# 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.

• Please determine your model. Without the model , we can not talk about this issue. – user16651 Aug 29 '16 at 13:57
• @BehrouzMaleki: edited the question. – Medan Aug 29 '16 at 14:12
• By no means does this constitute a proper answer but did you have a look at this paper by Derman: emanuelderman.com/media/insoutbarriers1.pdf, the graphs (especially the delta/gamma profile) may provide you with some hints as to why these options are not easy to hedge in practice + excellent answer of @Mats Lind – Quantuple Aug 29 '16 at 21:10

## 2 Answers

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

• this is an into to the options and not answering my hedging question. But I think I can find an answer in the links you have provided. Thanks. – Medan Aug 29 '16 at 15:10

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.