Static Hedging and Risk-Neutral Pricing for a Barrier Payoff

Summary

The document examines a security paying one dollar if IBM reaches a specified price, with the stock initially below that level and no dividends or transaction costs. It contrasts a risk-neutral probability argument, which would imply a payoff value of one dollar if the stock is certain to reach the threshold, with a proposed static hedge holding a fraction of a share. That hedge costs the same fraction of the stock price and can fund the promised payment once the threshold is reached.

The discussion identifies a missing assumption: the problem does not specify the stock’s dynamics or establish that the market is arbitrage-free. Without those conditions, the risk-neutral argument cannot determine a unique price, and the suggested stock holding depends on the payoff mechanics and assumptions. The example illustrates why pricing arguments require a coherent model and market assumptions; it does not provide a general valuation method for barrier claims.

Key ideas

• A risk-neutral valuation requires assumptions about the market and the underlying price process.
• A static stock position can be proposed as a hedge for a threshold-triggered payment.
• Different pricing arguments may conflict when the problem leaves key assumptions unspecified.

Tags

# Simple pricing example confusion


# Simple pricing example confusion












This it taken from "Heard on the Street", Section B.

> Consider a market with $0$ risk-free rate, no transactions costs etc. The IBM stock costs \$75 and does not pay dividends. Design a security which pays \$1 if IBM stock reaches \$100. What does it cost?

The answer is \$.75 which can be proved by no-arbitrary considerations. At the same time, risk-neutral pricing suggests that the price is $$ \mathsf P\{\text{IBM has hit \$100 at least once}\} = 1. $$

## Answer by Probilitator (score 2, accepted)

https://quant.stackexchange.com/a/10850

As I see it the question does not enforce that the market is free of arbitrage. This is why you can get to contradicting prices. Thus you can't actually apply a risk-neutral argument here without making additional assumptions.

You yourself provide the example of such an arbitrage. If the underlying process had a B&S dynamics you could just borrow money for free, buy the stock and wait until it hits the $100 mark.

Seeing how one does not know the dynamics of the underlying process it makes sense to enter into the static hedge of holding 0.75 of the stock.

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