Latex for Financial Engineering Mathematics Formula MonteCarlo Simulations and Interest Rate Models
rockingdingo 20230521 #financial engineering #mathematics #financeLatex for Financial Engineering Mathematics Formula MonteCarlo 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 MonteCarlo Simulations and Interest Rate Models. We will cover important topics including MonteCarlo Simulations, Bonds and Interest Rates, BlackDermanToy (BDT) model and CoxIngersollRoss (CIR) model.
 1. Simulations and Variance
 MonteCarlo Simulations
 Bonds and Interest Rates
 BlackDermanToy BDT
 CoxIngersollRoss CIR
1. Simulations and Variance

MonteCarlo Simulations
Financial,EconomicsEquation
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})(Tt) + \sigma (Z(T)  Z(t))} \\ \text{Variance} \\ e^{2rT} \times \frac{s^{2}}{n} \\ s^{2} = \frac{1}{n1} \sum [(g(S_{i})  \bar{g})]^{2}
Explanation
Latex code for the MonteCarlo Simulations of stock prices. I will briefly introduce the notations in this formulation. MonteCarlo 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 , convert to normal variable . The variance of the MonteCarlo estimate is calculated as .
 : The stock price at time T
 : The stock price at time t, which is nearer to final stage stock price S(T)
 : The ith simulated payoff
 : The variance of stock price
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Bonds and Interest Rates
Financial,EconomicsEquation
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 syear zero is P(0, S). The forward bond price formula is calculated as . And the noncontinuous annualized rate is .
 : Price of an syear zero.
 : Forward Bond Price
 : Noncontinuous annualized rate
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BlackDermanToy BDT
Financial,EconomicsEquation
Latex Code
\text{First Node: 1year 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 BlackDermanToy 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 shortterm interest rates, with a flexible enough structure to match the data. Black, Derman, and Toy describe their tree as driven by the shortterm rate, which they assume is lognormally distributed. Constructing the blackDermanToy tree, the first node is given by the prevailing 1year rate R0. The yield volatility for period3 is .
 : 1year interest rate.
 : 1year bond price
 : year1 price of a 1year bond, depending on the movement of the interest rate moving up and down.
 : Observed year1 price of a 1year bond
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CoxIngersollRoss CIR
Financial,EconomicsEquation
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(Tt)e^{rB(Tt)} \\ \gamma = \sqrt{(a\bar{\phi})^{2} + 2 \sigma^{2}} \\ q(r, t, T) = \sigma \sqrt{r} B(Tt) \\ \text{yield to maturity} \\ \frac{2ab}{ a  \bar{\phi} + \gamma}
Explanation
Latex code for the CoxIngersollRoss model.
 : 1year interest rate.
 : 1year bond price
 : year1 price of a 1year bond, depending on the movement of the interest rate moving up and down.
 : Observed year1 price of a 1year bond
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