Q1. How is the implied volatility of GBPUSD and USDGBP related to each other mathematically? Please explain this intuitively as well.
Q2. How is the historical volatility of GBPUSD and USDGBP related to each other mathematically?
My take ( Please correct if i am wrong):
"Historical Volatility or Realized Volatility is calculated as the standard deviation of log returns."
Let the price of GBPUSD at time t1, t2, t3 ....tn be S1, S2, S3...Sn. So, its return would be Log(S2/S1), Log(S3/S2)....Log(Sn/Sn-1). Suppose the standard deviation of these be σ1.
Now, the price of USDGBP at time t1, t2, t3....tn would be 1/S1, 1/S2, 1/S3.....1/Sn. So, its log return would be log(S1/S2), Log(S2/S3)....Log(Sn-1/Sn). These return have same magnitude as that of GBPUSD but the sign is opposite. So, its standard deviation would remain same as σ1.
Based on these calculations is it fair to assume that historical/released volatility would be same for GBPUSD and USDGBP?