Forward Prices Under Continuous Rates and Dividend Yield

Summary

The document explains the interest rate convention used in an implied volatility approximation and how to estimate an asset’s forward price. It treats the constant interest rate as the annualized risk-free rate, assuming the rate used in the approximation matches the risk-free rate used for forward pricing. The answer specifies continuous compounding rather than discrete compounding.

For a non-dividend-paying underlying, the forward price is the spot price multiplied by the exponential of the risk-free rate times the time to maturity. For a dividend-paying asset, the expression adjusts for the annualized continuous dividend yield, with the net carry represented by dividend yield less the risk-free rate in the source’s stated convention. The response is a concise clarification, not a full derivation; it assumes the relevant rates and yield are known and does not discuss discrete dividends, financing frictions, or other carry costs.

Key ideas

• The constant rate in the approximation is treated as the annualized risk-free rate.
• The forward price uses continuous compounding under the stated convention.
• For a non-dividend-paying asset, the forward grows from spot at the risk-free rate.
• A continuous dividend yield adjusts the forward price to account for dividends.
• The formulas assume the interest rate and dividend yield inputs are appropriate for the underlying and maturity.

Tags

# Question about Forward Price + Constant Interest Rate Approximations


# Question about Forward Price + Constant Interest Rate Approximations












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 What is the "constant interest rate" ? Is it the annualized risk-free rate? If not, how can I approximate/solve for it? 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 by Kermittfrog (score 1)

https://quant.stackexchange.com/a/53863

- yes.

- 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.