Barrier Option Payoffs and Delta Hedging Near the Barrier

# hedging barrier options


# hedging barrier options












Consider Black Scholes dynamics for the stock price $$dS_t=\mu S_tdt+\sigma S_t dW_t$$ I have "heard" it is difficult hedging barrier options if the payoff suddenly is set to zero by the boundary condition so close to the expiry if the stock price is close to the barrier, hedging issues arise.

For example, down and out put option where the strike $K$ is above the barrier $L$. But why is that? The partial derivative doesn't exist or the magnitude is very large? What are the work arounds if any? I assume this is the property of the function and has nothing to do with a numerical method used to calculate the solution.

## Answer by user16651 (score 1, accepted)

https://quant.stackexchange.com/a/29890

As you know, Barrier options are extensions of vanilla options in the sense that they have a barrier level which activates or deactivates the option's pay-off upon hitting the barrier. The barrier can be hit when the option is in-the-money or out-of-money. Barrier options which are activated upon hitting the barrier are called Knock-in barrier options or simply Ins and those that are instead deactivated are known as Knock-out barrier options or Outs.

In knock-out options, if the barrier is not hit by the underlying price from the time of issuance of the option to its maturity, then the option holder receives an equivalent pay-off of a vanilla option.Knock-in options only provide a possibility of a positive pay-off after the barrier has been hit.

When a barrier option knocks-in, it becomes an equivalent vanilla option and thus, offers the same pay-off whereas a knock-out is equivalent to the corresponding vanilla option as long as the barrier is not hit until maturity (exercise time).

Mathematically speaking

Let $M_T=\max\{S_t\, ,\, 0\le t\le T\}$ and $m_T=\min\{S_t\, ,\, 0\le t\le T\}$ then the payoffs of down-and-out-call and down-and-out put respectively are given by $$(S_T-K)^+ \mathbb{I}_{\{m_T>L\}}$$ $$(K-S_T)^+ \mathbb{I}_{\{m_T>L\}}$$ and the payoffs for the up-and-out call and up-and-out put are given as follows $$(S_T-K)^+ \mathbb{I}_{\{M_T>L\}}$$ $$(K-S_T)^+ \mathbb{I}_{\{M_T>L\}}$$ Knock-in options give a payoff equivalent to that of an equivalent vanilla option at maturity only when the barrier is hit otherwise the payoff is zero.

A portfolio consisting of one knock-in call and one knock-out call is equivalent to an ordinary call option,that is, $$\text{up-and-out call + up-and-in call}=\text{vanilla call}$$

Similarly, for the other barrier options we have the following relationships,

$$\text{down-and-out call + down-and-in call}=\text{vanilla call}$$ and $$\text{up-and-out put + up-and-in put}=\text{vanilla put}$$ $$\text{down-and-out put + down-and-in put}=\text{vanilla put}$$ Reference

- Lishang Jiang, Mathematical modelling and methods of option pricing, World Scientific Publishing Co. Pte. Ltd., 2003.

- Michael Kateregga,Pricing and Hedging of Barrier Options,Thesis · June 2011.

- Michael Suchaneck, The Pricing and Hedging of Barrier Options and Their Applications in Finance and Life Insurance, 2008.

- Nassim Taleb,Dynamic Hedging: Managing Vanilla and Exotic Options.

## Answer by Mats Lind (score 2)

https://quant.stackexchange.com/a/29889

Since you have tagged your question with delta-hedging I assume we delta hedge the option with the underlying. Here difficulties would arise if the amount of underlying contracts we need to exchange to keep the delta near zero is large compared to illiquidity and transaction costs. With these together with volatility about constant, transaction costs would increase with gamma. As gamma gets very high, typically but not only, for plain vanilla options at the money near expiry, delta hedging gets very expensive.