Financial Applications of Stochastic Calculus 
                                                                                                                                                                    Nunez, N.M. 
                                                                                                                                                     Cape Fear Community College 
                                                                                                                                       The University of North Carolina Wilmington 
                                                               Abstract
             I studied the concepts and principles of stochastic calculus to understand its implementations in 
             financial markets. The study began with probability theory and a review of important topics in calculus 
             and statistics including Taylor series expansions, partial derivatives, differential equations, and 
             cumulative standard normal distribution. Following this was study of stochastic processes and their 
             characteristics such as Martingales and Markov chains. Previous knowledge was pieced together with 
             the introduction of Brownian motion, the mean square limit theorem, and the stochastic integral. The 
             mathematical research culminated in the development of an understanding of Ito’s Lemma. The 
             research then transitioned to an exploration and investigation of the applications and ubiquity of 
             stochastic calculus in finance. The Black-Scholes options pricing model served as the prime point of 
             inquiry and evaluation although other options pricing models were also observed. Quantitative 
             methods involving stochastic calculus were evaluated in addition to less mathematical methods such as 
             fundamental analysis and the study of behavioral finance. The research concluded with a resolution 
             regarding the comparative importance and prevalence of stochastic calculus in finance.
                                                      Research Questions 
             •   What are the basic principles and ideas of stochastic calculus?                                                                                    Figure 2. Itô’s Lemma 
             •   What are the implementations of stochastic calculus in finance?                                                                                                                                                             Figure 3. Various stochastic processes. Note the image does not contain the 
             •   Relative to other financial methods, how prevalent are implementations of                                                                                    Results                                                        full detail of the processes as their movements are infinitely precise. 
                 stochastic calculus among professionals in the industry? 
                                                                                                                                               I found that the basic concepts and principles of stochastic 
                                                                                                                                               calculus center around the derivation of Ito’s lemma. Ito’s lemma                                                                         Conclusion
                                                                                                                                               is the stochastic calculus equivalent of the fundamental theorem 
                                                                                                                                               of calculus for differential calculus. Ito’s lemma states that a 
                                                                                                                                               second degree Taylor series approximation must be created for                              Stochastic calculus is a branch of calculus designed to analyze movement of random processes. Due 
                                                               Methods                                                                         the chain rule of a stochastic process because of the mean                                 to the infinite randomness of stochastic processes, they must be squared in order to contain finite 
                                                                                                                                               square limit. Stochastic processes are infinite in variation, due to                       values, and this necessitates second degree Taylor approximations. The implementations of stochastic 
             In order to find the answers to my research questions, I first consulted papers written by the founder                            Brownian motion, but finite when squared due to the mean                                   calculus are for financial forecasting and asset modeling. Professionals utilize stochastic calculus 
             and discoverer of many principles of stochastic calculus. Thus I learned the from the source the                                  square limit. Since the process is squared in order to be finite,                          within quantitative methods alongside qualitative methods and both offer valid, essential strategies 
             essence of stochastic calculus. I then read various different textbooks and journals that discussed and                           the chain rule of differential calculus will not apply with a first 
             explained the role of stochastic calculus in modeling financial instruments. Each time I encountered a                            order Taylor series approximation. Instead, a second order Taylor                          to companies, and thus stochastic calculus is relatively important and prevalent in finance. 
             term in a source that was not explained therein, I separately researched that term to understand it                               series approximation must be used because the coefficients do 
             entirely before continuing my research of the original source. This often required me to use the sources                          not cancel with a first order approximation. 
             cited within the original source. Figure 1 is a simplified graphic representation of this process. 
                                                                                                                                               Stochastic calculus is of use to quantitative analysts in that it                                                                        Works Cited  
                                                                                                                                               enables them to craft mathematical models to predict the 
                                                                                                                                               movement of processes that would otherwise be infinitely                                   Capiński, M., Kopp, P.E., Traple, J. (2012). Stochastic Calculus for Finance. Cambridge University Press
                                              Source                            Source                                                         unpredictable due to their variation. Companies often use 
                                                                          Source                                                               quantitative methods based on stochastic calculus in 
                                           Sources of                                                                                                                                                                                     Jumadilova, S., Silaubekov, N., & Kunanbayeva, D. (2017). Company’s Financial State Forecasting:   
                                            source                            Source                                                           combination with qualitative methods such as fundamental                                                                    Methods and Approaches. Investment Management and Financial 
                                              Sources of                                                                                       analysis for the most accurate possible results. Stochastic                                                                                               https://doi.org/10.21511/imfi.14(3).2017.09
                                              sources of                                                                                                                                                                                                                   Innovations, 14(3), 93-101. 
                                               source                                                                                          calculus plays a large role in financial forecasting, and it is notably 
                                                                            Findings                                                           implemented in options pricing models such as the Black-Scholes                            Itô, K. (1967) The Canonical Modification of Stochastic Processes. Journal of the Mathematical Society 
                                                                                                                                               model and the binomial model.                                                                                               of Japan, 20(1-2), 130-150. 10.2969/jmsj/02010130
                          Figure 1. A graphical representation of my research process 
                                                                                                                                                                                                                                          Reid, S. R. (2017, April 7). Random walks down Wall Street, Stochastic Processes in Python [Graph]. 
             Upon finding the answer to my first two research questions, I found material regarding the third                                                                                                                                                              Turing Finance. http://www.turingfinance.com/random-walks-down-
             question and studied to reach a conclusion.                                                                                                                                                                                                                   wall-street-stochastic-processes-in-python/
                                                                                                                                                                                                                                          Ito’s Lemma. (n.d.). [Equation]. Derivatives Academy. http://sp-finance.e-
                                                                                                                                                                                                                                                                           monsite.com/medias/images/ito-lemma-3.png