# Interest rates impact on forward prices

In stock option markets, rising interest rates will increase the froward price, causing call values to rise and put values to fall.

But my understanding is that rising interest rate will cause stock prices to go down generally. So shouldn’t forward price be inversely impacted?

• In the first statement we are assuming that "other things stay the same", in particular stock prices. It is a static analysis of a particular mathematical formula varying one parameter only. In the second statement you are making a (very reasonable though not universally accepted) statement about how i.r. and stock prices are related in the real economy. The two stmts are not necessarily in contradiction. (In fact IMHO both are right). Jul 7 '19 at 19:43
• That’s fair. So all else stay the same (I.e. the underlying stock price), an increase in interest rate will increase the funding costs of writing a forward ? Hence the increase in the price of the synthetic ? Is that the right way to look at this
– AMY
Jul 7 '19 at 19:43
• Yes, that is right IMO. Jul 7 '19 at 19:45

To answer this question, you need to know how forward prices are derived : non arbitrage argument. Thanks to it, we will show that we necessarily have $$F_T = S_0e^{rT}$$ where $$F_T$$ is the price of the forward for delivery at time $$T$$, $$r$$ the risk-free continuous interest rate and $$S_0$$ the current price of the asset.

If $$F_T > S_0e^{rT}$$:

I enter in a forward to sell the asset. I borrow the equivalent of $$S_0$$ at time 0 (and so I owe $$S_0e^{rT}$$ at time $$T$$), I buy the asset and wait for the delivery. At maturity I get a riskless profit of $$F_T - S_0e^{rT}$$. As everyone will do this trade, the price of the forward should decrease and the price of the asset increase. Thus, the arbitrage opportunity will disappear.

If $$F_T < S_0e^{rT}$$:

In this situation, I can borrow the stock, sell it (for $$S_0$$) dollars) and gain the risk free rate on the cash ( I will get $$S_0e^{rT}$$ at maturity). In the same time, at time $$0$$ I enter into a forward to buy the stock for $$F_T$$ dollars at time $$T$$. So, at $$T$$, I use the cash to pay the forward $$F_T$$ as agreed and give back the stock to the person from who I borrowed it. I get the riskless profit $$S_0e^{rT} - F_T$$. With the same argument than above, this situation should immediately disappear if it occurs.

Conclusion : You have this formula $$F_T = S_0e^{rT}$$ that tells you rising interest rates imply rising forward prices. As it comes from a non arbitrage argument, it is "meaningless" to find an economic reason to it except that : if this relationship does not hold, some persons could become infinitely rich without risks.

Yes, you have two competing and inverse effects of interest rates on stock forward prices. One is the "fair" futures price as Jean explains in another answer (simply the future value of the current stock price). The other is the effect on the spot price. Yes, in general, rising interest rates tend to lower the price of most equities for various reasons. So this would lower the forward price as well.

So the actual effect on forward prices is a combination of these two effects, one of which can be difficult (if not impossible) to quantify.