Deriving a Stock Forward Price from No-Arbitrage

# Calculate theoretical forward price of a stock


# Calculate theoretical forward price of a stock












The current price of a stock is USD400 per share and it pays no dividends. Assuming a constant interest rate of $8% $ compounded quarterly, what is the stock's theoretical forward price for delivery in $9$ months ?

I am taking the Financial Engineering and risk management course on Coursera. The above question was in the quiz and I got a wrong answer on it.

Shouldn't the answer be: $$400\times\left(1+\frac{0.08}{4}\right)^3 = 424.48\, ?$$

## Answer by wsw (score 4)

https://quant.stackexchange.com/a/22375

Let's use a no-arbitrage argument. Assume that the (continuously compounding) dividend yield is $q$ while the interest rate is $r$.

For portfolio 1, we go long 1 forward contract with maturity $T$ and delivery price $K$. The payoff at time $T$ is $S_T - K$.

For portfolio 2, we go long $e^{-qT}$ unit of a stock (while reinvest all dividends) and short $K e^{-rT}$ unit of a bond. The payoff at time $T$ is also $S_T - K$.

At time $t = 0$, the present value (PV) of portfolio 1 is 0, because we just entered the trade. The PV of portfolio 2 at time $t = 0$ is $S_0 e^{-qT} - K e^{-rT}$. Assuming that there is no arbitrage, we conclude that the PV at time $t = 0$ of portfolios 1 and 2 must be the same: $ S_0 e^{-qT} - K e^{-rT} = 0$. Hence $\boxed{K = S_0 e^{(r-q)T}}$. Your answer of $400 (1+0.08/4)^3=424.48$ is correct.

## Answer by Piyush Shandilya (score 0)

https://quant.stackexchange.com/a/22214

It should be just $400*(1+0.02)^3$ Where $0.02$ is quarterly compounded rate for a quarter, and $n$ is 3 quarters, hence the exponent 3

## Answer by LorenzQF (score 0)

https://quant.stackexchange.com/a/24302

It is correct. J.Hull's book explains it clearly in Chapter 3, paragraph 3.5 "Forward price for an investment asset". It is also shows the arbitrage strategy if the price does not match.

## Answer by PK K (score 0)

https://quant.stackexchange.com/a/51608

Based on No Arbitrage equation for Forward Asset combination: $S_0 e^{−qT} − K e^{−rT} = 0$

i.e. $K = S_0 e^{(r-q)T} \\ = 400 * e^{(0.02 - 0) * 3} \\ = 400 * 1.061837 \\ = 424.7346$

## Answer by sparkle (score -1)

https://quant.stackexchange.com/a/22191

Assuming 8% is the quartely interest rate

I think it's: $400*(1+0.08)^3 = 503.8$

$0.08$ is the quarter interest rate and you compound for 3 quarters

Or

$ r_{year} = (1+0.08)^4 - 1 = 0.36 $

$400*(1+0.36)^(9/12) = 503.8$

Because there are 9 months in a 12-month year.