23
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 ...
- 14.5k
18
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 ...
- 2,218
14
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 ...
- 7,999
9
votes
Accepted
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
(...
- 715
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 ...
- 101
8
votes
Accepted
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
Accepted
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 ...
- 5,662
7
votes
Accepted
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}...
- 3,936
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 ...
- 5,959
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 ...
- 241
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
- 341
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 ...
- 16.6k
6
votes
Accepted
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 ...
- 7,999
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 ...
- 802
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
Accepted
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 ...
- 624
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 ...
- 6,435
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 ...
- 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}...
- 14.5k
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:
$...
- 7,999
5
votes
Accepted
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 ...
- 1,356
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 ...
- 9,332
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 ...
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 ...
- 14.7k
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:
$$N = Δ(S+dS,t+dt)- Δ(S,t)≈ Γ*dS$$
...
- 749
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
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 ...
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4
votes
Accepted
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 ...
- 5,959
4
votes
Accepted
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 ...
- 7,999
4
votes
Accepted
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)^+.
\...
- 21k
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