Choosing Log-Moneyness for Implied Volatility Surfaces

# Creating Implied Volatility surface using log moneyness


# Creating Implied Volatility surface using log moneyness












When creating the implied volatility surface using $\left(T,log\left(\frac{K}{S_0}\right)\right)$ as $(x,y)$ axis, do the inputs for the implied vol calculation need to be logged too? In other words, when using `py_vollib` to calculate the IV, does the K input need to be changed? Or is it only scaled afterwards for plotting. In python that would be:

```
from py_vollib_vectorized import vectorized_implied_volatility
import numpy as np
import matplotlib.pyplot as plt

"""
Version 1: 
"""
T = np.linspace(0.5,2,10)
K = np.linspace(60,120,10)
r=0.05
q=0.02
S_0 = 100
call_prices = np.empty(100)
flag=[]
i=0

while i <100:
    call_prices[i] = np.random.normal(10,2)
    if call_prices[i]<3:
        continue
    if call_prices[i] <7:
        flag.append('c')
    else:
        flag.append('p')
    i+=1
Ti,Ki = np.meshgrid(T,K)
T=Ti.flatten()
K=Ki.flatten()

imp_vol = vectorized_implied_volatility(call_prices,S_0,K,T,r,flag,q,model='black_scholes_merton',return_as='numpy')
imp_vol=imp_vol.reshape(10,10)

# moneyness scale
Ki = np.log(Ki / S_0)

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(Ti, Ki, imp_vol, cmap='viridis')

"""
Version 2:
"""
r=0.05
q=0.02
S_0 = 100
T = np.linspace(0.5,2,10)
# Changing K
eps=0.0001
K = np.log(np.linspace(60,120,10)/(S_0+eps))

call_prices = np.empty(100)
flag=[]
i=0

while i <100:
    call_prices[i] = np.random.normal(10,2)
    if call_prices[i]<3:
        continue
    if call_prices[i] <7:
        flag.append('c')
    else:
        flag.append('p')
    i+=1
Ti,Ki = np.meshgrid(T,K)
T=Ti.flatten()
K=Ki.flatten()

imp_vol = vectorized_implied_volatility(call_prices,np.log(S_0),K,T,r,flag,q,model='black_scholes_merton',return_as='numpy')
imp_vol=imp_vol.reshape(10,10)

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(Ti, Ki, imp_vol, cmap='viridis')
```
```

## Answer by jherek (score 1)

https://quant.stackexchange.com/a/77898

Both ways are possible, and are not the same thing. There is a difference between using log-moneyness (or some other convention) for the input quotes, and using log-moneyness (or some other convention) for interpolation. You argue there may be a third possibility for plotting/presentation.

For equities, input quotes are typically strike based. But this is not always the case depending on where exactly this input comes from. BBG or intermediate systems may give you vols by log-moneyness as input directly.

As the question being asked only mentions creating a vol surface in log-moneyness, the most logical would be to use log-moneyness for both. This means you interpolate your quotes on log-moneyness. So yes, you would use (log(K/S), vol_K) as input to your interpolator.

This is different from interpolating on (K, vol_K).

There is also the question of forward log-moneyness vs. spot log-moneyness which will play a role across option maturities.

The plotting question is not so relevant, as it is really just a visual representation and should not distort the true interpolation being used. You have the choice to use log axis or not there, but be careful plotting lines with a sparse number of points. It may distort the reality.

Finally, the question applies only if your vol representation is a simple interpolation. In some cases, there is a underlying model, which will have its own convention (for example SVI, SABR, Heston). In the latter case, there is no choice on how to interpolate.