# 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.

• Please use Tex for the formulas. Can you provide the chart or the code that produced the chart? – Ric Dec 9 '14 at 10:39
• Sure Richard. Apologies since I am first time user and didn't know how to write formulas here. Thanks for your suggestion. – ForumWhiner Dec 19 '14 at 4:53

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

• Can you please tell me what do you mean by "terminal FX rate is lognormally distributed"? I am not using log OU process so skew is rather surprising to me. – ForumWhiner Dec 19 '14 at 4:52
• the process is a discretization of the log-normal SDE i.e. geometric Brownian motion. So if you run several steps per path you will get a distribution that is approximately log-normal. – Mark Joshi Dec 19 '14 at 8:08