## Multiscale Stochastic Volatility Asymptotics (2003)

Venue: | SIAM J. MULTISCALE MODELING AND SIMULATION |

Citations: | 28 - 11 self |

### BibTeX

@ARTICLE{Fouque03multiscalestochastic,

author = {Jean-Pierre Fouque and George Papanicolaou and Ronnie Sircar and Knut Solna},

title = {Multiscale Stochastic Volatility Asymptotics},

journal = {SIAM J. MULTISCALE MODELING AND SIMULATION},

year = {2003},

volume = {2},

pages = {22--42}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical Black-Scholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index say and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena makes it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. In previous work, see for instance [3], we considered the situation when the volatility is fast mean reverting. Using a singular perturbation expansion we derived an approximation for option prices. We also provided a calibration method using observed option prices as represented by the so-called term structure of implied volatility. Our analysis of market data, however, shows the need for introducing also a slowly varying factor in the model for the stochastic volatility. The combination of regular and singular perturbations approach that we set forth in this paper deals with this case. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parametrization of the implied volatility surface. In particular, the introduction of the slow factor gives a much better fit for options with longer maturities. We use option data to illustrate our results and show how exotic option prices also can be approximated using our multiscale perturbation approach.

### Citations

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Citation Context ...e introduced in the literature in the late 1980s by Hull & White [11], Wiggins [18] and Scott [16]. One popular class of models builds on the Feller process model introduced in this context by Heston =-=[10]-=- because call option prices can be solved for in closed form up to a Fourier inversion. Typically a lot of emphasis is placed on fitting the models very closely to observed implied volatilities (see S... |

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Citation Context ...eutral or equivalent martingale measure that is used for pricing of options. 2.1. Background. Volatility models built on diffusions were introduced in the literature in the late 1980s by Hull & White =-=[11]-=-, Wiggins [18] and Scott [16]. One popular class of models builds on the Feller process model introduced in this context by Heston [10] because call option prices can be solved for in closed form up t... |

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Citation Context ...losely to observed implied volatilities (see Section 4 for the definition), and not surprisingly, models with more degrees of freedom perform better in this regard. For example, the models studied in =-=[2, 4]-=- include jumps in stochastic volatility on top of a Heston-type model. 2However, little attention is paid to the stability of the estimated parameters over time, and it is usual practice in the indus... |

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Citation Context ... which will be described below. Observe that when σt is constant then Xt is a geometric Brownian motion and corresponds to the classical model used in the Black-Scholes theory. We refer the reader to=-= [11]-=- for details concerning diffusion processes and the related stochastic calculus, and a brief review 2sof this calculus and the Black-Scholes pricing theory can also be found in [3]. In the class of mo... |

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Citation Context ...le measure that is used for pricing of options. 2.1. Background. Volatility models built on diffusions were introduced in the literature in the late 1980s by Hull & White [11], Wiggins [18] and Scott =-=[16]-=-. One popular class of models builds on the Feller process model introduced in this context by Heston [10] because call option prices can be solved for in closed form up to a Fourier inversion. Typica... |

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Citation Context ...er frequencies over longer time periods primarily picks up a slower time-scale of fluctuation and cannot identify scales of length comparable to the sampling frequency. Another recent empirical study =-=[1], this tim-=-e of exchange rate dynamics, finds "the evidence points strongly toward two-factor [volatility] models with one highly persistent factor and one quickly mean-reverting factor". 2.2. Model un... |

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Citation Context ...valent martingale measure that is used for pricing of options. 2.1. Background. Volatility models built on diffusions were introduced in the literature in the late 1980s by Hull & White [11], Wiggins =-=[18]-=- and Scott [16]. One popular class of models builds on the Feller process model introduced in this context by Heston [10] because call option prices can be solved for in closed form up to a Fourier in... |

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Citation Context ...lity factor associated with # small corresponds the slow factor Z t . In the case of a single slow volatility factor such an expansion has been considered in [7], [9] and [12], for instance. See also =-=[10]-=- and [8] for related regular perturbation expansions, and [13] for approximations based on large strike-price limits. Definition 3.1. The leading order term P # 0 is defined as the unique solution to ... |

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Citation Context ...losely to observed implied volatilities (see Section 4 for the definition), and not surprisingly, models with more degrees of freedom perform better in this regard. For example, the models studied in =-=[2, 4]-=- include jumps in stochastic volatility on top of a Heston-type model. 2However, little attention is paid to the stability of the estimated parameters over time, and it is usual practice in the indus... |

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Citation Context ...or associated with # small corresponds the slow factor Z t . In the case of a single slow volatility factor such an expansion has been considered in [7], [9] and [12], for instance. See also [10] and =-=[8]-=- for related regular perturbation expansions, and [13] for approximations based on large strike-price limits. Definition 3.1. The leading order term P # 0 is defined as the unique solution to the prob... |

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Citation Context ... (3.1) Recall that the volatility factor associated with # small corresponds the slow factor Z t . In the case of a single slow volatility factor such an expansion has been considered in [7], [9] and =-=[12]-=-, for instance. See also [10] and [8] for related regular perturbation expansions, and [13] for approximations based on large strike-price limits. Definition 3.1. The leading order term P # 0 is defin... |

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Citation Context ...itivities) of the leading order Black-Scholes price. The accuracy of this approximation is given in Theorem 3.6, the main result of this section. The proof is a generalization of the one presented in =-=[6]-=- where only the fast scale factor was considered. In Section 4 we recall the concept of implied volatility and we deduce its expansion in the regime of fast and slow volatility factors. This leads to ... |

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Citation Context ...· · · . Recall that the volatility factor associated with δ small corresponds the slow factor Zt. In the case of a single slow volatility factor such an expansion has been considered in [9], [13] and =-=[17]-=-, for instance. See also [14] and [12] for related regular perturbation expansions, and [19] for approximations based on large strike-price limits. Definition 3.1. The leading order term P ε 0 is defi... |

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Citation Context ... 1 + δP ε 2 + · · · . Recall that the volatility factor associated with δ small corresponds the slow factor Zt. In the case of a single slow volatility factor such an expansion has been considered in =-=[9]-=-, [13] and [17], for instance. See also [14] and [12] for related regular perturbation expansions, and [19] for approximations based on large strike-price limits. Definition 3.1. The leading order ter... |

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Citation Context ... # 2 + . (3.1) Recall that the volatility factor associated with # small corresponds the slow factor Z t . In the case of a single slow volatility factor such an expansion has been considered in [7], =-=[9]-=- and [12], for instance. See also [10] and [8] for related regular perturbation expansions, and [13] for approximations based on large strike-price limits. Definition 3.1. The leading order term P # 0... |

8 |
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(Show Context)
Citation Context ...applied. 2.1.3. Empirical Evidence. Empirical evidence of a fast volatility factor (with a characteristic mean-reversion time of a few days) was found in the analysis of highfrequency S&P 500 data in =-=[5]-=-. Many empirical studies have looked at low-frequency (daily) data, with the data necessarily ranging over a period of years, and they have found a slow volatility factor. This does not contradict the... |

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Citation Context ...r Z t . In the case of a single slow volatility factor such an expansion has been considered in [7], [9] and [12], for instance. See also [10] and [8] for related regular perturbation expansions, and =-=[13]-=- for approximations based on large strike-price limits. Definition 3.1. The leading order term P # 0 is defined as the unique solution to the problem # 1 # L 0 + 1 # # L 1 + L 2 # P # 0 = 0 (3.2) P # ... |

5 |
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Citation Context ...hich will be described below. Observe that when # t is constant then X t is a geometric Brownian motion and corresponds to the classical model used in the Black-Scholes theory. We refer the reader to =-=[11]-=- for details concerning di#usion processes and the related stochastic calculus, and a brief review 2 of this calculus and the Black-Scholes pricing theory can also be found in [3]. In the class of mod... |

3 |
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Citation Context ...points strongly toward two-factor [volatility] models with one highly persistent factor and one quickly mean-reverting factor”. 2.2. Model under risk-neutral measure. No arbitrage pricing theory (se=-=e [2]-=-, for example) states that option prices are expectations of discounted payoffs with respect to an equivalent martingale measure. A brief review of this in the present stochastic volatility context is... |

3 | Maturity Cycles in Implied Volatility
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(Show Context)
Citation Context ...ting is at the level of the shortest maturities (the left-most strand). One way to handle these using a periodic scale corresponding to the monthly expiration cycles of traded options is presented in =-=[6]-=-. 0.38 LMMR Fit to Residual 0.36 0.34 0.32 δ−adjusted Implied Volatility 0.3 0.28 0.26 0.24 0.22 0.2 0.18 −2.5 −2 −1.5 −1 −0.5 0 0.5 LMMR Fig. 5.4. δ-adjusted implied volatility I − b δ τ − a δ (LM) a... |

2 |
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Citation Context ...points strongly toward two-factor [volatility] models with one highly persistent factor and one quickly mean-reverting factor". 2.2. Model under risk-neutral measure. No arbitrage pricing theory =-=(see [2]-=-, for example) states that option prices are expectations of discounted payo#s with respect to an equivalent martingale measure. A brief review of this in the present stochastic volatility context is ... |

1 |
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Citation Context ... + #P # 2 + . (3.1) Recall that the volatility factor associated with # small corresponds the slow factor Z t . In the case of a single slow volatility factor such an expansion has been considered in =-=[7]-=-, [9] and [12], for instance. See also [10] and [8] for related regular perturbation expansions, and [13] for approximations based on large strike-price limits. Definition 3.1. The leading order term ... |