# Tag Info

13

This is an interesting and not so easy question. Here's my 2 cents: First, you should distinguish between mathematical models for the dynamics of an underlying asset (Black-Scholes, Merton, Heston etc.) and numerical methods designed to calculate financial instruments' prices under given modelling assumptions (lattices, Fourier inversion techniques etc.). ...

11

Along with Gatheral's book, I'd recommend reading Lorenzo Bergomi's "Stochastic Volatility Modelling". The first 2 chapters are available for download on his website. That being said, let me try to give you the basic picture. Below we assume that the equity forward curve $F(0,t)=\Bbb{E}_0^\Bbb{Q}[S_t]$ is given for all $t$ smaller than some relevant ...

10

Here's a research note devoted to pricing of CMS by means of a stochastic volatility model. The authors indicate in the Introduction that an analysis of the coupon structure leads to the conclusion that CMS contracts are particularly sensitive to the asymptotic behavior of implied volatilities for very large strikes. Market CMS rates actually drive the ...

10

Okay just to wind things down here, I think an important clarification is needed if readers might come and seek to a similar solution. The Geometric Brownian Motion (GBM) is a model of asset prices dynamics which is usually given as follows: $$dS_t = \mu S_t dt + \sigma S_t dB_t$$ where $B_t$ is a standard brownian motion which has several important ...

9

To recover the Black-Scholes pricing equation, you should first express the standard normal cdf in terms of its characteristic function analogous to the Heston solution: $$N(x) = \frac{1}{2} - \frac{1}{\pi} \int_0^{\infty} Re [\frac{e^{-i\phi x} f(\phi)}{i\phi}] d\phi$$ where $f(\phi)$ is the characteristic function of the standard normal distribution: $$... 9 The way that I understand your question is that you are looking to fit the market prices of European plain vanilla options of a single maturity and then back out the corresponding implied probability density function. There are multiple ways that you could approach your problem. 1) Modelling the Market Prices The market prices of European plain vanilla ... 9 1. What does it mean by the vol surface is the current view of vol? The local volatility model is calibrated to vanillas prices (and equivalently their implied volatilities), which reflect the market's view of the volatility, in order to use it to use it to price other options that one will hedge with the vanillas. Where a Black-Scholes model (no smile) ... 8 X_t being a stochastic process, one cannot use ordinary calculus to express the differential of a (sufficiently well-behaved) function f of t and X_t. Instead one should turn to Itô's lemma, one of the key results of stochastic calculus, which stipulates (assuming X_t is here a continuous, square integrable stochastic process)$$ df(t,X_t) = \frac{...

7

The SABR model has an overly fat right tail. If you do the CMS replication using cash-settled swaptions you find that you need ridiculously high strikes.

7

[Short answer] No closed-form formula in general. You need to resort to numerical methods. Monte Carlo is preferred by most practitioners but you could also use Finite Difference schemes (and sometimes even Fourier inversion techniques depending on the model used and the instruments to be priced). [Long answer] One usually distinguishes between 2 classes ...

7

Some Notations It's easy to get lost so let's introduce some notations and let $$\sigma : (t, S, K, \tau) \to \sigma(K,\tau; S, t)$$ denote the implied volatility smile prevailing at time $t$ when the spot price is $S_t=S$ for an option with strike level $K$ and time to expiry $\tau=T-t$. From here onward, we drop the $t$ argument to keep notations ...

6

Let $dS_t = \mu_tS_tdt + \sigma_tS_tdW_t$ be the underlying GBM (Geometric Brownian Motion)-like dynamics as in the question. Let $B_t$ a Brownian motion such that $d[B,W]_t = \rho dt$, $\rho\in[-1,1].$ CIR (Cox-Ingersoll-Ross) for $\sigma_t^2$ (when combined with GBM-like underlying dynamics, it is the popular Heston SV model) $$d\sigma_t^2 = \kappa(\... 6 Whenever you use any model to price anything, all you need to do is make sure you model the underlying dynamics that the product you're pricing actually depends on. Any product will be dependent on numerous facets, to varying degrees - this is the same with modelling anything. The modelling that happens in pricing financial derivatives is an integration ... 6 I'm guessing, and correct me if I'm wrong, you want to create a number of possible paths the stock price could follow with the local volatilty given by GARCH depending on the simulated history, or in pseudocode: N <- numberOfPaths T <- numberOfSteps for (i in 1:N) { newSeries <- pastPrices for (t in 1:T) { epsilon <- normrnd(0,1) ... 6 You have to ask yourself what the ultimate purpose of this parameterization is. In their case, they imply the "end-goal is martingale pricing or maximum-likelihood estimation", both of which are ultimately about capturing long-period dynamics rather than intraday or interday behavior. For this reason, the fact that intraday variance may, ahem, vary around ... 6 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 ... 6 This effect is coming from the supply and demand in the options markets. Many portfolio managers want (or need) to buy out of the money put options, and many are willing to sell out of the money call options (thereby funding the purchase of put options). Now, when the market goes down, dealers find themselves short vol and they need to buy options to cover ... 6 In the context of option pricing, "implied volatility" always refers to the equivalent diffusion coefficient in the geometric Brownian motion (GBM) dynamics that is necessary to match an observed European plain vanilla price for a given strike and maturity. When talking about "model implied volatility smile", what is meant is that: You choose some pricing ... 6 The following paper is helpful for understanding the point you raise: Hagan et al.: Managing Smile Risk, January 2002, Wilmott 1:84-108 The main point is given in the paper: [...] the dynamics of the market smile predicted by local vol models is opposite of observed market behavior: when the price of the underlying decreases, local vol models ... 6 Well, what you find is that the introduction of stochastic vol changes the delta of your options. So what does this mean? If the new delta reduces the variance of your hedged portfolio versus the pure local vol model , then it means that the introduction of stochastic vol has resulted in a better description of market dynamics versus the pure local vol ... 5 For pricing, there are a few products whose prices are sensitive to the forward smile and when you compute that with just local vol, it is not realistic. So if you are a seller, you go to the next church and find something that looks kindof reasonable, and that kind of can reconstruct a reasonnable forward smile structure. The game in pricing is to not ... 5 I am going to supply an answer that is quite similar to SRKX's (which is very very good) because I want to discuss in more detail a few important things. First, you cannot use a stochastic volatility model for the SDE that you've provided as that's GBM with constant diffusion. However, based on what you've said it's obvious you wish to model a discretized ... 5 You could read it like this: The typical change in equity value is equal to the typical change in asset value, adjusted for the probability of the assets surviving. Note that the formula is not specific to Merton models, it's also true for regular options and their underlyings. It's just that volatility of option prices isn't typically a concern in "... 5 It's really quite simple. It's just a matter of the fact that we can change measure on the stochastic volatility while not changing the fact that the stock is a martingale. Once we can do this, we have payoffs that have different values under different measures, so the market can't be complete. For clarity, just consider a stock S, a money market account ... 5 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 ... 5 I am not sure if I understood your question correctly but I will try to answer it anyway. If you have a standard normal random vector z \sim N(\mathbb{0},I_n) (where z,0 \in \mathbb{R}^{n\times1} and I_n \in \mathbb{R}^{n\times n} is the identity matrix) and you want to transform it into a multivariate normal x \sim N(\mu,\Sigma) you do it the ... 5 Because vanilla derivatives with European exercise depend only on total variance , not on it's dynamics in time. If you have a simpler model (like interpolation of these total variances from your volatility surface) you don't have as much of unobservable parameters stochastic volatility models have. Having more parameters (which many times would need to be ... 5 I think you did something wrong in translating the input to numerics. As pointed out by dm63 normal vols are quoted in basis points. Using equation A.67a) from the Hagan paper you linked we see (setting \beta = 0)$$\sigma_N(K) = \alpha\frac{\xi}{x(\xi)}\left[1+\frac{2-3\rho^2}{24}\nu^2\tau_{exp}\right] where $\tau_{exp} = 0.25$ in your example and ...

5

If I got this right, you get a SABR model from fitting market implied volatilites (from market price via Black) to SABR volatilites (from SABR parameters via formula above). Then you step back and think the SABR distribution needs improvement because it is not arbitrage free. Instead you use the collocation method to replace it with its projection onto a ...

5

This depends largely on the model as well as the market, so there is no one-size-fits-all approach. Let us take the Stochastic Alpha Beta Rho (SABR) model, which has four parameters, as an example: $\alpha$, the initial instantaneous volatility of an ATM option. This is calibrated frequently and often intra-day due to the importance of ATM options. $\beta$, ...

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