Latex for Financial Engineering Mathematics Formula Monte-Carlo Simulations and Interest Rate Models

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Latex for Financial Engineering Mathematics Formula Monte-Carlo Simulations and Interest Rate Models

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

Latex for Financial Engineering Mathematics Formula Monte-Carlo Simulations and Interest Rate Models

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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 Monte-Carlo Simulations and Interest Rate Models. We will cover important topics including Monte-Carlo Simulations, Bonds and Interest Rates, Black-Derman-Toy (BDT) model and Cox-Ingersoll-Ross (CIR) model.

1. Simulations and Variance

Monte-Carlo Simulations

Bonds and Interest Rates

Black-Derman-Toy BDT

Cox-Ingersoll-Ross CIR

1. Simulations and Variance

Monte-Carlo Simulations
Financial,Economics
Equation

$S(T) = S(0) e^{(a - \delta - \frac{\sigma^2}{2})T + \sigma \sqrt{T} z} \\ S(T) = S(t) e^{(a - \delta - \frac{\sigma^2}{2})(T-t) + \sigma (Z(T) - Z(t))} \\ \text{Variance} \\ e^{-2rT} \times \frac{s^{2}}{n} \\ s^{2} = \frac{1}{n-1} \sum [(g(S_{i}) - \bar{g})]^{2}$

Latex Code
```
            S(T) = S(0) e^{(a - \delta - \frac{\sigma^2}{2})T + \sigma \sqrt{T} z} \\
            S(T) = S(t) e^{(a - \delta - \frac{\sigma^2}{2})(T-t) + \sigma (Z(T) - Z(t))} \\
            \text{Variance} \\
            e^{-2rT} \times \frac{s^{2}}{n} \\
            s^{2} = \frac{1}{n-1} \sum [(g(S_{i}) - \bar{g})]^{2}
        
```
Explanation

Latex code for the Monte-Carlo Simulations of stock prices. I will briefly introduce the notations in this formulation. Monte-Carlo simulation simulates stock prices, calculate the payoff the option for each of those simulated prices, find the average payoff, and then discount the average payoff. Firstly, we start with iid uniform numbers u_{1} to u_{n}, calculate standard normal variable z_{i} as $z_{i}=N^{-1}(u_{i})$ , convert to normal variable $r_{i} = \mu + \sigma z_{i}$ . The variance of the Monte-Carlo estimate is calculated as $e^{-2rT} \times \frac{s^{2}}{n}$ .
- $S(T)$ : The stock price at time T
- $S(t)$ : The stock price at time t, which is nearer to final stage stock price S(T)
- $g_{i}$ : The i-th simulated payoff
- $e^{-2rT} \times \frac{s^{2}}{n}$ : The variance of stock price
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Bonds and Interest Rates
Financial,Economics
Equation

$P(0, S) = \frac{1}{[1 + r(0, s)]^{s}} \text{or} e^{-r(0,s)s} \\ \text{Forward bond price} \\ F_{t,T}[P(T, T+s)] = \frac{P(t, T+s)}{P(t, T)} \\ P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)$

Latex Code
```
            P(0, S) = \frac{1}{[1 + r(0, s)]^{s}} \text{or} e^{-r(0,s)s} \\
            \text{Forward bond price} \\
            F_{t,T}[P(T, T+s)] = \frac{P(t, T+s)}{P(t, T)} \\
            P(t, T)[1 + r_{t}(T, T+s)]^{-s} = P(t, T+s)
        
```
Explanation

Latex code for the Bonds and Interest Rates. The price of an s-year zero is P(0, S). The forward bond price formula $F_{t,T}[P(T, T+s)]$ is calculated as $F_{t,T}[P(T, T+s)] = \frac{P(t, T+s)}{P(t, T)}$ . And the non-continuous annualized rate is $P(t, T)[1 + r_{t}(T, T+s)]^{-s}=P(t,T+s)$ .
- $P(0, S)$ : Price of an s-year zero.
- $F_{t,T}[P(T, T+s)]$ : Forward Bond Price
- $P(t, T)[1 + r_{t}(T, T+s)]^{-s}=P(t,T+s)$ : Non-continuous annualized rate
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- Bond and Interest Rate
- Understanding Interest Rates, Inflation, and Bonds
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Black-Derman-Toy BDT
Financial,Economics
Equation

$\text{First Node: 1-year bond price} \\ P_{0} = \frac{1}{1 + R_{0}} \\ \text{Second Node} \\ P_{1} = \frac{1}{1+R_{0}} [\frac{1}{2} P(1,2,r_{u}) + \frac{1}{2} P(1,2,r_{d})] \\ = \frac{1}{1+R_{0}} [\frac{1}{2(1 + R_{1}e^{2\sigma_{1}})} + \frac{1}{2(1 + R_{1})}] \\ R_{0} = \frac{1}{2} \ln (\frac{R_{1} e^{2\sigma_{1}} }{R_{1}})$

Latex Code
```
            \text{First Node: 1-year bond price} \\
            P_{0} = \frac{1}{1 + R_{0}} \\
            \text{Second Node} \\
            P_{1} = \frac{1}{1+R_{0}} [\frac{1}{2} P(1,2,r_{u}) + \frac{1}{2} P(1,2,r_{d})] \\
            = \frac{1}{1+R_{0}} [\frac{1}{2(1 + R_{1}e^{2\sigma_{1}})} + \frac{1}{2(1 + R_{1})}] \\
            R_{0} = \frac{1}{2} \ln (\frac{R_{1} e^{2\sigma_{1}} }{R_{1}})
        
```
Explanation

Latex code for the Black-Derman-Toy BDT model. The BDT model is a commonly used interest rate model. The basic idea of the BDT model is to compute a binomial tree of short-term interest rates, with a flexible enough structure to match the data. Black, Derman, and Toy describe their tree as driven by the short-term rate, which they assume is lognormally distributed. Constructing the black-Derman-Toy tree, the first node is given by the prevailing 1-year rate R0. The yield volatility for period-3 is $P(r_{u}, h, 3h)$ .
- $R_{0}$ : 1-year interest rate.
- $P_{0}$ : 1-year bond price
- $P(1, 2, r_{u}), P(1, 2, r_{d})$ : year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.
- $P_{1}$ : Observed year-1 price of a 1-year bond
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Cox-Ingersoll-Ross CIR
Financial,Economics
Equation

$\mathrm{d} r(t) = a[b - r(t)] \mathrm{d} t + \sigma \sqrt{r(t)} \mathrm{d} Z(t) \\ P(r, t, T) = A(T-t)e^{-rB(T-t)} \\ \gamma = \sqrt{(a-\bar{\phi})^{2} + 2 \sigma^{2}} \\ q(r, t, T) = \sigma \sqrt{r} B(T-t) \\ \text{yield to maturity} \\ \frac{2ab}{ a - \bar{\phi} + \gamma}$

Latex Code
```
            \mathrm{d} r(t) = a[b - r(t)]  \mathrm{d} t + \sigma \sqrt{r(t)} \mathrm{d} Z(t) \\
            P(r, t, T) = A(T-t)e^{-rB(T-t)} \\
            \gamma = \sqrt{(a-\bar{\phi})^{2} + 2 \sigma^{2}} \\
            q(r, t, T) = \sigma \sqrt{r} B(T-t) \\
            \text{yield to maturity} \\
            \frac{2ab}{ a - \bar{\phi} + \gamma}
        
```
Explanation

Latex code for the Cox-Ingersoll-Ross model.
- $R_{0}$ : 1-year interest rate.
- $P_{0}$ : 1-year bond price
- $P(1, 2, r_{u}), P(1, 2, r_{d})$ : year-1 price of a 1-year bond, depending on the movement of the interest rate moving up and down.
- $P_{1}$ : Observed year-1 price of a 1-year bond
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