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24 votes
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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 ...
Quantuple's user avatar
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14 votes
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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 ...
Daneel Olivaw's user avatar
9 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 ...
AlRacoon's user avatar
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9 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 ...
OGC's user avatar
  • 271
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 ...
HowtoETF101's user avatar
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*} ...
Gordon's user avatar
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7 votes
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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 ...
LocalVolatility's user avatar
7 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 ...
Bram's user avatar
  • 812
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
eltigrechino's user avatar
6 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 ...
Daneel Olivaw's user avatar
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.
dm63's user avatar
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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 ...
Soumirai's user avatar
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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 ...
Kermittfrog's user avatar
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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 ...
julien's user avatar
  • 51
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}...
Quantuple's user avatar
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5 votes
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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: $...
Daneel Olivaw's user avatar
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 ...
Ivan's user avatar
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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 ...
Alex C's user avatar
  • 9,382
5 votes

Gamma PnL Formula and Break-Even volatility

Good question! The answer to this is no. Let us work through a simple example to see why. Assume that the Gamma is $10$ and that the break-even move is $1$. For simplicity, also assume that, these are ...
Misha Wolynski's user avatar
5 votes

Why does volatility increase the expense of delta-hedging?

The key here is to observe that the volatility at the time the option is written is not exactly equal to the volatility that the markets actually experience during the option's lifetime. The seller ...
Brian B's user avatar
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5 votes
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How do we hedge option vega practically?

If you are a market maker, your primary Vega hedge is to sell Vega to other clients. You do this by being the best offered side price in the market, so you will attract the next piece of business. ...
dm63's user avatar
  • 17.1k
5 votes

Optimal delta-hedging frequency when gamma scalping

The model I quite like as a base-case/rule of thumb is the Hoggard, Whalley, and Wilmott (1994) model. Assuming GBM - the number of shares, $N$, per interval is: $$N = Δ(S+dS,t+dt)- Δ(S,t)≈ Γ*dS$$ ...
Newquant's user avatar
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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 ...
amdopt's user avatar
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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 ...
LocalVolatility's user avatar
4 votes

Derivation of BS PDE problem using Delta hedging

this is the d on the delta problem. (In my book concepts I discuss this. Sections 5.6 and 5.7 ) Essentially we hold delta constant across time steps and so ignore the d on the delta because there is ...
Mark Joshi's user avatar
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4 votes
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Delta Hedging: Clarification example of the book "Hull, Options, Futures, and Other Derivatives"

We denote by $C(S_0, K)$ the price for a call option with payoff $(S_T-K)^+$ at the option maturity $T.$ Here $S_0=100$ is the spot stock price. Generally, \begin{align*} C(S_0, K) \ne (S_0-K)^+. \...
Gordon's user avatar
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4 votes
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Example of delta one products

As indicated by the name, delta one products have a delta of exactly 1 (at least theoretically) with respect to the underlying; moreover, AFAIK the delta has to be constant, i.e. a product with ...
Daneel Olivaw's user avatar
4 votes

Dynamic Delta Hedging And a Self Financing Portfolio

for a self financing portfolio, you have a holding in stocks and one in bonds. If we want to do a hedging simulation, at the start of each step, work out the total value of the hedger's holding (...
Mark Joshi's user avatar
  • 6,973
4 votes

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

There are serious issues with how this graph is drawn, which impede understanding. The y axis is unlabeled and should be labeled "profits" or $\pi$ on a hedged position. The other axis should be ...
Alex C's user avatar
  • 9,382
4 votes
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Interpertation of delta hedge error in Black Scholes

Let's derive the proof. Consider you are short an option and long its self-financing delta hedge. You get the portfolio $\Pi$ whose $t$-value verifies $$ \Pi_t = \underbrace{- V_t}_{\text{Short option}...
Quantuple's user avatar
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