Economics-Engineering A Bridge between Economics and Engineering

Economics-Engineering First i wish to point out that the following Project is not to be considered as a mathematical research from a strictly point of view, rather a didactic tool or an interdisciplinary instrument used to clarify the role of Mathematics as fundamental key to understand the strong connection between engineering and economics models. Further, it’s crucial to remember that all the mathematical models used in Finance are, from a certain point of view, endogenously weak, and for this reason, every time somewhere in the world, a financial crash occurs, it is very difficult to prevent it. The main effort in this field, in the next future, is to try to improve these Models using more and more sophisticated mathematical instruments; for example from Functional Analysis, Probability, Stochastic Processes and even from Quantum Mechanics.

Economics-Engineering Unfortunately, Economic Science is not like Physics, Biology, Chemistry, or Mathematics, which are all governed by the laws of the nature and its equations. Today it is more complicated to fully understand all the possible economic Scenarios all over the world; Traders, Brokers and Market Makers must be very careful on using the financial instruments, because is not so easy ”Making Money” without risk. For this reason this project would like to be an useful instrument for the students who want to try to understand the mechanisms of Financial Mathematics, its rules, and most of all, its weaknesses.

Economics-Engineering Among the universe of all the Partial Differential Equations, the PDE’s, a fundamental role is played by the Heat Equation. This particular kind of equation is used in Mathematical Physics and in many other fields of Mathematics. It solves diffusion problems arising from Physics, Biology, and Chemistry etc. The Heat Equation is a second order PDE, it is also called “Parabolic Equation”. To start with, we consider the heat equation in one space variable, plus time.

Economics-Engineering We derive the fundamental solution and show how it is used to solve the Cauchy problem with the Dirichlet condition : Using the well known Fourier Transform we obtain:

Economics-Engineering The solution of the above equation is: To reach the solution written in the original x variable, we must now use the inverse Fourier transform, so the time evolution is no longer described by simply multiplication by a function, but by means of an integral operator with Heat Kernel or Green’s Function equal to:

Economics-Engineering Remark: It is worth outlining the fact that, without invoking a deep knowledge of functional analysis, it is always better to work with functions operating on Hilbert-Spaces, that is to say that the Fourier Transform of such functions, is well-defined, square-integrable function, operating on the Hilbert-Space L2(Rn). In this case are of crucial importance in many applications, e.g. in Physics and Economics, the Formulas of Parseval and Plancherel.

Economics-Engineering SDE: Stochastic Differential Equation dXt = μ(t)Xt dt +  (t)Xt dWt (μ: expected return; : standard deviation) (Ito’s Formula):   Stochastic Table

Economics-Engineering A call option is a contract between two parties in which the holder of the option has the right (not the obligation) to buy an asset at a certain time in the future for a specific price, called the strike price. A put option is a contract between two parties in which the holder of the option has the right (not the obligation) to sell an asset at a certain time in the future for a specific price, also called the strike price.

Economics-Engineering (Black-Scholes Equation) Final Condition P(S,T)=max(0, E-S)=[E-S]+ Pay-off (Black-Scholes Formula)

Economics-Engineering Example: Graph of a Put / Call with r = D0 = 0.05 (Future Contract)

Economics-Engineering Example: In the next example we want to evaluate the Implied Volatility of a European Put Option. This important calculator is based on a particular algorithm, that is to say a modified version of Newton’s algorithm in order to improve its accuracy.

Economics-Engineering CONCLUSIONS: The aim of this project is to be regarded as a guide to all the students, which are curious on how the universe of Finance is strongly correlated with the mathematical models arising from Engineering, where Mathematics is the Language. Using the Laboratory “A Bridge between Economics and Engineering”, the students can experience how Engineering is connected to Economics and vice-versa. I wish to remark that Economics is not an exact Science, for this reason all the models developed in the last years are more and more complicated due to all the shocks that have involved the universe of the financial markets. The new models are built in order to prevent in some way these shocks, most of all under the stochastic point of view. It is worth to remember that the new models are based on the Jump diffusion dynamics for the options pricing, as the models with stochastic volatility; see e.g. Levy, Chiarella, Heston et al.