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

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

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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

Geometric Brownian Motion SDEs

Stock Prices as Geometric Brownian Motion

Ito Lemma

Sharpe Ratio

Risk-Neutral Valuation and Power Contracts

1. Continuous-Time Finance

Standard Brownian Motion
Financial,Economics
Equation

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

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
- $Z(t)$ : Value of Z at time stamp t
- $Z(t+s) - Z(t)$ : Stationary increments of Standard Brownian Motion
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Geometric Brownian Motion

Financial,Economics

Equation

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

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.

$Y(t)$ : Observed value Y(t) at time stamp t
$U$ : Any normal random variable
$\mu$ : Drift coefficient
$\sigma$ : Volatility

Ito Lemma

Financial,Economics

Equation

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

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.

$X$ : Diffusion
$\mathrm{d}X(t)$ : Stochastic differential equation for X(t)

Stock Prices as Geometric Brownian Motion

Financial,Economics

Equation

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

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.

$S(t)$ : Observed stock price S(t) at time stamp t
$U$ : Any normal random variable
$\mu$ : Drift coefficient
$\sigma$ : Volatility

Risk-Neutral Valuation and Power Contracts

Financial,Economics

Equation

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

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.

$S^{a}(T)$ : Payoff a power contract at time T
$F^{p}_{t, T}(S^{a})$ : Price of the power contract
$V(S(t), t) = e^{-r(T-t)} E^{*}[V(S(T), T) | S(T)]$ : Risk-neutral equations

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1. Continuous-Time Finance

Standard Brownian Motion

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Geometric Brownian Motion

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Geometric Brownian Motion SDEs

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Ito Lemma

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Stock Prices as Geometric Brownian Motion

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Sharpe Ratio

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Risk-Neutral Valuation and Power Contracts

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