Stochastic calculus4/30/2023 ![]() Steele, Stochastic Calculus and Financial Applications. Yor, Continuous Martingales and Brownian Motion, Springer, 1999. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1998. Springer, 2016.Īdditional references for stochastic calculus: Le Gall, Brownian Motion, Martingales, and Stochastic Calculus. References marked * are available for free electronically through. Homework will be announced and posted on Stellar. Please include "18.676" in the subject line of all emails. TAs: Morris Ang (angm at #) and Vishesh Jain (visheshj at #). Instructor: Nike Sun (nsun at #), office hours Mondays 1-3pm. (The fall 2019 page contains a summary of topics covered.) We are not going to do the derivation here as it is too technical.Prerequisite: 18.675. 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 ![]() 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: Price of option = cost of hedging with stock and cash. Thus, the cost of this hedging process should be the price that option is worth. The main intuition is that the price of an option is the cost of hedging it.īy hedging, we mean that we can separately create a combination of stocks and cash to mimic the market exposure of the option. 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). Some of the assumptions are there for the convenience of mathematical modelling.īlack Scholes Model – Application to Finance There are assumptions that may not hold in real-life. Stochastic calculus as applied to finance, is a form of pseudo science. In reality, the stock prices may not be random and log-normally distributed in the long run.In reality, there are sudden jumps in prices.In GBM, the volatility is assumed to be constant. In reality, the randomness and volatility changes over time.However, those points above are debatable. Calculations with GBM processes are relatively easy.Similarly, it is said that the expected value of the stock price in the next time period has nothing to do with the last time period Expected value of the data in the next time period has nothing to do with the last time period.The GBM process has only positive values.They won an Nobel Prize in Economics for it.Įssentially, these mathematicians argue that GBM can be used to model stock prices because it is said that: His theory is later built upon by Robert Merton and Paul Samuelson in their work on options pricing. ![]() In 1900, Louis Bachelier, a mathematician, first introduced the idea of using geometric Brownian motion (GBM) on stock prices. This is where we relate everything we’ve just said to finance. 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).Ĭonstant A and Constant B are usually derived by analyzing historical market data. Which means the change in the stock price = current stock price multiplied by some constant value over time +Ĭurrent stock price + change due to randomness multiplied by another constant. Let S be stock price.Įxplanation: Change in S = Constant A * Current S * change in time + Constant B * Current S * change due to randomness as modeled by GBM In this case, we try to link the equation to finance. Let’s replace X (a regular variable) with S (stock price) so that you can visualize this better. More info on the derivation of Itô’s lemma: Derivation of Itô’s lemma by Math PartnerĪ variation of Itô’s lemma that uses GBM is:īefore we explain it. ![]() Which means the change in the value of a variable = some constant value over time + change due to randomness multiplied by another constant. This equation takes into account Brownian motion.Įxplanation: Change in X = Constant A * change in time + Constant B * change due to randomness as modeled by Brownian motion. The main equation in Itô calculus is Itô’s lemma. The main aspects of stochastic calculus revolve around Itô calculus, named after Kiyoshi Itô. Without a smooth curve, we can’t draw those slope lines productively. We can keep zooming in but we will not be able to find a smooth curve. If we zoom in, we see that it looks… somewhat the same. ![]()
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