# Equity Forward Price calculation

In the book of John Hull, the price of an equity forward on a dividend paying stock is formulated as: $$F_0 = (S_0 - I)e^{rT}$$ where $$r$$ is the risk free rate and $$I$$ is present value of the stream of dividend payments over the life of the forward.

In practice, what is the risk-free rate used for forward contracts? Would the correct rate to use be the repo rate or OIS rate? Furthermore, are dividends discounted using the same rate?

## 5 Answers

There is no real "risk-free" rate.

Now to answer your question, $$r$$ is time-dependent and should correspond to the repo rate corresponding to the maturity of your forward. In $$I$$, dividends should be "discounted" using the same time-dependent repo rate. Contrary to what others have suggested here, the use of an OIS rate or some other rate is not appropriate, otherwise arbitrage is possible.

Each dividend may not be discounted at the same rate, but the discounting will correspond to an interpolation of the equity repo rates.

As @ilovevolatility mentioned, the logic behind what I describe above is proved in Piterbarg paper Funding Beyond Discounting published in Risk Magazine, really a must read paper on the subject.

Finally, the price of an equity forward is an ambiguous terminology. What Hull refers to is the forward price. In reality, the NPV for an equity forward will include a strike price and an additional discounting, typically using OIS rate $$r_c$$:

$$NPV = e^{-r_c (T-t)}\left(e^{r (T-t)}(S(t)-I) - K \right)$$

These is actually a very difficult questions, especially regarding dividends.

1. What is the risk free rate? Theoretically OIS (with in Europe EONIA as the overnight rate) is the best estimate of risk free. However, other rates are used in practice, in the past is was quite common to look at government bond yields (e.g. Germany, USA). In equity world it is not uncommon to just use 0 at the moment instead of negative rates.

2. Are dividends discounted at the same rate? Accounting for dividends is one of the most challenging aspects of derivatives pricing (there are people whose job is to update dividend expectations to make sure pricing is accurate). It is the uncertainty of the dividend that makes it challenging. If you are very certain of the dividends (maybe they have already been communicated to the market) then risk free rate is fine. If the future has a long maturity I actually prefer accounting for the dividends in the discount rate (adjusting the risk free rate with the expected dividend yield).

1. What is the risk free rate?

In current practice the market repo rate is used. Each market firm faces a slightly different cost of funding and their internal rates will vary from one-another. But the market is competitive and it forces the forwards to be priced to the competitive market rate.

Forwards themselves don't really trade a lot. Most of the time futures or options are used for this kind of exposure. But calculation of a forward rate is critical since it's the base input for all other derivatives.

1. Are dividends discounted at the same rate?

I think you are mixing two concepts. 1: What rate do you use to discount a dividend. 2: How do you handle the uncertainty of the dividends?

There are a few approaches taken with the actual discounting.

• No discounting
• Discounting at the repo rate
• Discounting at OIS
• Discounting at your own cash rate (most firms have a cash rate that is OIS+spread)

In lower rate environments the difference are pretty small.

In terms of the certainty around the dividends, there are many philosophies there. It often depends on the reference underlyer.

1. The discount rate is NOT "risk-free", except in textbooks.

If a few brokers provide the majority of liquidity to the futures market, it's their funding cost that will be effective cost of capital for the futures, and associated options. Which will be inter-bank (Eurodollars, EURIBOR rather than OIS, EONIA etc).

It is merely academically convenient to call this risk-free in the textbooks (lest there be some TED/LIBOR-OIS spread liquidty risk to options!) And in practice, the impact is tiny.

1. Dividends need to be discounted at forward rates. Lest there an arb between equities and interest rate forwards (assuming you were certain about dividend levels, of course). Which then begs questions about what "forward riskless" looks like ;-)

The answer here too is interbank. Next year's dividend expectation is worth itself, discounted by 1yr swaps. The next 1y1y, then 2y1y, 3y1y etc. The credit spread over OIS does not matter if it's applicable to all parties' funding of their derivatives books. Nobody actually lends to anyone else at OIS.

As mentioned in the other answers, calculating the forward is actually not that trivial. Here is a link to a nice note on equity financing costs / repo:

https://www.globalvolatilitysummit.com/wp-content/uploads/2015/10/A-New-Normal-in-Equity-Repo-BNP-Paribas.pdf

How all this affects option pricing you might want to read Piterbarg's Risk paper "Funding beyond discounting".

But if you are still in the learning phase, just assume for now there is an unambigous risk-free rate denoted by $$r$$ that everybody agrees on.