22 votes
Accepted

Delta-Hedging Exotic Options

Consider reading Lorenzo Bergomi's excellent book -- or at least the first chapter available here for download --, it will help you clarify things. Some remarks as to your original question: It is ...
  • 14.1k
17 votes

Delta of binary option

The value of European binary call, paying \$1 if $S_T > K$ or nothing otherwise, is $$c_t=e^{-r(T-t)}N(d_2)$$ where, $d_2=\frac{ln(S_t/K)+(r-\sigma^2/2)(T-t)}{\sigma \sqrt{T-t}}$ Delta of your ...
  • 2,168
13 votes
Accepted

delta-hedging is failing

Regarding your 1st question, jumps are indeed unhedgeable. From a theoretical point of view, you might want to look at Merton's "Option pricing when underlying stock returns are discontinuous", the ...
11 votes
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Effect of volatility on the delta of a call option

Options have an asymmetric payoff profile: The payoffs are zero for almost all cases and positive else (as we well know). If the option is OTM, most of its payoffs are zero. A rise in volatility will ...
  • 5,669
9 votes
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Delta Hedging with fixed Implied Volatility or floating Implied Volatility?

Generally speaking, in the real world, you'd always want to use the correct implied vol. But you should think of your question in terms of: (1) Vega mark-to-market (m2m) PnL vs. theta/gamma profile (...
  • 705
9 votes

When should we delta hedge?

By delta hedging you are saying that you have a view on the path and the volatility of the option you are trading, but not on its direction; in your case, that being short delta. From a theoretical ...
8 votes
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Derivation of BS PDE problem using Delta hedging

This question has been asked many times and some clarifications appear needed. As pointed out in an answer to this question, the portfolio \begin{align*} \Delta_t^1 S_t + \Delta^2_t C, \end{align*} ...
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8 votes
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What really is Gamma scalping?

Assuming all else remains equal (implied vol has not changed and very little time decay has occurred), Gamma scalping can best be explained by Gamma (or realized volatility) enhancing the value of a ...
  • 5,452
7 votes
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derivation of the hedging error in a black scholes setup

The differential equation has a trend due to the interest rate. When you discount you take this trend away: $$ \frac{d}{dt} (e^{-rt}Z_t) = -re^{-rt}Z_t + e^{-rt} \frac{d}{dt}Z_t = e^{-rt}\frac{1}{2}...
  • 3,856
7 votes
Accepted

Proof of gamma profit formula

Assume you buy a plain vanilla call option at the price $V$ and the spot $S$. You immediately delta hedge buy selling $\partial V / \partial S$ units of the underlying asset. The underlying asset now ...
7 votes

What really is Gamma scalping?

Gamma scalping (being long gamma and re-hedging your delta) is inherently profitable because you make 0.5 x Gamma x Move^2 across the move from your option. (You get shorter delta on downmoves, so you ...
  • 221
6 votes

How do market makers hedge VIX index options?

Due to the lack of a carry arbitrage, VIX futures are actually the direct hedge for VIX Index options
6 votes
Accepted

Delta of binary option

If it wasn't clear from the previous answers, the answer they want is that the delta becomes infinite. That's because a tiny move in the stock will change the payout by $100 so your delta hedge must ...
  • 14.3k
6 votes

What really is Gamma scalping?

As long as you live in a world where implied and realized vol are the same, there is no net profit (or loss) from gamma scalping. However, if they are different, then you make a gain or loss which is ...
  • 794
6 votes

FX option trading questions

Yes, in the sense that it is assumed that the delta will be passed between participants at time of execution. Not necessarily. A non delta neutral trade may be used for speculation , or for hedging.
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6 votes
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Hedging strategy for payoff $\int_0^T\log S_u\mathrm{d}u$

I assume you want to price a derivative product that pays $\int_0^T\ln S_tdt$ at maturity time $T$, from time $t=0$. I'll ignore generalization to time $t$ because it is trivial (split the integral in ...
  • 614
6 votes

Effect of Implied volatility on option delta

In the Black-Scholes-Merton model, with model option price $V$ as a function of underlying price $S_t$, strike price $X$, continuously compounded risk-free rate $r$, continuously compounded dividend ...
  • 5,903
5 votes

The role of Gamma in replicating a put

If you could hedge continuously with zero transaction costs, the gamma would be irrelevant: you would perfectly replicate with delta hedging and be done. In practice, hedging is discrete and there ...
  • 6,763
5 votes

Why is this delta-hedging/P&L example on a variance swap call correct?

To answer your questions: Is the trading p&l meant to be the delta-hedging p&l? Yes, in his example it concerns delta hedged pnl. how come p&l is raising steadily even when stock price ...
  • 1,411
5 votes
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Dynamic Delta Hedging And a Self Financing Portfolio

Main references As explained in my comments, the correct approach to derive the hedging portfolio would be the one described in Gordon's answers to the following questions: Derivation of BS PDE ...
5 votes

Delta Hedging with fixed Implied Volatility to get rid of vega?

You should have a look at the following paper: Ahmad, Riaz and Paul Wilmott (2005) "Which free lunch would you like today, Sir? Delta hedging, volatility arbitrage and optimal portfolios," Wilmott ...
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5 votes

Hedging error in a stochastic volatility model

Let's assume that at time $t$ you become long an option, which you wish to price and risk-manage under the BS framework. The delta-hedged portfolio at time $t$ reads $$ \Pi_t = (V^{BS}(t) - \Delta_{BS}...
  • 14.1k
5 votes
Accepted

Gamma/delta dynamics in the Black Scholes model and it's relation to PnL (Basic of option theory)

We work in a Black-Scholes world. Consider the following delta-hedged portfolio: $$ \Pi_t=V_t-\frac{\partial V}{\partial S}S_t$$ We assume the portfolio is self-financing$^{\text{(a)}}$, therefore: $...
5 votes
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Deriving Delta Hedge error in the B-S setup (part 2)

The paper could be clearer indeed. It is a slightly confusing topic, but the important step here is to understand the consequence of the derivative $C$ in the portfolio being priced at the assumed ...
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5 votes

Options Delta Meaning of Term

It is true that when FX options are traded, the delta is often traded as well. That is a practice specific to the FX option market. It is called an "exchange of delta". You can undo it by selling the ...
  • 9,167
5 votes
Accepted

Delta Hedging with a Different Underlying

I'd tackle this using some handwavery application of the bivariate conditional normal distribution to the geometric brownian motions: For a multivariate normally distributed random variable $X$, ...
  • 5,903
4 votes

Delta of binary option

Delta of a digital (or binary) option is like the normal distribution probability function , approaching 0 at far OTM / ITM conditions and representing a very high peak at ATM. The peak at ATM ...
  • 308
4 votes

Why/How does a hedged portfolio make profits?

An Investment Bank earns a profit by selling you an option at a slightly higher price than the theoretical price, or buying it back from you at a slightly lower price. They call this "earning a spread"...
  • 9,167
4 votes

What is the effect of increasing volume depth to stock volatility?

Answers to your 3 questions: Empirically, there is no effect. I understand that this is not logical but it is reality. Adding thickness to an order book does not necessarily make it harder or easier ...
  • 3,880
4 votes
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Basic practical question about Delta hedging

As has been remarked in the comments already, the standard deviation of your hedging error should approach zero as your re-hedging frequency (the number of time steps) increases. Here is a sample ...

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