Risk-neutralize simulations

Posted on September 3, 2023 by T. Moudiki in R bloggers | 0 Comments

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A risk-neutral probability (in Quantitative Finance) is a probability measure in which, on average, all the risky assets return the risk-free rate. Risk-neutral probabilities are widely used for the pricing of derivative products. There are other advanced mathematical definitions of a risk-neutral probability that won’t be discussed here.

In R package ahead, it is possible to obtain simulations of risky assets returns both in historical and risk-neutral probability.

Table of contents

  • 0 – Install ahead

  • 1 – Get and transform data

  • 2 – Risk-neutralize simulations

  • 3 – Visualization

0 – Install ahead

ahead is released under the BSD Clear license. Here’s how to install the R version of the package:

  • 1st method: from R-universe

    In R console:

    options(repos = c(
    techtonique = 'https://techtonique.r-universe.dev',
    CRAN = 'https://cloud.r-project.org'))

install.packages("ahead")

  • 2nd method: from Github

    In R console:

    devtools::install_github("Techtonique/ahead")

    • Or

    remotes::install_github("Techtonique/ahead")

    Using ahead:

    library(ahead)

    1 – Get and transform data

    data(EuStockMarkets)

EuStocks <- ts(EuStockMarkets[1:100, ], 
               start = start(EuStockMarkets),
               frequency = frequency(EuStockMarkets))

EuStocksLogReturns <- ahead::getreturns(EuStocks, type = "log")

print(head(EuStocksLogReturns))

    2 - Risk-neutralize simulations

    2 - 1 Yield to maturities (fake risk-free rates)

    ym <- c(0.03013425, 0.03026776, 0.03040053, 0.03053258, 0.03066390, 0.03079450, 0.03092437)

freq <- frequency(EuStocksLogReturns)
(start_preds <- tsp(EuStocksLogReturns)[2] + 1 / freq)
(ym <- stats::ts(ym,
                 start = start_preds,
                 frequency = frequency(EuStocksLogReturns)))

    2 - 2 Risk-neutralized simulations

    obj <- ahead::ridge2f(EuStocksLogReturns, h = 7L,
                      type_pi = 'bootstrap',
                      B = 10L, ym = ym)

    Checks

    rowMeans(obj$neutralized_sims$CAC)

print(ym)

rowMeans(obj$neutralized_sims$DAX)

print(ym)

    3 - Visualization

    par(mfrow = c(2, 2))

matplot(EuStocksLogReturns, type = 'l', 
     main = "Historical log-Returns", xlab = "time")

plot(ym, main = "fake spot curve", 
     xlab = "time to maturity",
     ylab = "yield", 
     ylim = c(0.02, 0.04))

matplot(obj$neutralized_sims$DAX, type = 'l', 
     main = "simulations of \n predicted DAX log-returns ('risk-neutral')", 
     ylim = c(0.02, 0.04), 
     ylab = "log-returns")

ci <- apply(obj$neutralized_sims$DAX, 1, function(x) t.test(x)$conf.int)
plot(rowMeans(obj$neutralized_sims$DAX), type = 'l', main = "average predicted \n DAX log-returns ('risk-neutral')", col = "blue", 
     ylim = c(0.02, 0.04), 
     ylab = "log-returns")
lines(ci[1, ], col = "red")
lines(ci[2, ], col = "red")

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