# Futures and Forward Contracts: Pricing, Arbitrage, and Applications

## Futures and Forwards (II)

### The No-Arbitrage Principle

Consider a forward contract that obliges us to hand over an amount K at time T to receive the underlying asset. Today’s date is t, and the price of the asset is currently S(t). This is the spot price, the amount for which we could get immediate delivery of the asset.

When we get to maturity, we will hand over the amount K and receive the asset, whose worth is S(T), which we cannot know until time T.

Let’s analyze a scenario:

- Enter into the forward contract. This costs us nothing.
- Simultaneously sell the asset (going short). This is possible in many markets, but with some timing restrictions.
- We now have S(t) in cash from the sale, a forward contract, and a short asset position, resulting in a net position of zero. We then deposit the cash in the bank to receive interest.
- At maturity, we hand over the amount K and receive the asset, canceling our short position.
- We are left with a guaranteed -K in cash and the bank account containing the initial investment of S(t) with added interest.

Since we began with a portfolio worth zero and end up with a predictable amount, that amount should also be zero.

Therefore, we can conclude that:

Ft = St * e^{(r(T-t))}

This equation represents the relationship between the spot price (St) and the forward price (Ft or K).

#### Example: Forward Price Calculation

Suppose you enter into a 6-month forward contract on a non-dividend-paying stock with a current price of 100 euros. The risk-free interest rate (with continuous compounding) is 12%. What is the forward price?

Solution:

Ft = 100 * e^{(0.12 * 0.5)} = 106.1837 euros

#### Arbitrage Opportunity

If this relationship is violated, there will be an arbitrage opportunity. For instance, if Ft is less than St * e^{(r*(T-t))}, one could:

- Enter into the deals as explained above.
- At maturity, you will have St * e
^{(r*(T-t))}in the bank, a short asset, and a long forward. - The asset positions cancel when you hand over the amount F, leaving you with a profit of St * e
^{(r*(T-t))}– Ft.

If Ft is greater than St * e^{(r*(T-t))}, you would enter into the opposite positions, going short on the forward, again resulting in a riskless profit.

### Futures Contracts

Futures contracts are similar to forward contracts but are traded through an exchange, making them more liquid and subject to regulations. Some basic concepts include:

**Available Assets:**A futures contract specifies the underlying asset, crucial for commodities due to variations in type and quality. It also specifies the quantity to be delivered.**Delivery and Settlement:**The contract specifies the delivery date. Most futures contracts are closed out before delivery by taking the opposite position. If not closed, physical delivery or cash settlement occurs.**Margin:**Daily settlement of changes in the value of futures contracts is called marking to market. To mitigate default risk, exchanges require a margin deposit. Initial margin is deposited at the start, and the total margin must stay above a maintenance margin, requiring additional deposits if it falls below.

### Commodity Futures

Commodity futures involve additional factors:

- Storage Costs (s): Futures contract holders must compensate commodity holders for storage.
- Convenience Yield (c): Commodity holders often benefit from holding the physical asset.

Therefore, the forward price for commodities is:

Ft = St * e^{(r – c + s)(T-t)}

### Forex Futures

Similar to commodity futures, the interest received on the foreign currency (rf) acts as a convenience yield. The forward price is:

Ft = St * e^{(r – rf)(T-t))}

#### Example: Forex Forward Price

Suppose the 1-year interest rates in Australia and the USA are 2% and 1% per annum (continuously compounded), respectively, and the spot exchange rate is 0.98 USD per AUD. What is the 1-year forward price (considering USD as the local currency)?

Ft = 0.98 * e^{(0.01 – 0.02)(1-0)} = 0.97

### Known Income

When the investment asset provides a known income (I), the forward price is:

Ft = (St – I) * e^{(r*(T-t))}

Note that income can be negative, representing costs like storage.

#### Example: Forward Price with Dividends

Consider a 10-month forward contract on a stock with a current price of 100 euros. The risk-free interest rate (continuously compounded) is 5% per annum. Dividends of 0.8 euros per share are expected after 3, 6, and 9 months. The present value of the dividends is:

I = 0.8 * (e^{(-0.05 * 0.25)} + e^{(-0.05 * 0.5)} + e^{(-0.05 * 0.75)}) = 2.340866

The corresponding forward price is:

Ft = (100 – 2.340866) * e^{(0.05 * 10/12)} = 101.8142

### Value of Futures/Forward Contracts

A forward contract has zero value initially but can gain positive or negative value over time. Non-arbitrage arguments determine the fair value:

**Long Forward Contract:**(Ft – K) * e^{(-r*(T-t))}**Short Forward Contract:**(K – Ft) * e^{(-r*(T-t))}

where K is the delivery price and Ft is the forward value.