Risk-neutralize simulations

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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'))
  • 2nd method: from Github

    In R console:

  • Or


    Using ahead:


    1 – Get and transform data

    EuStocks <- ts(EuStockMarkets[1:100, ], 
                   start = start(EuStockMarkets),
                   frequency = frequency(EuStockMarkets))
    EuStocksLogReturns <- ahead::getreturns(EuStocks, type = "log")

    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)



    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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