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14 votes
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Barrier option (autocallable) Vega profile

You have a multidimensional problem - there isn't an answer of "this is what the greeks look like" for all cases, because it depends on the various levels of the different parameters. For example, if ...
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8 votes

Delta hedging on Barrier/Digital Options

You're right that the "real" greeks of a digital option are unwieldy, e.g. delta is zero everywhere except at the barrier where it is an impulse. So sell-side trading desks model/price digital options ...
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6 votes
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Pricing 'Down and In' claims

Let \begin{align*} \tau = \inf\{t: t \ge 0, S_t \le L \}. \end{align*} Then the down-out-call option has payoff \begin{align*} (S_T-K, 0)^+\pmb{1}_{\tau >T}, \end{align*} and the down-out version ...
  • 20.5k
6 votes

Delta hedging on Barrier/Digital Options

I nearly agree with @phlsmk's answer, but with some small differences. First off, the delta of a digital is not "zero everywhere except at the barrier where it is an impulse". This is what it is at $...
  • 2,446
6 votes
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Pricing a double barrier option using Monte Carlo (C++ & Python code included)

Here are at least three mistakes in your code: p += s0 * exp(...) should be p *= exp(...). Your volatility and rates are per ...
6 votes

How to hedge a perpetual barrier option?

Presumably the option can be exercised for intrinsic at any point. Note the interviewer asked for a static hedge using the stock, not a dynamic hedge. Hence you must find a buy and hold portfolio that ...
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5 votes
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Conditional probability of Brownian motion (with drift and scaling) hitting barrier

For part 1 of your question, the short answer is no, calculating conditional density is a looong way of doing it. Possible but not the easiest. Here is the sketch for a shorter version. We note that $(...
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4 votes
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Price of a double barrier option

As specified I will assume your option is perpetual; I will also assume that it is written on an asset whose price $(S_t)_{t \geq 0}$ follows a Geometric Brownian Motion (GBM) with drift coefficient $...
4 votes

How to hedge a barrier option with vanilla options?

there are a number of ways to do this. You do have to make some modelling assumptions, however. eg continuity, BS model holds, or log stock price process is independent of level. The most common way ...
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4 votes
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What is the probability that a OU process hits an upper barrier U before a lower barrier L?

Assuming $\theta>0$ (take $\tilde{X}=\mu-X$ if it is not the case) Let us denote $\text{erfi}(x)$ the imaginary error function Let us denote $\tau_L$,resp.$\tau_U$ the hitting time of $L$resp.$U$ ...
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4 votes
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PDE of barrier and lookback options

The difference is that the barrier option is weakly path dependent while the lookback option is strongly path dependent. In case of a knock-out barrier option, conditional on the option being alive ...
4 votes

For single barrier options, why is a plot of gamma so scattered compared to other greeks?

Your gamma seems to be "quantized" like if your calculation happens to be at the machine limit in term of precision. Maybe you aren't using a "dS" large enough if you compute ...
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3 votes
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How to use reflection principle to solve the analytic solution of double barrier-out-call

No, the pricing of a double barrier knock-out option cannot be decomposed into single barrier options. Here are a few references that apply the method of images to the valuation of double barrier ...
3 votes

First passage probability formula

Their formula looks correct. As is usually the case, there are multi ways to derive this result. I will outline two of them here. Reflection Principle & Measure Change The solution to the risk-...
3 votes

Valuation of barrier options in Jump diffusion model

The error is, you are not storing the random numbers for the same path at the end: ...
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3 votes
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Importance sampling for barrier option like pricing by Monte carlo

Since there is a closed form in the BS case for continuous barrier options, you probably won't find a huge amount of work on this since it's not needed. In the discrete case, I did a paper with Tang: ...
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3 votes
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how to price barrier option under local vol model using QuantLib

From a cursory look, the FdBlackScholesBarrierEngine seems to do what you want; when the localVol parameter is set to ...
3 votes
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Fair value of a binary cash-or-nothing option with a barrier

As Daneel mentioned in his comment, you can't simply split your expectation of product into a product of two expecations as the two quantities are far from being independent... Now, to answer your ...
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3 votes

Graph of a down-and-in barrier option

If you put some numbers into down-in/out barrier call option formulae that can be found in many books, you will see that the down-in curve is not symmetric. It just looks like it in that plot. Below ...
2 votes

Counting random paths

As I mentioned above, I am not sure what the variable $r$ is. If we ignore that, or assume the questioner wanted to say its the risk free interest rate, then it has no effect on the number of paths. ...
2 votes

Importance sampling for barrier option like pricing by Monte carlo

I'd recommend M. Joshi and T. Leung "Using Monte Carlo simulation and importance sampling to rapidly obtain jump-diffusion prices of continuous barrier options". Though it assumes jump-diffusion ...
2 votes

Why is H always* the letter used to describe the level of a barrier?

I do not have any reference, but I think $H$ is for hitting.
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2 votes

The PDE of the probability hitting the barrier before T

May be I have overlooked something, but I believe that \begin{align*} Q(t, S) = \mathbb{P}\left(\tau_{B} \le T \mid \mathcal{F}_t\right). \end{align*} Then $\{Q(t, S), \, 0<t < T\}$ is a ...
  • 20.5k
2 votes
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Pricing Secured Barrier Call 2

Formally, let \begin{align*} \tau = \inf\{t: t \ge 0, S_t \ge 50 \}. \end{align*} Then \begin{align*} \text{Payoff} &= \left(S(31)-33 \right)^+\pmb{1}_{\tau >31} + 50\times \pmb{1}_{\tau \le 31}...
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2 votes
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Pricing Secured Barrier Call

The goal of this exercise is to replicate the payoff of the Secured Barrier Call by a linear combination of the known products: European up-out call (cost 12), digital strike 33 (cost 0.73) and ...
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2 votes
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Double knockout binary pricing?

Assume that $H_1 < S_0 < H_2$. let \begin{align*} \tau_1 = \inf\{t: \, t>0 \text{ and } S_t \le H_1 \}, \end{align*} and \begin{align*} \tau_2 = \inf\{t: \, t>0 \text{ and } S_t \ge H_2 \}...
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2 votes
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Barrier option on a basket with arbitrary stochastic process

There are a few issues that need to be separated here. Issue "zero" is whether your MC is able to correctly represent the dynamics you've chosen for your assets. If you implement your MC properly, ...
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2 votes
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Price Down and In Barrier Option Using Local Vol and Monte Carlo

For the first question, you can just plug in t for T and S for K: $\sigma^2 \left(t, S \right)=\left. \sigma^2 \left(T,K\right) \right|_{T=t,K=S}$ For the Monte Carlo part, the barrier would apply ...
2 votes

What is the go-to method for numerical pricing of discrete barriers?

The ‘classical’ would be PDE based, say Crank Nicolson with Rannacher time marching for local vol based approach, and ADI scheme for Stochastic local vol.
2 votes
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Deltas on Barrier options vs Vanilla options

Standard call options are trivially more expensive than up/down and out call options. However, for high strikes, down and out options will very likely never be knocked out, therefore their prices ...
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