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9

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 $\sigma_I : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ ...


9

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 ...


4

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

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.


3

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 ...


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.


3

There are lots of papers online and here are a few I would suggest math.umn riskworx G. Dimitroff, J. de Kock Nowak, Sibetz I you have matlab there is an step step example to calibrate SABR model. Since it uses the financial toolbox of matlab for a few functions I dont think you can replicate it in any other language. There must be C++ code available ...


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

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 ...


2

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 ...


2

If you believe the process $Y_t$ to be stationary, you can try to profit from it via a mean-reversion strategy or any other way that exploits the stationarity. It doesn't matter whether $Y_t$ is obtained as a cointegrational combination of a few non-stationary processes, or as a linear combination of some processes that are stationary themselves. In the ...


2

If the security has negative correlation with other assets that enjoy attractive risk-free rates, then it can be attractive at a return rate under the risk-free level. It would, of course, never be attractive in a single-security portfolio.


2

Peter Jaeckel wrote a paper just on how to solve this problem: By Implication (July 2006; Wilmott, pages 60-66, November 2006). Probably the most complicated trivial issue in financial mathematics: how to compute Black's implied volatility robustly, simply, efficiently, and fast downloadable from jaeckel.org In my experience the most important thing is to ...


2

You should always think: I buy the one which is to cheap and sell the one that is too expensive and figure it out. The figuring out in this case is noting that: $C\geq 0$ since it will never cost you money The option is strictly better than $S-K$ so has a higher price. Now to your strategy: You buy $C(T_2)$ (the cheap) and sell $C(T_1)$ (the ...


2

CRR is just a numerical approximation to Black--Scholes. Its main use is in getting American option price. There is no real difference other than slight inaccuracy when using it for Europeans. So no it wouldn't do what you ask. Your questions are philosophical. What is the purpose of the model? if you estimate the volatility from a time series then you can ...


2

If you want to estimate volatility from historical data, the only best linear unbiased estimator (BLUE) is $$\sigma=\sqrt{\frac{1}{T-1}\sum_{i=1}^T (r_i-E(r_i))^2}$$ Any other estimator will hence either be biased or not consistent. Another approach could be to estimate volatility via a GARCH model, which has shown good empirical results in the past. It is ...


2

All option pricing formulas except this one and this one use some sort of historical volatility . I can't see how you can use the Black Sholes framework and not use some sort of historical volatility uses an order book uses geometric shapes and volume


2

In my mind volatility (SD) of a stock and implied volatility (IV) are two quite different things: volatility is usually measured backwards looking. The common methods (empirical, GARCH, ..) look into the past. Measuring the risk of owning the stock in the future is often based on these backwards looking observations. We try to measure risk in the real ...


2

Based on the example you gave, it seems that indeed your inputs are inconsistent. The intrinsic call value is $S-e^{-rT}K = 286.52355\dots$, which is higher than the market value, implying that there exists an arbitrage. Instead, one of your inputs is probably wrong. Even if the interest rate is set to $0$, the intrinsic call value is still above your bid, ...


1

(1) No, the stochastic differential equation for Heston model does not have an explicit solution. What does exist is an explicit formula for the Fourier transform of a call option price. See e.g. http://www.zeliade.com/whitepapers/zwp-0004.pdf for a decent survey. (2) Yes, implied vol always exists. You can check that the Black-Scholes price of an option ...


1

For Black-Scholes, $\Delta_C=\partial_{S} C=N(d_1)$, $d_1= \frac{\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)}{\sigma\sqrt{T - t}}$ You may fit the volatility $\sigma$ to this term by $$\Delta_C({\hat{\sigma}})=0.25$$Note that $\Delta_P=1-\Delta_C$ by Put-Call-Parity.


1

You need to use log of prices, because log of returns are normally distributed. So or where x is return- $$ x=-\frac{1}{\tau} ln(\frac{S_{t+\tau}}{S_{t}}) $$ The annualized standard deviation can be scaled as +/-$ n\frac{\sigma}{\sqrt{\tau}} $ where n is your multiple. You can either ignore or estimate drift. or look at it another way, S refers to the index ...


1

The skew plays an important part for pricing binaries. In S&P the VIX increases on declines and decreases on rises. We can explain a part of the premium by assuming the Black-Schole Call option captures the underlying volatility. Let us represent call option as $C(K,\sigma)$ and binary $V_{Binary}(K,\sigma)$ Then we can write $$ V_{Binary}(K,\sigma) = ...


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

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 $$ ...



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