Common mistake 1 with Forwards, futures & swaps

Mixing continuous and annual compounding in one calculation.

Common mistake 2 with Forwards, futures & swaps

Quoting an FX forward the wrong way round. Here the spot is domestic currency per unit of foreign currency.

Common mistake 3 with Forwards, futures & swaps

Forgetting to discount an FRA settlement from the end of the period back to the fixing date.

Common mistake 4 with Forwards, futures & swaps

Treating a swap's notional as money at risk. Only the net interest payments are exchanged.

EconStatsCraft › Forwards, futures & swaps

Forwards, futures & swaps

No-arbitrage pricing of the linear derivatives: a forward on an asset with income or storage costs, a currency forward from interest parity, a forward rate agreement and a plain-vanilla swap from a zero curve - with the arbitrage trade spelled out when a market price is off.

The intuition

If you want to own an asset in a year, you can buy it now and borrow the money, or agree today to buy it in a year. Both leave you owning the asset in a year, so they must cost the same, or someone could buy the cheap one, sell the expensive one and keep the difference. That single argument prices forwards on shares, commodities and currencies, and, applied to interest rates, FRAs and swaps.

When to use it

Pricing or valuing forwards, futures, FX forwards, FRAs and swaps, spotting cash-and-carry arbitrage, and the derivatives readings of CFA Levels I and II and FRM Part I.

Where the estimator comes from

Buy the asset for S_0, financed at r, receive income worth I and pay storage u: the cost of delivering it at T is (S_0-I)e^{(r+u-q)T}, which must equal the forward price. For currencies, investing one unit of domestic currency at home must equal converting it, investing abroad and converting back forward: F=S\frac{(1+r_d)^T}{(1+r_f)^T}. A forward rate solves (1+r_1t_1)(1+f\,(t_2-t_1))=(1+r_2t_2). A swap's fixed rate sets the fixed leg equal to a floating leg worth par: c\sum Z_k + Z_n = 1.

The model

F_0 = (S_0 - I)\,e^{(r+u-q)T},\qquad F_{FX} = S\left(\frac{1+r_d}{1+r_f}\right)^T,\qquad \text{swap rate} = \frac{1-Z_n}{\sum_{k=1}^n Z_k}

What it assumes

You can borrow and lend at the risk-free rate and short the underlying asset.
No transaction costs, taxes or counterparty default.
Forward and futures prices are treated as equal, which holds when interest rates are not random.
The swap uses annual payments and discount factors from the zero curve you enter.

Statistical properties

A forward is worth zero when agreed and afterwards worth the discounted change in forward price.
Income on the asset lowers the forward price; storage costs raise it.
The higher-interest currency trades at a forward discount.
A swap is equivalent to a portfolio of FRAs, or to being long one bond and short another.

How to read the output

The forward is not a forecast of the future spot price. It is the spot price carried forward at the cost of financing, less any income.
A currency with a higher interest rate trades at a forward discount, exactly offsetting its rate advantage.
A forward is worth zero when written; later its value to the long is the change in forward price, discounted.
A pay-fixed swap gains when market swap rates rise above its contract rate, like a short bond position.

A worked reading

A share at 100 with a 2% continuous dividend yield and a 5% risk-free rate. The one-year forward is 100 × e^(0.05 − 0.02) = 103.05. If the market quotes 105, the forward is too expensive: borrow 100, buy the share, sell the forward at 105, and at delivery hand over the share, repay 103.05 net of dividends and keep about 1.95. A long forward written earlier at 101 is now worth (103.05 − 101) × e^(−0.05), about 1.95. In swap mode, a five-year curve rising from 3.0% to 4.0% gives a par swap rate of 3.97%.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Forward far from the market price	Income or storage not entered	Add the dividend yield or storage cost
FX forward looks inverted	Quote convention reversed	Spot here is domestic per unit of foreign
Swap rate above every zero rate	Curve entered as decimals	Enter rates in percent
FRA payment seems too large	Not discounted from t2 to t1	The settlement already discounts it

Common mistakes

Mixing continuous and annual compounding in one calculation.
Quoting an FX forward the wrong way round. Here the spot is domestic currency per unit of foreign currency.
Forgetting to discount an FRA settlement from the end of the period back to the fixing date.
Treating a swap's notional as money at risk. Only the net interest payments are exchanged.

Where to read this properly

Hull, Options, Futures and Other Derivatives (2021), ch. 4, 5 and 7.
CFA Institute, Derivatives: Pricing and Valuation of Forward Commitments (Levels I and II).
GARP, FRM Part I, Financial Markets and Products.

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Forwards, futures & swaps

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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