Calculating Spot Delta
Consider a 1-year option in USD/CHF on an underlying of USD 1 million. The spot rate is 0.9668 and the forward delta is 50% (USD 500,000). 1-year deposit rates in USD are 4%, while 1-year deposit rate of CHF are 5%. Both are quoted on an actual/360 basis.
What is the present value of the USD 500,000?
Spot delta of USD
1. The forward value needs to be discounted through the standard equation:
\(e^{-rt}\) where
r=risk-free rate for domestic currenty
t=time to expiry of option
2. The continuously compounded equivalent of an annual 1-year 4% desposit rate is 3.975%.
The following formula is used to transform deposit rates to continuousley compounded equivalents:
\(\ln\left(1+\left(\frac{r}{m}\right)\right)^m\) where
r = periodic interest rate
m = number of periods (for instance, two for semi-annual)
where money market rates are quoted on an actual/360 basis, r needs to be adjusted by multiplying by 365/360. Therefore, in given example
\(\ln\left(1+\frac{0.04\left\{\frac{365}{360}\right\}}{1}\right)^{^1}=0.039754759=3.975\%\)
Therefore, \(e^{-rt}=0.96103\).
3. The forward delta of 50% is transformed to a spot delta of \(0.50\ \times0.96103=0.4805\%\) or USD 480,500.
Spot delta in CHF
The present value calculated as USD 480,500 can be checked by discounting the forward value of the CHF.
1. Using the rates above, the 1-year forward FX rate is 0.9762. Therefore, \(e^{-rt}=0.95175\)
The forward FX rate is calculated as follows: