Pricing Discretely Sampled Variance Swaps

A variance swap is a financial derivative arose in the1990s, whose value depends on the underlying asset’s volatility level in the future. Compared with traditional derivatives like options, it offers a more direct and more pure risk exposure of the underlying asset. Variance swaps, therefore, are not only used as an important way to gain the benefits, but also as an effective tool to hedge volatility risk. In the markets, the realized variance of a variance swap is calculated based on the sampling at discrete times. However, most of the existing pricing models assume continuous sampling to alleviate the mathematical difficulties associated with dealing with the discreteness of the sample data. This assumption will result in a systematic bias for the price of a variance swap. Consequently, trying to address the "discretely-sampled" nature of a variance swap is very essential. This paper price discretely sampled variance swaps and generalized variance swaps under mean-reverting Gaussian volatility model.Two typical definitions of the realized variance are and where Sti is the closing price of the underlying asset at the i-th observation time ti, and there are altogether N observations, AF is the annualized factor. Hereafter we refer to the contracts with realized variance σ2R.d1and σ2R.d2as proportional variance swaps and logarithmic variance swaps, respectively. We assume equally spaced discrete observations in this paper without losing any generality.We describe the price of an underlying asset as a mean-reverting Gaussian volatil-ity process, that is, under the risk-neutral measure, the underlying asset St and its volatility vt follow the stochastic differential equations where r denotes the risk-free interest, κ*and (?)*are risk-neutral parameters, B_t~s and B(?) are two Brownian motions with correlation p.In Chapter3, we present an analytical pricing formula for discretely sampled proportional variance swaps.Theorem1If St follows the dynamics described in (0.0.2) and the parameters satisfy then the fair strike price for discretely sampled proportional variance swaps is whereCompared with the previous methods, our analytical formula shows substantial advantage, in terms of both accuracy and efficiency. Our formula can be used to derive all the hedging ratios of a variance swap and the methods can also be used to price derivatives based on higher moments, such as skewness swaps and kurtosis swaps.We also give some numerical tests and discussions in Chapter3, illustrating our results from various aspects. In addition, we discuss the connection between the for-mulas under Heston model and ours, the restrictions of parameter space and perform some sensitivity tests on the key parameters in our models.In Chapter4, we derive an analytical pricing formula for logarithmic variance swaps under mean-reverting Gaussian volatility model. Theorem2:The fair slrike price for discrelely sampled logarilhmic variance swaps with mean-reuerting Gaussian volatility is where for all τ≥0. The constants α1,...,α8,β1,...,β7and γ1,...,γ6are given by We also give a theorem provides a suffcient condition for the validity of our solu-tion in the parameter space in Chapter4.Theorem3Let p=(Υ,κ~*,θ~*,σ,ρ)Τ (R-)4×「-1,1」be a parameter vector of mean-reverting Gaussian volatility model. Set△t*p=min({△t>0l A4(△t;p)A5(△t;p)A6(△t;p)=0}υ{∞｝), where A4(T;P),A5(T;P),A6(T;P) are given by Theorem2.Then,we have the following assertions:(1)△t*p is either strictly positive or infinite depending on p.(2)For any△t∈(0,△t*p),(i)0<Kd2(T,△t,υo;p)<∞ for allυo＞0;(ii)Kd2(T,△t,υo;p) is strictly increasing w.r.t. υo on(0,∞). Assertion(ii)in Theorem3implies that κd2is a monotonically increasing function of υo on(0,∞). This is in line with our intuition that a market practitioner should naturally expect a higher fair strike price for a variance swap when the volatility of the underlying is higher. We also give some numerical examples with some detailed discussions in Chapter4to illustrate the correctness of Theorem3.In addition,we obtain a pricing formula for continuous sampling variance swaps with mean-reverting Gaussian volatility and prove that our discretely sampled pricing formulas converge to their continuously monitored counterpart.Theorem4The fair strike price for continuous sampling variance swaps with mean-reverting Gaussian volatility is and it is the asymptotic limit of its discretely monitored counterparts, that is In Chapter5, we present pricing formulas for discretely sampled Gamma swaps, Corridor variance swaps and Conditional variance swaps.The realized variance of a Gamma swap is defined as and its pricing formula is given by the following theorem.Theorem5The fair strike price for discretely sampled Gamma swaps with mean-reverting Gaussian volatility is whereThere are two typical definitions for the realized variance of a Corridor variance swap andwhere1{.} is the indicator function. We can obtain the Corridor variance swaps with a one-sided barrier by taking L=0and U�'∞, respectively. They are called Downside variance swaps and Upside variance swaps. Note the property of indicator functions1{L<x≤∪}=1{x≤∪}-1{x≤L}, it suffices to consider pricing Downside variance swaps alone.Theorem6The fair strike prices for discretely sampled Downside variance swaps with mean-reverting Gaussian volatility are and where υ=In U, j denotes the imaginary unit, ω=ωr+jωi is a complex number satisfies ωi∈(－∞,0), q1,B1(Τ, q), B2(Τ, q), B3(Τ, q) are given by Theorem5andA Conditional variance swap is similar to a Corridor variance swap, except the following two aspects:(i) The accumulated sum of squared returns is divided by the number of observations D that the underlying asset price stays within the Corridor (L, U] instead of the total number of sampling observations N;(ii) The final payoff to the holder is scaled by the ratio D/N. Therefore, the strike price of Conditional variance swaps can be represented by the strike price of Corridor variance swaps.Theorem7The fair strike price for discretely sampled Conditional variance swaps (the counterpart of the first definition for a Corridor variance swaps) with mean reverting Gaussian volatility is where N is the total number of observations, KD,old is given by Theorem6and Hi(ω)=eB1(ti-1,q6)+B2(ti-1,q6)υ0+B3(ti-1,q6)υ02, q6=(-jω,0,0,0)Τ...