Question

Derive a formula for the value, at \(s\), of a forward contract on the \(T\)-claim \(X\), where the forward contract is made at \(t\), and \(t

Short answer

Answer: The formula for the value of a forward contract at time 's' is given by: V_s = P_s - Xe^{-r(T-s)}, where V_s represents the value of the forward contract at 's', P_s is the market price of the T-claim at time 's', X is the agreed upon price, r is the interest rate over the period [s, T], and T and s are time points such that t<s<T.

Step-by-step solution

Understand the Initial Conditions

The task provides us with several pieces of information. A forward contract is initiated at time 't' for a certain 'T'-claim 'X' with time 'T' being the maturity date. We want to find the value of this contract at a time point 's', where t<s<T.

Define the Symbol Notations

Denote the interest rate over the period [s, T] as 'r', the value of the forward contract at 's' as 'V_s', and the price of the T-claim at time 's' as 'P_s'.

Calculate the Price of T-Claim at 's' using Forward Price Formula

In any arbitrage-free market, the price at 's' (P_s) of the T-claim 'X' bought at 't' using a forward contract needs to be the present value of its pay-off discounted at the risk-free rate from 'T' to 's'. This can be represented with the following equation: \[ P_s = Xe^{-r(T-s)} \]

Calculate the Value of the Forward Contract at 's'

The value at 's' of a forward contract initiated at 't' is the difference between the market price of the T-claim at 's' and its calculated price as per the forward contract. Hence the formula would look like: \[ V_s = P_s - Xe^{-r(T-s)} \] Thus completing the derivation of the forward contract's value at time 's'. This formula explains that the value of a forward contract at any intermediate time point 's' depends on the market price of the asset, the agreed upon price 'X' and the interest rate 'r' till the time 'T'. It is always recommended to keep track of these parameters in real time, to know the worth of the contract at any given time.

Key concepts

Forward Contract Valuation

A forward contract is a financial agreement between two parties to buy or sell an asset at a predetermined price on a specified future date. Understanding its valuation is crucial as it helps investors determine its worth at any point before its expiration, specifically at a time after it has been agreed upon but before it matures. To find the valuation at an intermediate time point \( s \), we consider its market price relative to the agreed upon price.

• The agreed upon price is the price \( X \) set at the initiation of the contract.
• The market price of the asset at time \( s \) is crucial since it helps determine if the contract is profitable or not.

By calculating the difference between the market price of the asset at \( s \) and the price agreed upon in the contract \( X \), adjusted for interest rates, we can determine the contract's value.
This valuation at time \( s \) is usually denoted by \( V_s \), and it reflects how much the forward contract is worth factoring in all external conditions leading to \( T \).

Arbitrage Theory

Arbitrage theory is a fundamental concept in financial mathematics that allows investors to make profits without any risk. It relies on the principle that if two equivalent assets have different prices, one can generate a profit through simultaneous buying and selling, exploiting the price difference.

• This theory ensures that prices in financial markets are balanced and fair.
• In the context of forward contracts, it ensures that the forward price aligns with the expected value of an asset because any misalignment would lead to arbitrage opportunities.

The absence of arbitrage opportunities in a market confirms that the price of the forward contract is fairly adjusted to reflect the current market value of the asset and the risk-free interest rate.
This prevents discrepancies and aligns the market to ensure that no free profits can be made without assuming some risk, continuously balancing the market forces.

Risk-Free Interest Rate

The risk-free interest rate is a theoretical rate of return of an investment with no risk of financial loss. It is a crucial assumption in financial mathematics since it is used for discounting future cash flows.

• In context, it's what you would expect to earn by investing in a very safe asset, such as government bonds.
• It influences the valuation of forward contracts as it helps calculate the present value of future payments without the influence of possible risk.

When calculating the value of a forward contract, the expected future returns need to be discounted back to the present value (time \( s \)) using this risk-free rate.
The formula \( P_s = Xe^{-r(T-s)} \) utilizes the risk-free rate \( r \), transforming the future value into a present value. This discounting process ensures that future pay-outs are adjusted to reflect their value today.