Question

Consider a general down-and-out contract, with contract function \(\Phi\), as descibed in Section \(13.2 .1\). We now modifiy the contract by adding a fixed "rebate" \(A\), and the entire contract is specified as follows. \- If \(S(t)>L\) for all \(t \leq T\) then \(\Phi(S(T))\) is paid to the holder. \- If \(S(t) \leq L\) for some \(t \leq T\) then the holder receives the fixed amount \(A\). Derive a pricing formula for this contract.

Short answer

Answer: The pricing formula for the down-and-out contract is given by Price_of_contract = e^(-rT) × (P(Stay above L) × E[Φ(S(T)) | Stay above L] + P(Go below L) × A), where e^(-rT) is the discount factor, r is the risk-free interest rate, T is the time to maturity, P(Stay above L) represents the probability of the stock price staying above the barrier L, P(Go below L) represents the probability of the stock price going below the barrier L, and E[Φ(S(T)) | Stay above L] is the expected value of Φ(S(T)) given the stock price stays above the barrier L.

Step-by-step solution

Understand the Contract Payoff Structure

Here, the contract payout depends on the stock price S(t) at time t compared to the barrier level L. The contract payout structure is as follows: - If the stock price S(t) stays above the barrier level L for all times t <= T, the contract holder receives a payment of Φ(S(T)). - If the stock price S(t) goes below the barrier level L at any time t <= T, the contract holder receives a fixed amount A.

Determine the Expected Payoff Using Risk-Neutral Probability

In this step, we'll calculate the expected payoff of this contract under the risk-neutral probability framework. We can represent the expected payoff as follows: Expected Payoff = (Probability of Staying Above the Barrier) × (Expected Payoff Given Staying Above the Barrier) + (Probability of Going Below the Barrier) × (Expected Payoff Given Going Below the Barrier) Since the contract holder receives Φ(S(T)) when staying above the barrier and A when going below the barrier, the above equation can be written as: Expected Payoff = P(Stay above L) × E[Φ(S(T)) | Stay above L] + P(Go below L) × A Here, P(Stay above L) and P(Go below L) represent the probabilities of staying above and going below the barrier level L respectively, under the risk-neutral probability measure.

Derive the Pricing Formula

Finally, we need to discount the expected payoff to get the pricing formula for this contract. Let r be the risk-free interest rate and T be the time to maturity. The pricing formula will be: Price_of_contract = e^(-rT) × (P(Stay above L) × E[Φ(S(T)) | Stay above L] + P(Go below L) × A) This is the pricing formula for the down-and-out contract with the given conditions. In practice, we would need to calculate the risk-neutral probabilities of staying above and going below the barrier and the expected value of Φ(S(T)) given the stock stays above the barrier. After determining these values, we can plug them into the pricing formula to find the price of the contract.

Key concepts

Barrier Options

Barrier options are a type of financial derivative that are similar to standard options, but with a twist - their existence or expiration depends on whether the underlying asset's price hits a certain predetermined level called the "barrier". There are two main types of barrier options: **"knock-in" options**, which come into existence when the barrier is breached, and **"knock-out" options**, which expire when the barrier is breached.
In our example, we have a **"down-and-out"** option, which is a type of knock-out option. The contract specifies that if the stock price, denoted as \( S(t) \), stays above a certain barrier level \( L \) throughout the contract's life until expiration time \( T \), the holder receives a payout \( \Phi(S(T)) \). However, if the stock price falls below the barrier \( L \) at any time before expiration, the contract gets "knocked out" and only a fixed rebate \( A \) is paid to the holder.
This additional conditional element makes barrier options generally cheaper than similar regular options, as the barrier condition introduces a risk of early expiration.

Risk-Neutral Probability

The concept of risk-neutral probability is central to pricing derivative securities. In a financial model, the risk-neutral measure is a probability measure under which the present value of the expected future payoffs of a derivative is calculated, assuming the expected return of the underlying asset is the risk-free rate, \( r \).
Under this framework, the market is assumed to be "risk-neutral," meaning that investors are indifferent to risk. This simplifies the process of derivative pricing because, rather than calculating payoffs for various risk scenarios, we use expected values under this harmonized probability measure. This results in more straightforward mathematical models for calculating expected values.
In our barrier option scenario, we calculate the probabilities of the stock price staying above or going below the barrier. These probabilities are denoted as \( P(\text{Stay above } L) \) and \( P(\text{Go below } L) \) respectively. Using the risk-neutral probability, both the expected payoff of staying above the barrier and the expected rebate if the barrier is breached can be accurately valued.

Expected Payoff Calculation

The expected payoff calculation for our barrier option involves considering both the outcomes where the stock remains above the barrier and the outcomes where it falls below. Each has its own probability and associated payoff.
The formula to determine the expected payoff under the risk-neutral framework is:

• \( P(\text{Stay above } L) \times E[\Phi(S(T)) | \text{Stay above } L] \)
• \(+ P(\text{Go below } L) \times A \)

Here, \( E[\Phi(S(T)) | \text{Stay above } L] \) represents the expected payoff given that the stock price stays above \( L \) till maturity \( T \), and \( A \) is the constant rebate if the stock price breaches \( L \).
This means that the expected payoff calculation involves weighing these two contributions by their respective risk-neutral probabilities. Once we determine these values, we discount them to present value using the formula \( e^{-rT} \times \text{Expected Payoff} \), ensuring you account for the time value of money. This gives us the option's price today.