#newton-raphson

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Tarea 01: Derivadas y optimización

Solución de derivadas con R

Una de las ventajas de R es encontrar una solución analítica para derivadas comunes mediante lenguaje simbólico. La siguiente es una función que debe ser corrida antes:

mi.derivada <- function(funcion.matematica, var) { temp <- substitute(funcion.matematica) name <- deparse(substitute(var)) D(temp, name) }

Por ejemplo, la solución para la función:

\[ f(X) = aX^b \]

es la siguiente:

\[ f(X)' = abX^{b-1} \]

Para comprobar se debe ejecutar la siguiente instrucción

mi.derivada(a * X^b, X)

## a * (X^(b - 1) * b)

A usted se le pide obtener la derivada de las siguientes funciones: \[ f(x)=(x^2+5x)^3 \] \[ f(x)=exp(x^2) \] \[ f(x)=k*(1-x^q) \] \[ f(x)=log(1-x)^2 \] \[ f(x)=k*exp(-k*x) \] \[ f(x)=(1-exp(-k*x)) \] \[ f(x)=a*x*exp(-b*x) \] \[ f(x)=a*(1-exp(-(b*(x-c))))^3 \]

Newton-Raphson

Uno de los métodos más utilizados para buscar la solución numérica de derivativas es la formula de Newton-Raphson, que está representado por: \[ X_{i+1}=X_{i} - f(X)'/f(X)'' \] donde \( X_{i+1} \) es el valor de X actualizado, el subíndice \( i \) denota iteración, \( f(X)' \) es la primera derivada de la función y \( f(X)'' \) es la segunda derivada de la función.

Ejemplo

Se pide buscar el valor de X que minimiza la siguiente función \( f(x)=exp(-x)+x^4 \).

# Se escribe la función f <- function(x) { exp(-x) + x^4 } # Esta función tiene la siguiente forma: curve(f, from = -1, to = 3)

Sabemos que la primera derivada de la función es \( f(x)= -exp(-x)+4*x^3 \), y que la segunda derivada es \( f(x)''=exp(-x)+12*x^2 \).

Para buscar la solución, aplicamos Newton-Raphson en R:

# Primera y segunda derivada de la función: fprima <- function(x) -exp(-x) + 4 * x^3 fprima2 <- function(x) exp(-x) + 12 * x^2 # Para la solución con 6 iteraciones se necesita evaluar la funcion en # cada valor de x (se comienza con x=1), i.e., x <- c(1, rep(NA, 6)) feval <- rep(NA, 7) fprimaeval <- rep(NA, 7) fprimaeval2 <- rep(NA, 7) # Se realizan 6 iteraciones: for (i in 1:6) { feval[i] <- f(x[i]) fprimaeval[i] <- fprima(x[i]) fprimaeval2[i] <- fprima2(x[i]) x[i + 1] <- x[i] - fprimaeval[i]/fprimaeval2[i] } data.frame(x, feval, fprimaeval, fprimaeval2)

## x feval fprimaeval fprimaeval2 ## 1 1.0000 1.3679 3.632e+00 12.368 ## 2 0.7063 0.7424 9.161e-01 6.480 ## 3 0.5650 0.6703 1.529e-01 4.399 ## 4 0.5302 0.6675 7.678e-03 3.962 ## 5 0.5283 0.6675 2.276e-05 3.938 ## 6 0.5283 0.6675 2.019e-10 3.938 ## 7 0.5283 NA NA NA

Se pude observar que la solución es \( x=0.5283 \).

A usted se le pide encontrar el máximo rendimiento sostenido para una pesquería, sabiendo que la función de rendimiento es:

\[ f(x)=100*x-10*x^2 \]

Es una función cuadrática, i.e,

f <- function(x) { 100 * x - 10 * x^2 } curve(f, from = 0, to = 10)

A usted se le pide que busque la primera y segunda derivada, y encuentre el valor de x que maximiza la función con el método de Newton-Raphson.

Instrucciones

Trabaje en RStudio, script Markdown y entre los resultados (archivo *.Rmd) por email. No se olvide de incluir su nombre.

Curso Herramientas Cuantitativas en Ecología

Luis A. Cubillos

Biólogo Pesquero

blog: HCE

blog: code4fisheries

Question:

What is implied volatility? Write an implementation for calculating the implied volatility of an option.

Solution:

Implied Volatility:

Implied Volatility is obtained by plugging in quoted option price into Black-Scholes formula or your option pricing formula to solve for volatility. We can look at implied volatility as the volatility in the underlying as implied by the option prices - a market view or perceived future volatility of the underlying.

More theory on the relation between Implied Volatility and Option Prices

Implementation:

This implementation is part of the COptionPricingBlackScholes class, the rest of which can be found at Option Pricing Black Scoles

Still awake, solving our ChE 106 probset all night. 3/5 done! Yay! :) I was already feeling productive until I got stuck at Newton Raphson's Method for Simultaneous Nonlinear Equations which I thought would be the easiest one. :| I just don't quite get it right now. :((