FX Conventions

We list and derive the various FX conventions.

FX Volatility Smile -Malz quadratic function

One of the simplest approach to construct volatility smile function is to fit a quadratic function in delta space - i.e., volatility is a quadratic function of delta, or the distance of the delta from ATM delta to be more precise.

$$ \sigma_{\Delta}=a + b \left( \Delta - \Delta_{ATM} \right) + c \left( \Delta - \Delta_{ATM} \right)^2 $$

Different delta conventions will produce different results, but by making some simplifying assumptions, Malz derived an explicit formula for volatility as function of delta. In our settings, these assumptions translate into:

$$ \Delta_{ATM} = 0.5 $$ $$ \sigma_{75Delta-Call}=\sigma_{25Delta-Put} $$

Substituting \( \Delta_{ATM} = 0.5\) into the quadratic equation,

$$ \sigma_{\Delta}=a + b \left( \Delta - 0.5 \right) + c \left( \Delta - 0.5 \right)^2 $$

We have three unknowns, and substituting the three levels of delta into the above equation gives us three equations:

$$ \sigma_{ATM}=a + b \left( \Delta_{ATM} - 0.5 \right) + c \left( \Delta_{ATM} - 0.5 \right)^2 $$ $$ a=\sigma_{ATM} $$ $$ \sigma_{25C}=a + b \left( 0.25 - 0.5 \right) + c \left( 0.25- 0.5 \right)^2 $$ $$ \sigma_{25C}=a - \frac{1}{4}b + \frac{1}{16} c $$ $$ \sigma_{25P}=\sigma_{75C}=a + b \left( 0.75 - 0.5 \right) + c \left( 0.75- 0.5 \right)^2 $$ $$ \sigma_{25P}=a + \frac{1}{4}b + \frac{1}{16} c $$

We can solve the three equations for the three unknowns. We already have \( a=\sigma_{ATM} \), so we only need to solve for b and c:

$$ \sigma_{25C} + \sigma_{25P} = 2a - \frac{1}{4}b + \frac{1}{16} c + a + \frac{1}{4}b + \frac{1}{16} c $$ $$ \sigma_{25C} + \sigma_{25P} = 2a + \frac{1}{8} c $$ $$ \frac{1}{8} c=\sigma_{25C} + \sigma_{25P} - 2a $$ $$ \frac{1}{8} c=\sigma_{25C} + \sigma_{25P} - 2 \sigma_{ATM} $$ $$ \frac{1}{8} c=2 \left( \frac{\sigma_{25C} + \sigma_{25P}}{2}- \sigma_{ATM} \right) $$ $$ c= 16 \left( \frac{\sigma_{25C} + \sigma_{25P}}{2}- \sigma_{ATM} \right) $$ $$ c= 16 SS_{25 Delta} $$ $$ \sigma_{25C} - \sigma_{25P} = a - \frac{1}{4}b + \frac{1}{16} c - a - \frac{1}{4}b - \frac{1}{16} c $$ $$ \sigma_{25C} - \sigma_{25P} = - \frac{1}{2} b $$ $$ b= -2 \left( \sigma_{25C} - \sigma_{25P} \right) $$ $$ b= -2 RR_{25 Delta} $$

Thus.

$$ \sigma_{\Delta}=a + b \left( \Delta - \Delta_{ATM} \right) + c \left( \Delta - \Delta_{ATM} \right)^2 $$ $$ \sigma_{\Delta}=\sigma_{ATM} -2 RR_{25 Delta} \left( \Delta - \Delta_{ATM} \right) + 16 SS_{25 Delta} \left( \Delta - \Delta_{ATM} \right)^2 $$