Latex for Financial Engineering Mathematics Formula (Continuous-Time Finance)

Navigation

In this blog, we will summarize the latex code of most popular formulas and equations for Financial Engineering Formula and Equation (Continuous-Time Finance). We will cover important topics including Standard Brownian Motion (SBM), Geometric Brownian Motion (GBM), Ito Lemma, Stochastic Integrals, Solutions to Some Common SDEs, Brownian Motion Variation, Stock Prices as a GBM, Stock Prices are Lognormal, Sharpe Ratio and Hedging, The Black-Scholes Equation, Risk-Neutral Valuation and Power Contracts.

• 1. Continuous-Time Finance
• Standard Brownian Motion
• Geometric Brownian Motion
• Ito Lemma

• Standard Brownian Motion

  Financial,Economics

  Equation

  
  
  

  Latex Code

              Z(t) \sim N(0, t) \\
              Z(t+s) - Z(t) \sim N(0, s) \\
              Z(t+s) \sim N(Z(t), s)
          

  Explanation

  Latex code for the Standard Brownian Motion. I will briefly introduce the notations in this formulation. {Z(t)} has independent increments, and {Z(t)} has stationary increments such that Z (t + s) â?? Z (t) follows standard normal distribution

  • : Value of Z at time stamp t
  • : Stationary increments of Standard Brownian Motion
  
  

  Related Documents

  • Investopedia Simple Interest

  Related Videos

• Geometric Brownian Motion

  Financial,Economics

  Equation

  
  
  

  Latex Code

              Y(t) = Y(0)e^{X(t)} = Y(0)e^{[\mu t + \sigma Z(t)]} \\
              E(e^{kU}) = e^{kE(U) + \frac{1}{2}k^{2}\text{Var}(U)} \\
              E[Y^{k}(t)] = Y^{k}(0) e^{(k\mu + \frac{1}{2}k^{2}\sigma^{2})t} \\
              \ln Y(t) \sim N(\ln Y(0) + \mu t, \sigma^{2} t)
          

  Explanation

  Latex code for the Geometric Brownian Motion.

  • : Observed value Y(t) at time stamp t
  • : Any normal random variable
  • : Drift coefficient
  • : Volatility
  
  

  Related Documents

  • Geometric Brownian Motion
  • Geometric Brownian motion

  Related Videos

• Ito Lemma

  Financial,Economics

  Equation

  
  
  

  Latex Code

              \mathrm{d}X(t) = a(t, X(t)) \mathrm{d}t + b(t, X(t))\mathrm{d} Z(t) \\
              Y(t) = f(t, X(t)) \mathrm{d}t \\
              \mathrm{d} Y(t) = f_{t}(t, X(t)) + f_{x}(t, X(t))\mathrm{d} X(t) + \frac{1}{2} f_{xx}(t, X(t))[\mathrm{d}X(t)]^{2} \\
              [\mathrm{d} X(t)]^{2} = b^{2}(t, X(t))\mathrm{d} t
          

  Explanation

  Latex code for the Ito Lemma.

  • : Diffusion
  • : Stochastic differential equation for X(t)
  
  

  Related Documents

  • Geometric Brownian Motion
  • Geometric Brownian motion

  Related Videos