Question

Suppose that \(S\) is the price process of an asset in a standard Black-Scholes model, with \(r\) as the constant rate of interest, and fix a contingent \(T\)-claim \(\Phi(S(T))\). We know that this claim can be replicated by a portfolio based on the money account \(B\), and on the underlying asset \(S .\) Show that it is also possible to find a replicating portfolio, based on the money account and on futures contracts for \(S(T)\).

Short answer

Question: Show that it is possible to find a replicating portfolio based on the money account, B, and futures contracts for S(T) in the standard Black-Scholes model with a constant interest rate r and a contingent T-claim denoted by Φ(S(T)). Answer: It is possible to find a replicating portfolio based on the money account and futures contracts for S(T) in the Black-Scholes model using the concept of delta hedging. By finding a hedge ratio, θ(t), such that the combined risk-free portfolio has the same payoff as the contingent claim Φ(S(T)) at maturity, we can create a replicating portfolio. The hedge ratio can be expressed as θ(t) = (Φ'(S(T))S(T))/B(t), where Φ'(S(T)) is the derivative of the contingent claim with respect to the asset price.

Step-by-step solution

Understand the Black-Scholes Model

The Black-Scholes model is a widely used model in finance for option pricing. It assumes that the underlying asset follows a geometric Brownian motion, which implies the asset price is log-normally distributed. The model provides a formula to compute the price of European call and put options.

Define the terms

Let's define the terms used in this exercise: 1. \(S\): The price process of the underlying asset in the Black-Scholes model 2. \(r\): The constant interest rate 3. \(T\): The time to maturity of the contingent claim 4. \(\Phi(S(T))\): The contingent claim, which is a function of the price process \(S\) at time \(T\) 5. \(B\): The money account or risk-free bond 6. Futures contracts: Financial contracts to buy or sell an asset at a predetermined price on a specified future date

Replicating Portfolio

A replicating portfolio is a portfolio that generates the same cash flows as the security or derivative being replicated. In our case, the goal is to find a replicating portfolio based on the money account \(B\), and on futures contracts for \(S(T)\) to hedge the contingent claim \(\Phi(S(T))\).

Use Delta Hedging

One of the key concepts in the Black-Scholes model is the delta hedging strategy. Delta hedging is the process of adjusting the number of shares (or futures contracts) in the portfolio in order to keep the portfolio's delta neutral. The delta of a portfolio is the sensitivity of the portfolio's value to changes in the underlying asset's price. In the context of replicating portfolios, delta hedging means creating a risk-free portfolio by combining the underlying asset, the derivative, and the risk-free bond. This risk-free portfolio will have no exposure to the price changes of the underlying asset, effectively replicating the payoff of the derivative security.

Replicating the contingent claim with futures contracts

To replicate the contingent claim \(\Phi(S(T))\) using the money account \(B\) and futures contracts on \(S(T)\), we need to find a hedge ratio, denoted by \(\theta(t)\), such that the combined risk-free portfolio has the same payoff as the contingent claim. Using the concept of delta hedging from step 4, we can write the self-financing replicating portfolio as: $$ \Pi(t) = B(t)\theta(t) + F(t)S(T) $$ Here, \(\Pi(t)\) is the value of the replicating portfolio at time \(t\), \(B(t)\) is the value of the money account, \(\theta(t)\) is the hedge ratio, \(F(t)\) is the futures price at time \(t\) for the underlying asset \(S(T)\). The objective is to find the hedge ratio \(\theta(t)\) such that the replicating portfolio has the same payoff as the contingent claim \(\Phi(S(T))\) at maturity: $$ \Pi(T) = \Phi(S(T)) $$ By substituting the replicating portfolio equation into the objective function, we get: $$ B(T)\theta(T) + F(T)S(T) = \Phi(S(T)) $$ To find the hedge ratio, we need to differentiate the contingent claim with respect to the asset price, which gives us the delta of the claim. $$ \theta(t)B(t) = \Phi'(S(T))S(T) $$ To solve for \(\theta(t)\), we can express the futures price \(F(T)\) at time \(T\) in terms of the contingent claim payoff and the risk-free bond, as follows: $$ \theta(t) = \frac{\Phi'(S(T))S(T)}{B(t)} $$ Thus, we have found a hedge ratio \(\theta(t)\) using futures contracts on \(S(T)\) that allows us to create a replicating portfolio based on the money account and futures contracts.

Key concepts

Replicating Portfolio

A replicating portfolio is a fundamental concept in the Black-Scholes Model. In finance, it is a strategy used to replicate the payoffs of a specific asset or derivative, like an option, using a mix of other assets. Typically, it involves combining risk-free assets, such as bonds, with risky assets, like stocks, to mirror the returns of the security we're interested in replicating.
For the exercise, a particular claim, denoted as \( \Phi(S(T)) \), can be simulated using a combination of risk-free assets and futures contracts. The end goal of a replicating portfolio is to end up with the same cash flow or return as the claim at maturity without actually holding the specific financial instrument directly.
This technique is essential because it allows traders to manage risks effectively and capitalize on financial opportunities.

• A replicating portfolio ensures that at the end of the period, the return matches exactly the payoff of the derivative.


• It supports pricing and risk assessment for options without direct involvement with the actual underlying asset.

When executed accurately, replicating portfolios create a safe, predictable financial instrument outcome by aligning their value with the option or claim at a specific point in time.

Delta Hedging

Delta hedging is a critical concept in the realm of options trading and the Black-Scholes model. It involves adjusting the composition of the replicating portfolio to maintain a delta-neutral position. Delta, represented as \( \Delta \), measures how much the price of an option changes when there is a price change in the underlying asset.
For example, if a portfolio has a delta of 0.5, it suggests that for every dollar increase in the price of the underlying asset, the value of the option will increase by $0.50. The idea behind delta hedging is to counterbalance this sensitivity, keeping the portfolio's overall value stable, even when the market prices move.

• A delta-neutral portfolio implies that its value is unaffected by small movements in the underlying asset's price.


• It is achieved by buying or selling units of the underlying asset or equivalent financial derivatives to offset changes in the delta.

This strategy is especially useful for traders looking to minimize risk associated with fluctuations in stock prices. By continuously adjusting the portfolio to maintain neutrality, the trader aligns it with the underlying asset's price, essentially locking in a specific risk-free rate of return, much like a replicating portfolio.

Futures Contracts

Futures contracts play a crucial role in the Black-Scholes model when forming replicating portfolios. These are standardized agreements to buy or sell an asset at a predetermined price on a future date. They are typically used in the realm of derivatives trading for hedging or speculation purposes.
In the exercise provided, futures contracts on \( S(T) \), the price of the underlying asset at maturity time \( T \), are used to replicate the contingent claim \( \Phi(S(T)) \). This allows traders to forecast and manage their future payoffs without owning the physical asset.

• Futures contracts guarantee a fixed price for the asset, reducing uncertainty and helping in precise financial planning.


• They are leveraged instruments, meaning small futures price changes can lead to significant capital changes, offering both opportunities and risks.

By using futures, traders can create a strategy where they are indifferent to price changes of the asset in the market path up to maturity. The futures contract offers a substitute to directly holding an asset while still achieving the similar outcome in terms of payoff, aligning perfectly with the concept of a replicating portfolio. This serves to stabilize potential losses and optimize financial outcomes.