Question about Forward Price + Constant Interest Rate Approximations

Question

Sorry if this is an obvious question, but I'm reading the following paper An Explicit Implied Volatility Formula, Dan Stefanica, Rados Radoicic, International Journal of Theoretical and Applied Finance, Vol. 20, no. 7, 2017

I have some questions about the formula. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2908494

The formula has the following:

• $C_m$ = market price of call option
• $K$ = option strike
• $T$ = option maturity
• $F$ = forward price at T of underlying asset
• $r$ = constant interest rate

Output:

• $\sigma_{\text{imp,approx}}$ = implied volatility approximation

  1. What is the "constant interest rate" ? Is it the annualized risk-free rate? If not, how can I approximate/solve for it?
  2. Considering that I am solving for the implied volatility of stocks, can I approximate the forward price of an underlying asset with: $$ F_t = S_t×(1+r_f)^{T} $$

Answer 1

1. yes.
2. Assuming that $r$ and $r_f$ are the same, you should use continuous compounding:

$$ F(t,T)=S_te^{r_f(T-t)} $$

and if the underlying is a dividend paying instrument, then $$ F(t,T)=S_te^{(y-r_f)(T-t)} $$

with $y$ the annualized continuous dividend yield.