en>Kvng: Reverted good faith edits by MarcBoyerONERA (talk): Content probably better suited to Generalized processor sharing. (TW)

2014-01-11T20:14:05Z

Reverted good faith edits by MarcBoyerONERA (talk): Content probably better suited to Generalized processor sharing. (TW)

← Older revision		Revision as of 22:14, 11 January 2014
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	~~Hi there~~. ~~Let me start~~ by ~~introducing~~ the ~~writer~~, ~~her title~~ is ~~Sophia Boon but she never really favored~~ that ~~name~~. ~~Office supervising~~ is ~~my profession~~. ~~I am really fond~~ of ~~handwriting but I~~ can'~~t make~~ it ~~my profession really. Mississippi~~ is ~~exactly~~ where ~~her house~~ is ~~but her husband wants them to move~~.<br><br>~~Take a look at my blog~~ - ~~psychic love readings~~ ([http://~~cartoonkorea~~.com/~~ce002~~/~~1093612 More hints~~])		In [[mathematical finance]], the '''SABR''' model is a [[stochastic volatility]] model, which attempts to capture the [[volatility smile]] in derivatives markets. The name stands for "[[stochastic]] [[alpha]], [[beta]], [[rho]]", referring to the parameters of the model. The '''SABR''' model is widely used by practitioners in the financial industry, especially in the [[interest rate derivative]] markets. It was developed by [[Patrick Hagan]], [[Deep Kumar]], [[Andrew Lesniewski]], and [[Diana Woodward]].

			== Dynamics ==

			The '''SABR''' model describes a single forward <math>F</math>, such as a [[LIBOR]] [[forward rate]], a forward swap rate, or a forward stock price. The volatility of the forward <math>F</math> is described by a parameter <math>\sigma</math>. '''SABR''' is a dynamic model in which both <math>F</math> and <math>\sigma</math> are represented by stochastic state variables whose time evolution is given by the following system of [[stochastic differential equations]]:

			:<math>dF_t=\sigma_t F^\beta_t\, dW_t,</math>

			:<math>d\sigma_t=\alpha\sigma^{}_t\, dZ_t,</math>

			with the prescribed time zero (currently observed) values <math>F_0</math> and <math>\sigma_0</math>. Here, <math>W_t</math> and <math>Z_t</math> are two correlated [[Wiener process\|Wiener processes]] with correlation coefficient <math>-1<\rho<1</math>. The constant parameters <math>\beta,\;\alpha</math> satisfy the conditions <math>0\leq\beta\leq 1,\;\alpha\geq 0</math>.

			The above dynamics is a stochastic version of the [[Constant Elasticity of Variance\|'''CEV''' model]] with the ''skewness'' parameter <math>\beta</math>: in fact, it reduces to the '''CEV''' model if <math>\alpha=0</math> The parameter <math>\alpha</math> is often referred to as the ''volvol'', and its meaning is that of the lognormal volatility of the volatility parameter <math>\sigma</math>.

			== Asymptotic solution ==

			We consider a [[European option]] (say, a call) on the forward <math>F</math> struck at <math>K</math>, which expires <math>T</math> years from now. The value of this option is equal to the suitably discounted expected value of the payoff <math>\max\left(F_T-K,\;0\right)</math> under the probability distribution of the process <math>F_t</math>.

			Except for the special cases of <math>\beta=0</math> and <math>\beta=1</math>, no closed form expression for this probability distribution is known. The general case can be solved approximately by means of an asymptotic expansion in the parameter <math>\varepsilon=T\alpha^2</math>. Under typical market conditions, this parameter is small and the approximate solution is actually quite accurate. Also significantly, this solution has a rather simple functional form, is very easy to implement in computer code, and lends itself well to risk management of large portfolios of options in real time.

			It is convenient to express the solution in terms of the [[implied volatility]] of the option. Namely, we force the SABR model price of the option into the form of the [[Black model]] valuation formula. Then the implied volatility, which is the value of the lognormal volatility parameter in Black's model that forces it to match the SABR price, is approximately given by:

			:<math>
			\sigma_{\text{impl}}=\alpha\;
			\frac{\log\left(F_0/K\right)}{D\left(\zeta\right)}\;
			\left\{1+\left[\frac{2\gamma_2-\gamma_1^2+1/F_{\text{mid}}^2}{24}\;\left(\frac{\sigma_0 C\left(F_{\text{mid}}\right)}{\alpha}\right)^2+\frac{\rho\gamma_1}{4}\;\frac{\sigma_0 C\left(F_{\text{mid}}\right)}{\alpha}+\frac{2-3\rho^2}{24}
			\right]\varepsilon\right\},
			</math>

			where, for clarity, we have set <math>C\left(F\right)=F^\beta</math>. The value <math>F_{\text{mid}}</math> denotes a conveniently chosen midpoint between <math>F_0</math> and <math>K</math> (such as the geometric average <math>\sqrt{F_0 K}</math> or the arithmetic average <math>\left(F_0+K\right)/2</math>). We have also set

			:<math>
			\zeta=\frac{\alpha}{\sigma_0}\;\int_K^{F_0}\frac{dx}{C\left(x\right)}
			=\frac{\alpha}{\sigma_0\left(1-\beta\right)}\;\left(F_0^{1-\beta}-K^{1-\beta}\right),
			</math>

			and

			:<math>
			\gamma_1=\frac{C'\left(F_{\text{mid}}\right)}{C\left(F_{\text{mid}}\right)}
			=\frac{\beta}{F_{\text{mid}}}\;,
			</math>

			:<math>
			\gamma_2=\frac{C''\left(F_{\text{mid}}\right)}{C\left(F_{\text{mid}}\right)}
			=-\frac{\beta\left(1-\beta\right)}{F_{\text{mid}}^2}\;.
			</math>

			The function <math>D\left(\zeta\right)</math> entering the formula above is given by

			: <math>
			D\left(\zeta\right)=\log\left(\frac{\sqrt{1-2\rho\zeta+\zeta^2}+\zeta-\rho}{1-\rho}\right).
			</math>

			Alternatively, one can express the SABR price in terms of the normal Black's model. Then the implied normal volatility can be asymptotically computed by means of the following expression:

			:<math>
			\sigma_{\text{impl}}^{\text{n}}=\alpha\;
			\frac{F_0-K}{D\left(\zeta\right)}\;
			\left\{1+\left[\frac{2\gamma_2-\gamma_1^2}{24}\;\left(\frac{\sigma_0 C\left(F_{\text{mid}}\right)}{\alpha}\right)^2+\frac{\rho\gamma_1}{4}\;\frac{\sigma_0 C\left(F_{\text{mid}}\right)}{\alpha}+\frac{2-3\rho^2}{24}
			\right]\varepsilon\right\}.
			</math>

			It is worth noting that the normal SABR implied volatility is generally somewhat more accurate than the lognormal implied volatility.


			==See also==
			*[[Volatility (finance)]]
			*[[Stochastic Volatility]]
			*[[Risk-neutral measure]]

			==References==
			*[http://www.math.columbia.edu/~lrb/sabrAll.pdf Managing Smile Risk, P. Hagan et al.] – The original paper introducing the SABR model.
			*[http://www.lesniewski.us/papers/published/HedgingUnderSABRModel.pdf Hedging under SABR Model, B. Bartlett] – Refined risk management under the SABR model.
			*[http://arxiv.org/abs/0708.0998v3 Fine Tune Your Smile – Correction to Hagan et al.]
			*[http://www.riskworx.com/insights/sabr/sabr.html A SUMMARY OF THE APPROACHES TO THE SABR MODEL FOR EQUITY DERIVATIVE SMILES]
			*[http://arxiv.org/pdf/physics/0602102v1 UNIFYING THE BGM AND SABR MODELS: A SHORT RIDE IN HYPERBOLIC GEOMETRY, PIERRE HENRY-LABORD`ERE]
			*[http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1850709 Asymptotic Approximations to CEV and SABR Models]
			*[http://www.pricing-option.com/calibration_sabr.aspx Test SABR (with calibration) online]
			*[http://www.serdarsen.somee.com/Calbration%20SABR.aspx SABR calibration]
			*[http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2026350 Advanced Analytics for the SABR Model] - Includes '''exact''' formula for zero correlation case

			{{Derivatives market}}
			{{Volatility}}
			{{Stochastic processes}}

			[[Category:Mathematical finance]]
			[[Category:Options (finance)]]
			[[Category:Derivatives (finance)]]
			[[Category:Finance theories]]

en>EmausBot: r2.7.2+) (Robot: Modifying ru:Взвешенная справедливая очередь

2012-08-15T09:53:30Z

r2.7.2+) (Robot: Modifying ru:Взвешенная справедливая очередь

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Hi there. Let me start by introducing the writer, her title is Sophia Boon but she never really favored that name. Office supervising is my profession. I am really fond of handwriting but I can't make it my profession really. Mississippi is exactly where her house is but her husband wants them to move.<br><br>Take a look at my blog - psychic love readings ([http://cartoonkorea.com/ce002/1093612 More hints])

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en>Kvng: Reverted good faith edits by MarcBoyerONERA (talk): Content probably better suited to Generalized processor sharing. (TW)

en>EmausBot: r2.7.2+) (Robot: Modifying ru:Взвешенная справедливая очередь