# Tag Info

8

There is no "plain Black Scholes implied surface" because implied volatilities come from options market prices (calls and put). If you had a whole continuum of call prices $C : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$, $(T,K) \mapsto C(T,K)$ you would get a implied volatility function $\sigma_I : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ ...

8

Setting aside, that it's not pure riskless arbitrage, but rather statistical arbitrage: You can extract the profit by performing continuous delta hedging. If you constantly adjust your hedge position you gain/lose money by delta hedging. Being long option (gamma long), you sell at higher prices and buy at lower ones. Over the course of time you realize ...

5

My try to answer this question with some other questions: Is the BS model right? No. Is it useful: yes. Taking a traded price and the BS Model there is only one input factor that is not given by the market: the implied volatility. It is a measure to compare options across time and strike. Are there better models? yes. Those that you mention: The local vol ...

5

First of all, may I point out two big misperceptions that you may have: Implied Volatility (IV) is the input to any vanilla option pricing model (not just Black Scholes (BS) that impacts the pricing the most. You can verify this by flipping through the different risk exposures (greeks and higher order sensitivities) and study mean volatilities in such risk ...

4

In that white paper itself they quote where it came from: “More than you ever wanted to know about volatility swaps” by Kresimir Demeterfi, Emanuel Derman, Michael Kamal and Joseph Zou, Goldman Sachs Quantitative Strategies Research Notes, March 1999. This is a classic article which you should definitely read if you are trading volatility. While there might ...

4

You can see concavity in mean-reverting underlying assets where the option tenor is comparable to the characteristic reversion time of the asset. For a geometric brownian motion, all underlying prices are possible, so any mean reversion or other limitation on large changes that might occur in reality would ultimately appear as a skinny tail and negative ...

3

For an individual firm, a theoretical model of the capital structure was developed by Robert Merton in 1974. The simplest form of this model assumes the firm has zero-coupon debt maturing at some future time $T$. Default is defined as the condition where the value of the firm's assets fall below the outstanding debt. The firm equity is viewed as a call ...

3

VG belongs in the family of variance-mean mixture models. Given a horizon $T$ the distribution of log-returns $f$ is a mixture of Gaussians $f_G$ with randomised mean and variance. The randomisation density is $g$ and its mean and variance increase with $T$. For the VG process this randomised factor is Gamma-distributed. More concretely, denote with ...

3

The method described in Hallerbach (2004) always worked well for me. We derive an estimator for Black-Scholes-Merton implied volatility that, when compared to the familiar Corrado & Miller [JBaF, 1996] estimator, has substantially higher approximation accuracy and extends over a wider region of moneyness.

2

Negative excess kurtosis leads to a concave vol smile. By the way, no-arbitrage arguments are of theoretical nature: implied volatilities can exhibit no-arbitrate violations in the theoretical sense for extended periods given that such arbitrate cannot be traded due to other factors, such as liquidity, spreads, transaction related costs...not saying this ...

2

The lower bound is not just a BS-specific bound. It is a no-arbitrage bound and so if the price is lower than this, you have an arbitrage opportunity (some good explanation here). It doesn't mean it is present in the market necessarily, because mid price is not necessarily the price you can trade and when you take spread into account this is likely to go ...

2

So we have the identity $$g(S,\sigma, t, C,C_t,C_S,...)=g(S, t,\sigma, V,V_t,V_S,...)$$ where $S$, $\sigma$, and $t$ are independent variables and $V=V(S,\sigma,t)$, $C=C(S,\sigma,t)$ are some unknown functions. But we can also treat the above identity formally and assume that the functions $C,C_t,C_S,...,V,V_t,V_S,...$ are themselves independent ...

2

Calendar spreads have a number of disadvantages for trading Vega: Vega in different months are generally not additive, some traders use root-time-Vega but it does not remove the additional risk. You are trading time spread not just volatility, so be careful Calendar spreads are affected by dividends and rate changes - another source of risk. A ...

2

No, there is an upper limit to a binary option's value, based on the interest rate and how much of the distribution can be packed under the payoff region. Essentially $$C = e^{-rT} \int_K^\infty \psi(S_T) dS_T$$ for calls and $$P = e^{-rT} \int_0^K \psi(S_T) dS_T$$ for puts. Neither of the integrals can ever exceed 1.0 and often they take on a ...

2

There are no free resources that provide historical bid and ask prices for option chains. You should consider buying them from a data provider. However, you can start accumulating data from yahoo using the getOptionChain function of the {quantmod} R package.

2

we use implied vol for similar reasons why we use duration. we know that security prices are not linear functions of rates, yet we look at the duration, because it gives us an idea of sensitivity to a rate. implied vol gives you a measure of volatility, it doesn't perfectly describe it, but as long as we know this, it's still a valuable metric.

2

Are you sure you are using the correct pricing formula. For a binary (digital) call that pays $1$, the simple Black-Scholes price at time $t=0$ is $$C_d = e^{-rT}N(d_2)$$ $$d_2 = \frac{\text{ln}(F/K) - \frac1{2}\sigma^2T}{\sigma \sqrt{T}}$$ where $N$ is the standard normal distribution function, $F=Se^{(r-q)T}$ is the forward index price, $S$ is the spot ...

1

RRL's answer is entirely correct in terms of the theoretical reason underpinning the relationship between equity IV and CDS spreads. "CDS spreads are not “pure” default risk compensation" - no they are not since the ISDA Quoted Spreads assume a homogeneous Poisson process (implying that instantaneous default risk is a constant over the life of a contract) ...

1

The relationship between volatility and CDS is very interesting. Volatility in finance is synonym of risk. There are many aspects of volatility. There are 2 primary ways to find CDS premium, one is using structural model and the other is reduced form or intensity based model. Structural models use equity valuation, outstanding debt and equity volatility to ...

1

If you want to calculate the change of a greek, lets say Delta, from a change in the volatility, you would need Vanna: $$Vanna=\partial_\sigma\Delta=\partial_\sigma\partial_S C$$, which under Black-Scholes becomes: $$Vanna=\left(\sqrt{T-t+\frac{1}{\sigma}}\right)\phi\left(d_1\right)$$ where $\phi\left(\cdot\right)$ is the standardnormal density and $$... 1 What if the actual volatility during the following period is lower, so my bet was correct, but the implied volatility stays the same for the whole period anyway? If under those circumstances I liquidate the position, wouldn't the profit be 0? I think I know where your confusion comes from. 1) this isn't arb - it is not a risk free strategy ... 1 They are not referring to any implied volatility but actual volatility, i.e. statistical standard deviation. The price volatility is the annualized standard deviation of bond price changes and the yield volatility the annualized standard deviation of bond yield changes. These quantities are usually estimated using a historical estimator. If you have n ... 1 I'll try to anwswer too. 1) You seem to try to interpret implied volatility as having a statistical nature. In fact implied volatility is nothing but (today's) market prices except that you look at them through Black and Scholes "glasses". Why the Black-Scholes model? There are many reasons for that. this is the simplest sensible model: basically you ... 1 I am not a trader and will pass on question 1 for now. To answer your second question. You want you model to be able to reproduce market prices of certain vanilla instruments. This way you achieve consistency with the market. Thus if you want to price an exotict call option you will calibrate your model to liquid call option prices. If the option has more ... 1 Ofcourse, It is always possible to find the implied volatility. The value of binary call is$$ {e}^{-r(T-t)}N(d_2) $$where$$ d_2=\frac{ln(\frac{S}{E})+(r-D-\frac{\sigma^2}{2})\tau}{\sigma\sqrt\tau}  Now, there is nothing that can ever ever stop the newton raphson method to find a $\sigma$ for which the value of binary call is given and is positive ...

1

www.livevol.com seems to offer good services but it seems pretty expensive. It all depends on your needs. It is probably a good service for semi-professional option trader.

1

You can construct delta and gamma neutral option portfolio, but: It won't generally stay neutral forever, so you would still have to constantly rebalance it by trading additional options (thus paying more transaction costs and creating mess in the portofolio). Anything will break the neutrality - underlying move, time passage, implied volatility change ...

1

Most likely you are looking at bid prices which are lower that fair (theoretical) price. It is very common that bid price of an ITM option is below the lower bound as bid-ask spreads are wide. The IV of ITM call at theoretical price should match IV of OTM put at corresponding strike. If this does not happen then check your forward price, rates and dividends. ...

1

One is computed from historical stock time-series, that is, from observed past, the other is computed from traded option prices (prices of bets made on the stock with payoff at a future time), that is, from a view on the future paid for with cash. They are both equally important and useful (if one has enough data to compute them). Loosely speaking, ...

1

I think there are 2 approaches being a bit mixed up here. You can analyze the option market by looking at implied volatilities and apply Black-Scholes (BS), thus assuming that log-returns follow a Gaussian distribution. Implied volatilies are the parameters that bring together BS and market prices. Then you will observe a pattern of implied volatilies for ...

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