Laboratorio 5

library(pdfetch)
library(tidyverse)

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library(yuima)

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## This is YUIMA Project package v.1.15.22
## Why don't you try yuimaGUI package?
## Visit: http://www.yuima-project.com
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## Attaching package: 'yuima'
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#obtenemos los datos desde el sitio de yahoo
NVCRdata <- pdfetch_YAHOO("VOW3.DE",from = c("2019-01-01"),to = c("2020-01-01"), interval = '1d')
#Obtenemos la columna que es de nuestro interés
Novocure <- NVCRdata[,4]
length(Novocure)

## [1] 251

#Convertimos en una serie temporal
tsNovoCure <- ts(Novocure, start = c(2019,1),frequency=365)
#Graficamos
plot(tsNovoCure)

#Ver si estacionaria
#.-....
#Se calculan las diferencias de la serie de datos con logaritmo
l_NovoCure<-diff(log(tsNovoCure))
plot(l_NovoCure)

#Calculo parámetros iniciales manera 1
Delta <- 1/365
alpha <- mean(l_NovoCure)/Delta
sigma <- sqrt(var(l_NovoCure)/Delta)
mu <- alpha +0.5*sigma^2
x0<-tsNovoCure[1]

#Calculo parámetros iniciales manera 2
x <- tsNovoCure
gBm <- setModel(drift="mu*x", diffusion="sigma*x", xinit=x0)

## Warning in yuima.warn("Solution variable (lhs) not specified. Trying to use state variables."): 
## YUIMA: Solution variable (lhs) not specified. Trying to use state variables.

mod <- setYuima(model=gBm, data=setData(tsNovoCure, delta=Delta))
set.seed(123)
fit <- qmle(mod, start=list(mu=0, sigma=1),
            lower=list(mu=0.1, sigma=0.1),
            upper=list(mu=100, sigma=10))
summary(fit)

## Quasi-Maximum likelihood estimation
## 
## Call:
## qmle(yuima = mod, start = list(mu = 0, sigma = 1), lower = list(mu = 0.1, 
##     sigma = 0.1), upper = list(mu = 100, sigma = 10))
## 
## Coefficients:
##        Estimate Std. Error
## sigma 0.2758558 0.01238444
## mu    0.4135387 0.33331784
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## -2 log L: 1105.859

#comparación
coef(fit)

##     sigma        mu 
## 0.2758558 0.4135387

sigma

##               VOW3.DE.close
## VOW3.DE.close     0.2749526

mu

##               VOW3.DE.close
## VOW3.DE.close     0.4134311

gbm_vec <- function(nsim = 10000, t = 25, mu = 0, sigma = 0.1, S0 = 100, dt = 1./365) {
  # matrix of random draws - one for each day for each simulation
  epsilon <- matrix(rnorm(t*nsim), ncol = nsim, nrow = t)  
  # get GBM and convert to price paths
  gbm <- exp((mu - sigma * sigma / 2) * dt + sigma * epsilon * sqrt(dt))
  gbm <- apply(rbind(rep(S0, nsim), gbm), 2, cumprod)
  return(gbm)
}

gBm

## 
## Diffusion process
## Number of equations: 1
## Number of Wiener noises: 1
## Parametric model with 2 parameters

valores_simulados <- simulate(gBm, true.parameter = list(mu=mu, sigma=sigma))

## Warning in yuima.warn("'delta' (re)defined."): 
## YUIMA: 'delta' (re)defined.

plot(valores_simulados)

#PROBAR CON FORMA 1 Y FORMA 2
nsim <- 1000
t <- 256
mu <- 0.4135387 
sigma <- 0.2758558
S0 <- x0
dt = 1/365
gbm <- gbm_vec(nsim, t, mu, sigma, S0, dt)

gbm_df <- as.data.frame(gbm) %>%
  mutate(ix = 1:nrow(gbm)) %>%
  pivot_longer(-ix, names_to = 'sim', values_to = 'price')
gbm_df %>%
  ggplot(aes(x=ix, y=price, color=sim)) +
  geom_line() +
  theme(legend.position = 'none')

data.frame(price = gbm[253, ]) %>%
  ggplot(aes(x = price)) +
  geom_histogram(aes(y = ..density..), binwidth = 0.1) +
  geom_density() + 
  ggtitle('terminal price distribution')

## Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
## ℹ Please use `after_stat(density)` instead.

D <- gbm[256, ] %>%
  density()

D$x[which.max(D$y)]  #resultado del modelo

## [1] 163.3094