## Latex for Financial Engineering Mathematics Formula-Brownian Motion Ito Lemma Risk-Neutral Valuation

rockingdingo 2023-05-21 #financial engineering #mathematics #finance 2 0

Latex for Financial Engineering Mathematics Formula-Brownian Motion Ito Lemma Risk-Neutral Valuation

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

Financial,Economics

#### 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

• #### Geometric Brownian Motion

Financial,Economics

#### 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

• #### Geometric Brownian Motion SDEs

Financial,Economics

#### Latex Code

            \mathrm{d}Y(t) = \mu Y(t)dt + \sigma Y(t) \mathrm{d}Z(t) \\
\mathrm{d}[\ln Y(t)] = (\mu - \frac{\sigma^2}{2}) \mathrm{d}t + \sigma \mathrm{d}Z(t) \\
Y(t) = T(0) e^{(\mu - \frac{\sigma^2}{2})t + \sigma Z(t)}


#### Explanation

Latex code for the Geometric Brownian Motion.

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

• #### Ito Lemma

Financial,Economics

#### 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)

• #### Stock Prices as Geometric Brownian Motion

Financial,Economics

#### Latex Code

            \frac{\mathrm{d}S(t)}{S(t)} = (a - \delta) \mathrm{d}t + \sigma \mathrm{d}Z(t) \\
S(t) = S(0) e^{(a - \delta - \frac{\sigma^{2}}{2})t + \sigma Z(t)} \\
\mathrm{d}[\ln S(t)] = (a - \delta - \frac{\sigma^{2}}{2}) \mathrm{d}t + \sigma \sigma \mathrm{d} Z(t) \\
S(t) \sim \ln( \ln S(0) + (a - \delta - \frac{\sigma^2}{2})t, \sigma^{2}t)


#### Explanation

Latex code for Stock Prices as Geometric Brownian Motion.

• : Observed stock price S(t) at time stamp t
• : Any normal random variable
• : Drift coefficient
• : Volatility

• #### Sharpe Ratio

Financial,Economics

#### Latex Code

            \frac{\mathrm{d}X(t)}{X(t)} = m \mathrm{d}t + s \mathrm{d}Z(t) \\
\phi = \frac{m + \delta -r }{s} \\
\phi = \frac{a - r}{\sigma}


#### Explanation

Latex code for Sharpe Ratio.

• : Compounded Dividend Rate
• : Sharpe ratio
• : Sharpe ratio of any asset written on a GBM

• #### Risk-Neutral Valuation and Power Contracts

Financial,Economics

#### Latex Code

            \frac{\mathrm{d}S(t)}{S(t)} = (r - \delta) \mathrm{d}t + \sigma \mathrm{d} \tilt{Z}(t) \\
\tilt{Z}(t) = Z(t) + \phi t \\
V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)] \\
F^{p}_{t, T}(S^{a}) = S^{a}(t) e ^{ (-r + a(r-\delta) + \frac{1}{2} a(a-1)\sigma^{2})(T-t)}


#### Explanation

Latex code for Risk-Neutral Valuation and Power Contracts.

• : Payoff a power contract at time T
• : Price of the power contract
• : Risk-neutral equations