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Why Black-Scholes FX Simulations Produce a Skewed Rate Distribution

Article Quant Q&A · Author: ForumWhiner
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Summary

The question describes simulating an FX rate with a Black-Scholes-style process that includes the domestic and foreign short rates, volatility, and correlated random shocks. The simulation’s terminal-rate percentiles appear positively skewed, prompting a question about the source of the shape.

The answer points to the model’s lognormal terminal distribution: a lognormal variable is generally asymmetric, with a longer right tail, so positive skew can arise even when the underlying modeled shocks are normal. This is a distributional property of the multiplicative rate process rather than, by itself, evidence of a coding error. The brief reply does not analyze the implementation, the effects of rate correlations, or whether the simulated parameters and volatility surface are appropriate. It also does not establish whether the observed skew magnitude is correct; diagnosing that would require checking the model setup and comparing the empirical simulation output with the implied lognormal distribution.

Key ideas

  • A multiplicative Black-Scholes-style process leads to a lognormally distributed terminal FX rate.
  • Lognormal distributions are asymmetric and can show positive skew.
  • Positive skew in simulated FX levels is not by itself evidence of an implementation error.
  • The brief explanation does not validate the simulation parameters or the observed skew magnitude.

Tags

ForexVolatilityStatisticsDerivatives pricing
Full text
# Skew in Black Scholes model


# Skew in Black Scholes model












We are modeling Foreign exchange rates using Black Scholes model given below:

$$F_{t}=F_{t−1} + (r_d−r_f)F_{t−1}dt + \sigma F_{t−1}dW_t$$

Where:

$F_t$ and $F_{t−1}$ are FX rates at time $t$ and $t−1$

$r_d$ domestic short rate

$r_f$ foreign short rate

$dt$ is the change in time period

$\sigma$ is the volatility obtained from ATM volatility surface

$dW_t$ is correlated random number (correlation is between $r_d$, $r_f$, and FX rate)

I ran this model for $1000$ simulations and my percentile graphs show a skew on the positive side of distribution. Can someone please help me understand the skewness.

Thanks in advance.

## Answer by Mark Joshi (score 5)

https://quant.stackexchange.com/a/15844

Well the terminal FX rate is lognormally distributed and lognormals are skewed. So this is not surprising.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.