>\n\n\n\n Thus, in finance, we use geometric Brownian motion to model our stock prices. <\/p>\n\n\n\n Geometric Brownian motion (GBM) is essentially regular Brownian motion but with an upward drift.<\/p>\n\n\n\n The end result of all these GBM movements is a skewed bell shaped output.<\/p>\n\n\n\n This skewed bell-shaped curve no longer resembles a normal distribution. It now resembles a log-normal distribution. <\/p>\n\n\n\n Stochastic Calculus = Regular Calculus + Randomness<\/strong><\/p>\n\n\n\n When we zoom in on a curve chart, we get a nice curve line. We can then measure the rate of increase using those slopes.<\/p>\n\n\n\n Now let’s look at a chart with randomness.<\/p>\n\n\n\n If we zoom in, we see that it looks… somewhat the same.<\/p>\n\n\n\n We can keep zooming in but we will not be able to find a smooth curve. Without a smooth curve, we can’t draw those slope lines productively. <\/p>\n\n\n\n Thus, normal calculus will fail here. This is why we need stochastic calculus.<\/p>\n\n\n\n Stochastic Calculus Mathematics<\/strong><\/p>\n\n\n\n The main aspects of stochastic calculus revolve around It\u00f4 calculus, named after Kiyoshi It\u00f4.<\/p>\n\n\n\n The main equation in It\u00f4 calculus is It\u00f4’s lemma. This equation takes into account Brownian motion.<\/p>\n\n\n\n It\u00f4’s lemma: <\/p>\n\n\n\n <\/span> <\/span> Explanation: Change in X = Constant A * change in time + Constant B * change due to randomness as modeled by Brownian motion.<\/p>\n\n\n\n Which means the change in the value of a variable = some constant value over time + change due to randomness multiplied by another constant.<\/p>\n\n\n\n More info on the derivation of It\u00f4’s lemma: <\/em>Derivation of It\u00f4’s lemma by Math Partner<\/em><\/a><\/p>\n\n\n\n A variation of It\u00f4’s lemma that uses GBM is: <\/p>\n\n\n\n <\/span> <\/span> Before we explain it. Let’s replace X (a regular variable) with S (stock price) so that you can visualize this better.<\/p>\n\n\n\n <\/span> <\/span> In this case, we try to link the equation to finance. Let S be stock price. <\/p>\n\n\n\n Explanation: Change in S = Constant A * Current S * change in time + Constant B * Current S * change due to randomness as modeled by GBM<\/p>\n\n\n\n Which means the change in the stock price = current stock price multiplied by some constant value over time + That should intuitively make sense as over time, the change of the stock price is based on some overall trend (the Constant A part) and an element of randomness (the Constant B part and randomness part). <\/p>\n\n\n\n Constant A and Constant B are usually derived by analyzing historical market data. <\/p>\n\n\n\n Finance and Stochastic Calculus<\/strong><\/p>\n\n\n\n This is where we relate everything we’ve just said to finance.<\/p>\n\n\n\n In 1900, Louis Bachelier, a mathematician, first introduced the idea of using geometric Brownian motion (GBM) on stock prices. <\/p>\n\n\n\n His theory is later built upon by Robert Merton and Paul Samuelson in their work on options pricing. They won an Nobel Prize in Economics for it.<\/p>\n\n\n\n Essentially, these mathematicians argue that GBM can be used to model stock prices because it is said that:<\/p>\n\n\n\n However, those points above are debatable.<\/p>\n\n\n\n Stochastic calculus as applied to finance, is a form of pseudo science. There are assumptions that may not hold in real-life. Some of the assumptions are there for the convenience of mathematical modelling.<\/p>\n\n\n\n Black Scholes Model – Application to Finance<\/strong><\/p>\n\n\n\n The most famous application of stochastic calculus to finance is to price options (options are a special financial instrument that gives the holder the choice to buy or sell an asset at a certain price).<\/p>\n\n\n\n The main intuition is that the price of an option is the cost of hedging it. <\/p>\n\n\n\n By hedging, we mean that we can separately create a combination of stocks and cash to mimic the market exposure of the option.<\/p>\n\n\n\n Thus, the cost of this hedging process should be the price that option is worth.<\/p>\n\n\n\n Price of option = cost of hedging with stock and cash. <\/p>\n\n\n\n Now, we can calculate the price of the option if we assume that the stock can be modeled using Ito’s lemma, which brings us back to the equation above:<\/p>\n\n\n\n \n\n <\/span> <\/span> Using the above equation and the fact that the price of the option = cost of hedging with stock and cash, we can derive our Black-Scholes equation<\/p>\n\n\n\n Black-Scholes Equation<\/p>\n\n\n\n <\/span> <\/span> We are not going to do the derivation here as it is too technical. <\/p>\n\n\n\n Here is the derivation: <\/em>Paul Wilmott on Quantitative Finance, Chapter 5, Black-Scholes<\/em><\/a><\/p>\n\n\n\n Once you solve that equation and turn it into a form that we can plug in figures and use, you’ll get the Black-Scholes Formula: <\/p>\n\n\n\n <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> This is how you get from the equation to the formula: Links to Other Explanations<\/strong><\/p>\n\n\n\n Related Terms<\/strong><\/p>\n\n\n\n Definition Stochastic calculus is a way to conduct regular calculus when there is a random element. Regular calculus is the study of how things change and the rate at which they change. Description Think of stochastic calculus as the analysis of regular calculus + randomness. Regular Calculus Regular calculus studies the rate at which things […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_links_to":"","_links_to_target":"","footnotes":""},"categories":[7],"tags":[],"class_list":["post-517","post","type-post","status-publish","format-standard","hentry","category-data-science-resources"],"yoast_head":"\n
\n\n\n\n![\"Normal](\"http:\/\/algotrading101.com\/wiki\/wp-content\/uploads\/2019\/06\/Normal-Calculus-1-1024x602.png\")
![\"\"](\"https:\/\/algotrading101.com\/wiki\/wp-content\/uploads\/2019\/06\/Increasing-steepest-of-the-slopes.gif\")
![\"Randomness](\"http:\/\/algotrading101.com\/wiki\/wp-content\/uploads\/2019\/06\/Randomness-chart.png\")
![\"\"](\"https:\/\/algotrading101.com\/wiki\/wp-content\/uploads\/2019\/06\/Brownian-motion-normal-distribution.png\")
![\"geometric](\"http:\/\/algotrading101.com\/wiki\/wp-content\/uploads\/2019\/06\/geometric-brownian-motion-with-normal-distribution.gif\")
![\"normal](\"http:\/\/algotrading101.com\/wiki\/wp-content\/uploads\/2019\/06\/normal-vs-lognormal-distribution.png\")
![\"x](\"http:\/\/algotrading101.com\/wiki\/wp-content\/uploads\/2019\/06\/x-power-of-2-chart-2-1024x551.png\")
<\/figcaption><\/figure><\/div>\n\n\n\n![\"Increasing](\"http:\/\/algotrading101.com\/wiki\/wp-content\/uploads\/2019\/06\/Increasing-steepest-of-the-slopes-2.gif\")
The curve is smooth<\/figcaption><\/figure><\/div>\n\n\n\n![\"fractals](\"http:\/\/algotrading101.com\/wiki\/wp-content\/uploads\/2019\/06\/fractals-chart.png\")
![\"Randomness](\"http:\/\/algotrading101.com\/wiki\/wp-content\/uploads\/2019\/06\/vid264-chart-volatility1-1024x593.png\")
![\"\[ dX_t = \mu_t dt + \sigma dB_t\]\"](\"https:\/\/algotrading101.com\/wiki\/wp-content\/ql-cache\/quicklatex.com-0d00a1995fa042ff7cbd81b570bbf5bb_l3.png\" "\\"Rendered")
<\/p> <\/p>\n\n\n\n![\"\[ dX_t = \mu_t X_t dt + \sigma X_t dB_t\]\"](\"https:\/\/algotrading101.com\/wiki\/wp-content\/ql-cache\/quicklatex.com-f149fe3580e64a44b2cd0b1811bd4356_l3.png\" "\\"Rendered")
<\/p> <\/p>\n\n\n\n![\"\[ dS_t = \mu_t S_t dt + \sigma S_t dB_t\]\"](\"https:\/\/algotrading101.com\/wiki\/wp-content\/ql-cache\/quicklatex.com-551ceeb90a58881b0acfb916cee6b471_l3.png\" "\\"Rendered")
<\/p> <\/p>\n\n\n\n
current stock price + change due to randomness multiplied by another constant. <\/p>\n\n\n\n<\/p> \n\n<\/p>\n\n\n\n![\"\[ \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0 \]\"](\"https:\/\/algotrading101.com\/wiki\/wp-content\/ql-cache\/quicklatex.com-46952bfac796fef247258ec7bd73dda3_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[ \mathrm C(\mathrm S,\mathrm t)= \mathrm N(\mathrm d_1)\mathrm S - \mathrm N(\mathrm d_2) \mathrm K \mathrm e^{-rt} \label{eq:2} \]\"](\"https:\/\/algotrading101.com\/wiki\/wp-content\/ql-cache\/quicklatex.com-43719260ed158f44596dbce2e119cf8a_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[ \hspace*{-10cm} \text{where} \]\"](\"https:\/\/algotrading101.com\/wiki\/wp-content\/ql-cache\/quicklatex.com-ea92420ebc61e426daffcc05de4ddf3d_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[ \mathrm d_1= \frac{1}{\sigma \sqrt{\mathrm t}} \left[\ln{\left(\frac{S}{K}\right)} + t\left(r + \frac{\sigma^2}{2} \right) \right] \]\"](\"https:\/\/algotrading101.com\/wiki\/wp-content\/ql-cache\/quicklatex.com-e57df4ec39972564fde137703cc70881_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[ \mathrm d_2= \frac{1}{\sigma \sqrt{\mathrm t}} \left[\ln{\left(\frac{S}{K}\right)} + t\left(r - \frac{\sigma^2}{2} \right) \right] \]\"](\"https:\/\/algotrading101.com\/wiki\/wp-content\/ql-cache\/quicklatex.com-8fbc37f530b04d6f1e69db78f04ac39f_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\[ N(x)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} \mathrm e^{-\frac{1}{2}z^2} dz \label{eq:5} \]\"](\"https:\/\/algotrading101.com\/wiki\/wp-content\/ql-cache\/quicklatex.com-7c1836482af0d7ed9e06af2f29481525_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n
<\/em>Solution of the Black-Scholes Equation – University of Nebraska<\/em><\/a> (warning: It gets technical) <\/em><\/p>\n\n\n\n
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