9 votes
Accepted

Show that $\frac{\partial c(t))}{\partial \sigma^2 }>0 \text{ if and only if } S(t)<Xe^{-r(r+\frac{1}{2} \sigma^2 )(T-t)}.$

Hints: You know the vega of a digital call option formula: $V=-\frac{e^{-r(T-t)}}{\sigma} d_1 n\left(d_2\right)$ Where n is the standard normal density, which is positive. Sigma and exponential are ...
Magic is in the chain's user avatar
6 votes
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Binary Option in B-S model - technical question

$I_{\{S_{T}-K>0\}}$ is NOT independent of $\mathcal{F}_{t}$, since \begin{align*} S_T=S_t \, e^{(r-\frac{1}{2}\sigma^2)(T-t) + \sigma (W_T^*-W_t^*)}, \end{align*} where $S_t \in \mathcal{F}_t$, ...
Gordon's user avatar
  • 21.1k
6 votes
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Black-Scholes pricing of binary options

$S_T$ is log-normal distributed and therefore skewed. In particular $E[S_T]=S=110$ (no drift), but $Q(S_T>S)<Q(S_T<S)$. For example if S=K=100 you don't get a value of 0.5 as you might expect,...
Andrew's user avatar
  • 609
5 votes

Binary Option in B-S model - technical question

You can also infer the value of your binary option from the value of a European call option with the same strike and time-to-maturity by going long on a call with strike $K$ and time-to-maturity $\tau$...
BS.'s user avatar
  • 165
5 votes

Derivation of the formulas for the values of European asset-or-nothing and cash-or-nothing options

You can derive these formulae by tweaking the black scholes derivation. If you are using PDE method, you will use different boundary conditions. If you are using integration over the risk neutral ...
dm63's user avatar
  • 17k
5 votes
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Flaw in the following argument with Binary Options and Skew

Let $f_0(S_T) =f(S_T|S_0)$ be the risk-neutral PDF for the underlying asset price at time $T$ (conditional on the price $S_0$ at present time $t=0$). The probability that the price is above a strike ...
RRL's user avatar
  • 3,630
4 votes
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Pricing for an Odd Type of Asset or Nothing Option

The price is, under the risk-neutral measure, $$ P_t = e^{-r(T-t)}\mathbb E[S_T^1 \mathbb 1(S_T^2\le K)\mid \mathcal F_t].$$ Since the risk-neutral asset processes are independent geometric brownian ...
spaceisdarkgreen's user avatar
4 votes

Using a call-spread to hedge a digital option

There are a few extra things to consider here where you'll get a different answer if you ask a quant or a trader. If we have a european digital that pays \$1 if the underlying is above 120 ($S_0 = ...
will's user avatar
  • 2,561
4 votes
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Why represent a digital payoff as a call spread

An important practical reason is for hedging purposes. Consider a situation where the option is very close to maturity and the rate $R$ is fluctuating around the strike $K$, such that the option is ...
Daneel Olivaw's user avatar
4 votes

Why do Binary Options have a bad reputation?

Binary options are not traded on standard options exchanges and cleared OCC. Instead you are trading vs individual houses/web sites. These sites have been accused on not paying, freezing funds, ...
JoshK's user avatar
  • 2,613
4 votes

Why do Binary Options have a bad reputation?

Adding to the answer of @JoshK, many of the firms offering binary options often operate their scams by advertising their trading platforms on sites like Facebook. Their trading platform and website ...
Pleb's user avatar
  • 4,231
4 votes
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The greeks, vanillas and digitals

There is no difference in real world settings. Market practitioners usually always price a digital as a tight call spread to capture skewness. For example, setting strikes at $$𝐾± = 𝐾 ±1/2𝑑𝐾,$$ in ...
AKdemy's user avatar
  • 9,079
3 votes

Flaw in the following argument with Binary Options and Skew

Here is an intuitive explanation: you conclude that there is more chance of expiring otm than itm if there is skew but that isn’t correct. The atm volatility is unchanged vs the flat vol case, and atm ...
Ivan's user avatar
  • 1,376
3 votes
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How do binary options broker hedge themselves against losses?

If we are talking about brokers who making markets for https://en.wikipedia.org/wiki/Binary_option than I would guess that they aren't hedging at all. It's very common that maturities are in a ...
DataAdventurer's user avatar
3 votes

Barrier digital options and pricing

Disclaimer: This answer derives the prices of two different binary options within the Black/Scholes framework. Note that this is not an appropriate valuation model to use for non-European contracts in ...
LocalVolatility's user avatar
3 votes

Derivation of the formulas for the values of European asset-or-nothing and cash-or-nothing options

To add a bit to Will Gu's answer: Compute $\mathbb{E} \left[ \left. S_T \right| S_T > K \right]$ using the fact that $S_T$ is lognormally distributed with mean $ln(S_0) + (r - \sigma^2/2)T$ and ...
X Y's user avatar
  • 61
3 votes

Binary Option in B-S model - technical question

I think that I found correct answer to my question. We have the following theorem: Theorem. If $X$ is independent of $\mathcal{G}$, $Y$ is $\mathcal{G}$ - measurable and $\phi(x,y)$ is bounded ...
Leon's user avatar
  • 352
3 votes

Black-Scholes pricing of binary options

Just adding a few graphical explanations to Andrew's answer because I think thez could help in understanding the result. The Black Scholes price of a Digital corresponds to the discounted probability ...
AKdemy's user avatar
  • 9,079
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 ...
byouness's user avatar
  • 2,210
3 votes
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Price of Binary Option using Explicit Finite Difference Method not matching with closed Form solution

The price close to 0.93 is correct, here is a reimplementation of both FD and analytic using QuantLib: ...
StackG's user avatar
  • 3,016
3 votes

Intuitive explanation for the value of a binary option being lower when volatility skew is positive?

You can replicate the payout of a binary with a put spread with strike prices which are very close to one another. Higher skew makes the further out of the money put more expensive, which makes the ...
cholzer68's user avatar
3 votes

Digital and binary put/call options

Generally, I would say it is a bit difficult to look for put-call parity of something you do not know what it is in the first place. To give more intuition to what fesman wrote, look at your C and P ...
AKdemy's user avatar
  • 9,079
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. ...
Borun Chowdhury's user avatar
2 votes

How are bets and high-risk investments conceptually different, and how does this apply to binary options?

The main difference is that with a binary option you are betting on a real economic risk, that exists independently of the bet. (For example, even if options did not exist, if stock prices go down ...
Alex C's user avatar
  • 9,372
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 ...
mbison's user avatar
  • 1,568
2 votes
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Justification for Binary Option's Infinite Delta?

Your question is essentially the same as this. The approximation \begin{equation} V_{t + 1} \approx V_t + \Delta_t \left( S_{t + 1} - S_t \right) \end{equation} is only accurate when $S_{t + 1} - ...
LocalVolatility's user avatar
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 \}...
Gordon's user avatar
  • 21.1k
2 votes
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Where can I see the bid stack for FX?

The FX price bounce you see near expiration is generally because market makers are continually hedging their positions. When a market maker sells a binary option, she is left with a short gamma ...
Ramanujam Narayanan's user avatar
2 votes
Accepted

Binary option analytical formula

As you say, you simply differentiate with respect to $K$. Assuming your binary's maturity is $T$, note that in a Black-Scholes framework with constant risk-free rate $r$, by the Breeden-Litzenberger ...
Daneel Olivaw's user avatar
2 votes

Payoff of an odd indicator of one stock being greater than another

The price is $e^{-r(T-t)} \mathbb{P}(S_{T}^{1} > S_{T}^2) =e^{-r(T-t)} \mathbb{P}(S_{T}^{1} / S_{T}^2 >1) $ The crucial point is that the ratio of two log-normals is log-normal even when they ...
Mark Joshi's user avatar
  • 6,913

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