17
votes
Accepted
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 ...
- 2,591
9
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 ...
- 725
7
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 $...
- 2,591
6
votes
Accepted
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 ...
- 6,074
6
votes
Accepted
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 ...
- 21.2k
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 ...
- 1,406
5
votes
Accepted
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 $(...
- 940
4
votes
Accepted
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 $...
- 8,169
4
votes
Accepted
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 ...
- 6,074
4
votes
Accepted
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.$U$ ...
- 2,442
4
votes
Accepted
Estimating the knockout probability of a discretely observed autocall note
There is a closed-form formula for the probability $\mathbb{P}(\tau = t_i)$.
First, we remind that
$$S_t=S_0\cdot \exp\left(\left(\mu-\frac{1}{2}\sigma^2 \right)t+\sigma W_t \right)
$$
For $i=1$, it'...
- 1,033
3
votes
Accepted
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 ...
- 7,808
3
votes
Accepted
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 ...
- 2,230
3
votes
Accepted
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 ...
- 6,074
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-...
- 6,074
3
votes
Accepted
Graph of a down-and-in barrier option
Intuitively, underlying call keeps losing value as the spot goes down, but the barrier option value (which starts at almost nothing for high spot) keeps growing as the spot approaches the barrier ...
- 5,113
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 ...
- 241
3
votes
Accepted
Is the moneyness of a barrier option based on the strike value or the barrier when mapping to a volatility surface?
If your barrier is american and your market has any sort of volatility skew then trying to map some sort of moneyness measure to the vol surface will almost certainly fail. That is due to the fact ...
- 1,080
3
votes
Vega hedge of a barrier option
Too long for a comment. I find Bergomi's sentence vague, so here follows an equally imprecise attempt at an answer.
A claim that can be statically replicated in a model-free manner is in fact immune ...
- 2,153
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 ...
- 21.2k
2
votes
Accepted
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 ...
- 1,578
2
votes
Accepted
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}...
- 21.2k
2
votes
Accepted
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 \}...
- 21.2k
2
votes
Accepted
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, ...
- 1,406
2
votes
Accepted
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 ...
- 6,222
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.
- 6,222
2
votes
Accepted
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 ...
- 553
2
votes
Deltas on Barrier options vs Vanilla options
for an intuitive answer,
if we start with a vanilla call as our base, then with an up & out call, we would like the underlying to go up in price yes. But as the price increases, we also increase ...
- 2,591
2
votes
Accepted
Sample path simulation using two random variables
When you simulate a sample path of a standard Brownian motion, you are generating a sequence $(B_t)_{t \in \mathbb{\Pi}}$ where $\mathbb{\Pi} := \{t_0, ..., t_n\}$ is your time partition. You can view ...
- 2,506
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