Question

An "all-or-nothing" contract, with delivery date \(T\), and strike price \(K\), will pay you the amount \(K\), if the price of the underlying stock exceeds the level \(L\) at some time during the interval \([0, f]\). Otherwise it will pay nothing. Compute the price, at \(t

Short answer

Answer: The process of determining the price of an all-or-nothing contract at time t < T involves the following steps: 1. Analyze the all-or-nothing contract and focus on the time interval (t, f] in which the stock price might exceed the level L. 2. Define the probability of the stock price exceeding L, given that the stock price follows a geometric Brownian motion with a drift term and volatility term. 3. Use the Change of Measure technique and Girsanov's theorem to find the Radon-Nikodym derivative to account for the risk-neutral probabilities. 4. Calculate the probability P(t) under the Risk-Neutral Measure by computing the expected value of the Indicator function I. 5. Compute the price of the all-or-nothing contract at time t by discounting the expected payout of the contract at time T, multiplied by the probability of receiving the payout at T.

Step-by-step solution

Analyze the all-or-nothing contract

We know the contract will pay K if the stock price exceeds L in the specified interval [0, f]. Since we also know that the stock price will be less than L for all s ≤ t, our focus should be on the time interval (t, f]. As it's an "all-or-nothing" contract, we need to compute the probability of the stock price exceeding L at least once during (t, f].

Define the probability of the stock price exceeding L

Let P(t) be the probability that the stock price exceeds L during the time interval (t, f]. This probability depends on the stock price dynamics, which we will assume follows a geometric Brownian motion characterized by risk-neutral drift term \(\mu\) and volatility term \(\sigma\). This motion is given by: \(dS(s) = \mu S(s) ds + \sigma S(s) dW(s)\).

Change of Measure Technique

Change of measure technique will be used here in order to get the probability of the stock price exceeding L under risk-neutral probabilities during the time interval (t, f]. We can do this by finding the Radon-Nikodym derivative with Girsanov's theorem. Let's denote the Radon-Nikodym derivative as \(\Lambda(s)\) for some s ∈ (t, f]. The change of measure is given by: \(\Lambda(s) = \text{exp}(-\frac{1}{2} (\frac{\mu - r}{\sigma})^2 (s-t) - (\frac{\mu - r}{\sigma}) W(s-t))\), where r is the risk-free interest rate and W(s) denotes the Brownian motion.

Calculate the probability P(t) under the Risk-Neutral Measure

To calculate P(t), we need to compute the expected value of the Indicator function I under the risk-neutral measure. The Indicator function I is defined as \(I(s) = 1\), when the stock price exceeds L during the time interval (t, s] and zero otherwise. Let Q be the risk-neutral measure. Then, the probability P(t) can be computed as: \(P(t) = EQ[\frac{\Lambda(s)}{\Lambda(t)}I(s)]\).

Compute the price of the all-or-nothing contract

To compute the price of the all-or-nothing contract at time t, we need to discount the expected payout of the contract at time T, multiplied by the probability of receiving the payout at T. Thus, the contract price at time t is given by: \(Price(t) = e^{-r(T-t)} K \times P(t)\).

Key concepts

Probability of Stock Price Exceeding a Threshold

Understanding the probability of a stock price exceeding a threshold is crucial when evaluating financial derivatives like all-or-nothing contracts. In such contracts, the payout is dependent on whether the underlying asset's price crosses a certain level within a given period.

In our specific scenario, we look at the probability (\(P(t)\)) that the stock price will exceed a certain level (\(L\) before the expiration date (\(T\)). This is modeled using stochastic processes, typically assuming the stock price follows a geometric Brownian motion. Key parameters in this motion are the drift term, \(\mu\), representing the percentage change in the price over time, and the volatility term, \(\sigma\), which accounts for the price's fluctuations.

Calculating \(P(t)\) necessitates understanding these dynamics and often requires sophisticated mathematical techniques. This probability forms the basis of the contract's valuation and must be computed accurately for proper pricing.

Change of Measure Technique

The change of measure technique is a powerful mathematical tool used in financial modeling to facilitate the computation of probabilities under different scenarios. It involves changing the 'world' we are using to measure probabilities.

This transformation allows us to evaluate probabilities under the 'risk-neutral measure' instead of the 'real-world measure.' The risk-neutral world assumes that all investors are indifferent to risk, meaning securities are priced without any premium for risk. In the context of our problem, the technique helps transform the risk-neutral drift term from \(\mu\) to the risk-free interest rate \(r\), simplifying the calculation of the desired probability.

To perform this transformation, we use the Radon-Nikodym derivative, which effectively translates the probabilities from one measure to another. Calculating this derivative involves using the Girsanov's theorem, which adjusts the drift of the process to reflect the risk-free rate instead of the actual expected return.

Girsanov's Theorem

Girsanov's theorem is a cornerstone of modern financial mathematics. It allows us to change probability measures in a way that the transformed process, under the new measure, can be thought of as a standard Brownian motion with a different drift term.

When applied to our all-or-nothing contract, Girsanov's theorem helps in adjusting the stochastic process for the stock price so that it reflects the risk-neutral world. Essentially, it provides the mechanics for the change of measure technique by defining the Radon-Nikodym derivative (\(\Lambda(s)\)) and applying it to the original process.

By 'tilting' the probability distribution in this manner, it allows for the easier computation of risk-neutral probabilities, which are a critical aspect of financial modeling and derivative pricing. The theorem bridges the gap between the expected returns in the real market and the theoretical risk-neutral world used for pricing derivatives.

Risk-Neutral Measure

A risk-neutral measure, denoted as \(Q\), is an alternative probability measure used in financial economics where all investors are assumed to be risk-neutral. This implies that they do not require additional returns to compensate for risk when valuing future cash flows or assets.

Under the risk-neutral measure, the expected return of any investment equals the risk-free rate of interest. This greatly simplifies the process of valuing derivatives because it removes the need to estimate risk premia. By employing a risk-neutral measure, we effectively discount future prices or payouts at the risk-free rate, which provides a consistent pricing method that is arbitrage-free.

When calculating the price of our all-or-nothing contract, we discount the expected payoff by the risk-free rate, assuming we are operating under the risk-neutral measure Q. This leads to a valuation that is aligned with the rest of the market, where pricing is based on the assumption of risk neutrality.