利率相关性为何会使期货价格高于远期

# Future Versus Forward Price When Underlying Asset Price Positively Correlated with Interest Rate


# Future Versus Forward Price When Underlying Asset Price Positively Correlated with Interest Rate












I'm reading a book called a Practical Guide to Quantitative Finance Interview, and cannot make sense of the solution for a particular question, so I really appreciate your advice:

Question:

What is the difference between futures and forwards? If the price of the underlying asset is strongly positively correlated with interest rates, and the interest rates are stochastic, which one has higher price? futures or forwards? Why?

Solution: If the future price is positively correlated with the interest rate, Here is my first doubt: the question itself say underlying asset price instead of future price is positively correlated with interest rate, is it because future price is positively correlated with underlying price and future price is then positively correlated with interest rate? The increases of the future price tend to occur the same time when the interest rate is high. Because of the Mark to market feature. The investor who long the futures has an immediate profit that can be reinvested at a higher rate. The loss tends to occur when the interest rate is low, so that it can be financed at low rate. Here is my second doubt: I cannot make sense of the last sentence, when the interest rate is low, the loss tends to occur, what does it mean? It means lower than forward price or something else? Besides, what is the meaning of "can be financed at low rate"?

## Answer by McCabe (score 1, accepted)

https://quant.stackexchange.com/a/50550

While the answer seems to be clear, the reason why this correlation to interest rates was important is due to the posting of margin.

The book you're reading was written prior to Dodd-Frank, Swaps clearinghouses and collateral collection for all forward contracts.

Today, presuming both products are collateralized via margin, there will be no difference in futures versus forward. FRAs are collateralized.

## Answer by siou0107 (score 1)

https://quant.stackexchange.com/a/50524

For your first doubt: the futures price is proportional to the asset price, so they are perfectly correlated.

For your second doubt: if futures price is positively correlated to interest rates, the buyer of a futures contract will (tend to) make a gain when interest rates are higher. The gain is immediately realised through margin calls, and invested at high rates. Similarly, they will tend to make a loss when interest rates are lower. The immediately realised loss must be financed (since you have to cash out the margin call), yet that financing is made at a low rate.

That makes the futures contract attractive compared to the forward contract. Demand thus drives the futures price higher than the forward price.

## Answer by Phil H (score 1)

https://quant.stackexchange.com/a/50525

The easiest way to understand this issue is to consider a basket holding opposite positions in the two derivatives.

tl;dr: The futures have a linear profile whereas the forward is convex due to discounting, so there is a bias priced in by the market

### Building a simple, par basket

So we are long some interest rate futures and short some Forward Rate Agreement (FRA) - the FRA is exactly correlated with interest rates, so what applies to that we can also apply to something with less correlation. We choose a FRA with the same fixing date as the future so they depend on the same number.

Since FRA and entering a futures position are both par trades, we have a par (zero) value on the basket. So we set the notional amounts on the two trades such that they offset each other - the FRA pays out a discounted amount so its notional will be slightly to the future. This is then delta hedged at delivery; one trade will exactly pay for the other regardless of how interest rates move.

### Its value goes up when rates move in either direction

Consider now what happens immediately after we construct the basket, as interest rates move. Suppose they go up in parallel by 10 bp: the futures position will move 10 bp up, netting us 10 x \$25 per tick = \$250 per future.

What about the FRA? Its payoff is discounted at the real Libor rate, not at the rate we traded, so its payoff is now more heavily discounted. Note that we discount the payoff now because there is still time left before delivery; on the maturity date, the payoff will still match the futures position. For now, though, our basket has a net positive value.

What about when rates go down by 10 bp? The reverse happens, and our futures position loses \$250 per contract, and the FRA value moves up (it will pay out at maturity). But rates are lower, so the payout suffers less discounting than when we set it up, and thus its value increases by more than the \$250. So our basket has a positive value again!

### The market prices that in

If you can build a basket whose value goes up whether the underlying goes up or down, then why not do that as much as you're allowed and make use of tur free money between now and expiry?

Inevitably, then, the market does factor that in and thus the futures prices are discounted by an amount which reflects this bias for holding futures over FRAs.

The adjustment comes from this mismatch between the profiles of the instruments - we would say that the future has a linear payoff profile whereas the FRA has a convex profile, so the adjustment is labelled a convexity adjustment.

### The convexity adjustment depends on the rate volatility

You will note that the positive position we ended up with depended on how much rates move - the more they move, the higher the value, so we can see that the convexity adjustment will depend on the volatility of the rate - if it is expected to move more, we can expect it to make more money.

### Futures prices reflect this adjustment

If you look at futures prices, e.g. for interest rates, the effective rate embedded in the price is already adjusted by this bias, so to read the market's expectations of forward interest rates you must calculate the adjustment and apply it to those rates.

### Adjustments apply beyond rate instruments

The above all applies to futures and forwards on any instrument correlated to interest rates, because it is that action of changing the degree of discounting which makes one half of the basket convex with regard to rate movement.