Modeling and Pricing of Variance and Volatility Swaps for Local Semi-Markov Volatilities in Financial Engineering

We consider a semi-Markov modulated security market consisting of a riskless asset or bond with constant interest rate and risky asset or stock, whose dynamics follow gemoetric Brownian motion with volatility that depends on semi-Markov process. Two cases for semi-Markov volatilities are studied: local current and local semi-Markov volatilities. Using the martingale characterization of semi-Markov processes, we find the minimal martingale measure for this incomplete market. Then we model and price variance and volatility swaps for local semi-Markov stochastic volatilities.


Introduction
Financial engineering is a multidisciplinary field involving financial theory, the methods of financing, using tools of mathematics, computation, and the practice of programming to achieve the desired end results.The financial engineering methodologies usually apply engineering methodologies and quantitative methods to finance.It is normally used in the securities, banking, and financial management and consulting industries, or by quantitative analysts in corporate treasury and finance departments of general manufacturing and service firms.
One of the recent and new financial products are variance and voaltility swaps, which are useful for volatility hedging and speculation.The market for variance and volatility swaps has been growing, and many investment banks and other financial institutions are now actively quoting volatility swaps on various assets: stock indexes, currencies, as well

Volatilities and Swaps
Volatility σ is the standard deviation of the change in value of a financial instrument with a specific horizon.It is often used to quantify the risk of the instrument over that time period.The higher volatility, the riskier the security.
Historical volatility is the volatility of a financial instrument based on historical returns.It is a standard deviation uses historical daily, weekly, monthly, quarterly, yearly price data to empirically measure the volatility of a market or instrument in the past.
The annualized volatility σ is the standard deviation of the instrument's logarithmic returns in a year, where R i : ln S t i /S t i−1 , R : 1/n n i 1 ln S t i /S t i−1 , and S t i is a asset price at time t i , i 1, 2, . . ., n.
Implied volatility is related to historical volatility; however, the two are distinct.Historical volatility is a direct measure of the movement of the underlier's price realized volatility over recent history.Implied volatility, in contrast, is set by the market price of the derivative contract itself, and not the underlier.Therefore, different derivative contracts on the same underlier have different implied volatilities.Most derivative markets exhibit persistent patterns of volatilities varying by strike.The pattern displays different characteristics for different markets.In some markets, those patterns form a smile curve.In others, such as equity index options markets, they form more of a skewed curve.This has motivated the name "volatility skew".For markets where the graph is downward sloping, such as for equity options, the term "volatility skew" is often used.For other markets, such as FX options or equity index options, where the typical graph turns up at either end, the more familiar term "volatility smile" is used.In practice, either the term "volatility smile" or "volatility skew" may be used to refer to the general phenomenon of volatilities varying by strike.
The models by Black and Scholes 4 continuous-time B, S -security market and Cox et al. 5 discrete-time B, S -security market binomial tree are unable to explain the negative skewness and leptokurticity fat tail commonly observed in the stock markets The famous implied-volatility smile would not exist under their assumptions.
Given the prices of call or put options across all strikes and maturities, we may deduce the volatility which produces those prices via the full Black-Scholes equation Black  Level-dependent volatility e.g., CEV or Firm Model -function of the spot price alone.To have a smile across strike price, we need σ to depend on S: σ ≡ σ S t .In this case, the volatility and stock price changes are now perfectly correlated.
Black and Scholes 1973 made a major breakthrough by deriving pricing formulas for vanilla options written on the stock.The Black-Scholes model assumes that the volatility term is a constant.This assumption is not always satisfied by real-life options as the probability distribution of an equity has a fatter left tail and thinner right tail than the lognormal distribution see Hull 2 , and the assumption of constant volatility σ in financial model such as the original Black-Scholes model is incompatible with derivatives prices observed in the market.Stochastic volatility models are used in the field of quantitative finance to evaluate derivative securities, such as options, swaps.By assuming that the volatility of the underlying price is a stochastic process rather than a constant, it becomes possible to more accurately model derivatives.In the approach i , the volatility coefficient is independent of the current level of the underlying stochastic process S t .This is a deterministic volatility model, and the special case where σ is a constant reduces to the well-known Black-Scholes model that suggests changes in stock prices are lognormal distributed.But the empirical test by Bollerslev 16 seems to indicate otherwise.One explanation for this problem of a lognormal model is the possibility that the variance of log S t /S t − 1 changes randomly.
In the approach ii , several ways have been developed to derive the corresponding Black-Scholes formula: one can obtain the formula by using stochastic calculus and, in particular, the Ito's formula see, e.g., Shiryaev 17 .
A generalized volatility coefficient of the form σ t, S t is said to be level-dependent.Because volatility and asset price are perfectly correlated, we have only one source of randomness given by W 1 t .A time and level-dependent volatility coefficient makes the arithmetic more challenging and usually precludes the existence of a closed-form solution.However, the arbitrage argument based on portfolio replication and a completeness of the market remain unchanged.
The situation becomes different if the volatility is influenced by a second "nontradable" source of randomness.This is addressed in the approachs iii , iv , and v we usually obtains a stochastic volatility model, introduced by Hull and White 10 , which is general enough to include the deterministic model as a special case.
Hobson and Rogers 18 suggested a new class of nonconstant volatility models, which can be extended to include the aforementioned level-dependent model and share many characteristics with the stochastic volatility model.The volatility is nonconstant and can be regarded as an endogenous factor in the sense that it is defined in terms of the past behaviour of the stock price.This is done in such a way that the price and volatility form a multidimensional Markov process.
The Generalized Auto-Regression Conditional Heteroskedacity GARCH model see Bollerslev 16 is another popular model for estimating stochastic volatility.It assumes that the randomness of the variance process varies with the variance, as opposed to the square root of the variance as in the Heston model.The standard GARCH 1,1 model has the following form for the variance differential: 2.4 The GARCH model has been extended via numerous variants, including the NGARCH, LGARCH, EGARCH, GJR-GARCH, and so forth.
Volatility swaps are forward contracts on future realized stock volatility, variance swaps are similar contract on variance, the square of the future volatility, both these instruments provide an easy way for investors to gain exposure to the future level of volatility.
A stock's volatility is the simplest measure of its risk less or uncertainty.Formally, the volatility σ R is the annualized standard deviation of the stock's returns during the period of interest, where the subscript R denotes the observed or "realized" volatility.
The easy way to trade volatility is to use volatility swaps, sometimes called realized volatility forward contracts, because they provide pure exposure to volatility and only to volatility .
Demeterfi et al. 1 explained the properties and the theory of both variance and volatility swaps.They derived an analytical formula for theoretical fair value in the presence of realistic volatility skews, and pointed out that volatility swaps can be replicated by dynamically trading the more straightforward variance swap.
Javaheri et al. 19 discussed the valuation and hedging of a GARCH 1,1 stochastic volatility model.They used a general and exible PDE approach to determine the first two moments of the realized variance in a continuous or discrete context.Then they approximate the expected realized volatility via a convexity adjustment.
Brockhaus and Long 20 provided an analytical approximation for the valuation of volatility swaps and analyzed other options with volatility exposure.
Elliott and Swishchuk 12 studied option pricing formulae and pricing swaps for Markov-modulated Brownian with jumps.
In 3 we considered a semi-Markov modulated market consisting of a riskless asset or bond, B, and a risky asset or stock, S, whose dynamics depend on a semi-Markov process x.Using the martingale characterization of semi-Markov processes, we noted the incompleteness of semi-Markov modulated markets and found the minimal martingale measure.We priced variance and volatility swaps for stochastic volatilities driven by the semi-Markov processes.
In this paper, we study some extensions of the results obtained in 3 for models such as local current and local semi-Markov volatilities.We obtain the prices of variance Theorem 6.1 and volatility swaps Theorem 7.1 for these models.

Martingale Characterization of Semi-Markov Processes:
Definitions and Preliminary Results

Markov Renewal and Semi-Markov Processes
Let Ω, F, P be a probability space, X, X be a measurable space and Q x, B, t P x, B G x t , x ∈ X, B ∈ X, t ∈ R , be a semi-Markov kernel.Let us consider a X×R , X⊗Bvalued stochastic process x n , τ n ; n ≥ 0 , with Remark 3.6.If we expand the state space of the semi-Markov process to include a component that records the amount of time already spent in the current state, then this additional information in the state description makes the semi-Markov process Markovian.For example, the following process x t , γ t is a Markov process; see 21, 22 .
Definition 3.7.The compensating operator Q of the Markov renewal process is defined by the following relation: Lemma 3.8.The compensating operator of the Markov renewal process can be defined by the relation This statement follows directly from Definition 3.7.
Lemma 3.9.Let x n , τ n be the Markov renewal process, Q be the compensating operator, and F n σ{x k , τ k ; k ≤ n}.Then the process m n is a F n -martingale for any function f such that Mathematical Problems in Engineering 7 Let x t be a Markov process with infinitesimal generator Q.
This statement follows from the Dynkin formula see 23 .
Theorem 3.11.The quadratic variation of the martingale m t is the process (See [12]).

Jump Measure for a Semi-Markov Process
The jump measure for x t is defined in the following way see 22 : It is known see 12 that the predictable projection compensator for μ has the following form: where G x t 1 − G x t , g x t dG x t /dt, for all x ∈ X.

Pricing of Variance and Volatility Swaps for Semi-Markov Volatilities
In 3 we stated two main results on pricing of variance and volatility swaps for semi-Markov volatility.We considered the following B, S -security market B stands for risk-free security "Bond" and S stands for risky security "Stock" .

4.4
Here Q is defined as In this paper, we generalize these results for local semi-Markov volatilities.

Minimal Martingale Measure for Stock Price with Local Semi-Markov Volatility
We suppose that the stock price S t satisfies the following stochastic differential equation

5.4
The error of estimation will be where In this way, it is enough to consider the case with the local current stochastic volatility in 5.1 .

Minimal Martingale Measure
We consider the following B, S security market.Let the stock price S t satisfies the following equation: where μ ∈ R is the appreciation rate and σ S t , x t , γ t is the local current semi-Markov volatility, and the bond price B t is where r > 0 is the risk-free rate of return interest rate .
As long as we have two sources of randomness, Brownian motion w t and semi-Markov process x t , the above B, S -security market 4.1 -4.2 is incomplete see 12, Theorem 1 so there are many risk-neutral or martingale measures.We are going to construct the minimal martingale measure see 12 .With respect to this construction see 12, Lemma 4 , the minimal martingale measure P * is as follows.
Using Girsanov's Theorem see 17 , we obtain the following result concerning the minimal martingale measure in the above market 5.6 -5.7 .

s., the following holds:
1 There is a probability measure P * equivalent to P such that is its Radon-Nikodym density.

Pricing of Variance Swaps for Local Semi-Markov Stochastic Volatility
A variance swap is a forward contract on annualized variance, the square of the realized volatility.Its payoff at expiration is equal to where σ 2 R x is the realized stock variance quoted in annual terms over the life of the contract, K var is the delivery price for variance, and N is the notional amount of the swap in dollars per annualized volatility point squared.The holder of a variance swap receives, at expiry, N dollars for every point by which the stock's realized variance σ 2 R x has exceeded the variance delivery price K var .See 1, 24 .
Pricing a variance forward contract or swap is no different from valuing any other derivative security.The value of a forward contract F on future realized variance with strike price K var is the expected present value of the future payoff in the risk-neutral world: where r is the risk-free discount rate corresponding to the expiration date T , E denotes the expectation with respect to the minimal martingale measure P from now on we use simplier notation P instead of P * , and σ 2 R x is defined in 6.where Q is defined in 5.2 .

Pricing of Volatility Swaps for Local Semi-Markov Volatility
Volatility swaps are forward contracts on future realized stock volatility.A stock's volatility is the simplest measure of its risk less or uncertainty.Formally, the volatility σ R S is the annualized standard deviation of the stock's returns during the period of interest, where the subscript R denotes the observed or "realized" volatility for the stock S.
The easy way to trade volatility is to use volatility swaps, sometimes called realized volatility forward contracts, because they provide pure exposure to volatility and only to volatility see 1, 19, 20, 24, 25 .
A stock volatility swap is a forward contract on the annualized volatility.Its payoff at expiration is equal to where σ R S is the realized stock volatility quoted in annual terms over the life of contract, σ t is a stochastic stock volatility, K vol is the annualized volatility delivery price, and N is the notional amount of the swap in dollar per annualized volatility point.The holder of a volatility swap receives, at expiry, N dollars for every point by which the stock's realized volatility σ R has exceeded the volatility delivery price K vol .The holder is swapping a fixed volatility K vol for the actual floating future volatility σ R .We note that usually N αI, where α is a converting parameter such as 1 per volatility-square, and I is a long-short index 1 for long and −1 for short .Pricing a variance forward contract or swap is no different from valuing any other derivative security.The value of a forward contract F on future realized variance with strike price K var is the expected present value of the future payoff in the risk-neutral world: where r is the risk-free discount rate corresponding to the expiration date T , E denotes the expectation with respect to the minimal martingale measure P from now on we use simplier notation P instead of P * and σ R x is defined in 7.3 .Thus, for calculating variance swaps we need to know only E{σ 2 R S }, namely, mean value of the underlying variance.
To calculate volatility swaps we need more.From 20 we have the approximation which uses the second-order Taylor expansion for function √ x see also 19, page 16 : where V : σ 2 R S and Var{V }/8E{V } 3/2 is the convexity adjustment.Thus, to calculate volatility swaps we need both E{V } and Var{V }.

Pricing of Volatility Swap
From Brockhaus and Long approximation 20 we have see also 19  where Q is defined in 5.2 .

2 . 7 . 7 Taking into account 7 . 3 - 7 . 7 ,σ 2 0Theorem 7 . 1 .≈0
we obtain: P vol s, x e −rT Eσ R S − K vol S s , x s , γ s dt − K vol Eσ 2 S t , x t , γ t σ 2 S s , x s , γ s dt ds− 1/T T 0 e tQ σ 2 s, x, 3 -7.8, we have the following.The value of volatility swap for local current semi-Markov stochastic volatility σ S t , x t , γ t is P vol s, x Eσ 2 S t , x t , γ t σ 2 S s , x s , γ s dtds − 1 and Scholes 4 , Dupire 6 and Derman and Kani 7 .This function has come to be known as local volatility.Local volatility-function of the spot price S t and time t: σ ≡ σ S t , t see Dupire 6 formulae for local volatility .
Definition 3.1.A Markov renewal process is a two-component Markov chain, x n τ n ; n ≥ 0 , homogeneous with respect to the second component with transition probabilities is a σ-algebra generated by x n , τ n .Define the counting process of jumps ν t by ν t sup{n ≥ 0: τ n ≤ t}, that gives the number of jumps of the Markov renewal process in the time interval 0, t .Definition 3.2.A stochastic process x t , t ≥ 0, defined by x t x ν t is called a semi-Markov process, associated to the Markov renewal process x n τ n ; n ≥ 0 .Remark 3.3.Markov jump processes are special cases of semi-Markov processes with semi-Markov kernel Q x, B, t P x, B 1 − e −λ x t .Definition 3.4.The auxiliary process γ t t − τ ν t is called the backward recurrence time-the time period since the last renewal epoch before t or the current life in the terminology of reliability theory, or age random variable ; see 22 .
: σ{x s , τ ν s ; 0 ≤ s ≤ t}, where Q is the infinitesimal operator of Markov process x t , γ t : t t 0 Qf 2 x s , γ s − 2f x s , γ s Qf x s , γ s ds < ∞, ∀f 2 ∈ Domain Q .f t in 3.13 is a P -martingale (Doléans-Dade martingale).Here, m f t is defined in 3.11 and Q in 3.10 .
R is the appreciation rate and σ x t , γ t is the semi-Markov volatility, and the bond price B t is Price of Volatility Swap 3 .The value of a volatility swap for semi-Markov stochastic volatility σ x t , γ t is loc S t , x t , γ t depending on the process x t , which is independent on standard Wiener process w t , stock price S t , and the current life γ t t − τ ν t .We call the volatility σ S t , x t , γ t the local current semi-Markov volatility.
t S t rdt σ loc S t , x t , γ t dw t 5.1 with the volatility σ : σ with the volatility σ : σ S t , x t , t depending on the process x t , which is independent on standard Wiener process w t , stock price S t and current time t.Suppose also that σ S, x, t is differentiable function by t with bounded derivative.Then we can reduce the problem of calculating of swaps with local semi- Remark 5.4.Measure P * is called the minimal martingale measure.Remark 5.5.We note that under the P * measure the stock price S t satisfies the following equation:Remark 5.6.A sufficient condition Novikov's condition for the right-hand side of 5.8 to be a martingale is t 0 r − μ /σ S t , x s , γ s ds w t is a standard Brownian motion with respect to P * .t, x s , γ s dw * t .5.11 2 .Let us show how we can calculate EV x , where V x : σ 2 R x .For that we need to calculate Eσ 2 S t , x t , γ t .We note see Section 3.3, Lemma 3.12 that for σ x ∈ Domain Q the following process -mean martingale with respect to F t : σ{w s , x s , τ ν s ; 0 ≤ s ≤ t} and Q is the infinitesimal operator defined in 5.2 .
, page 16 : R S and Var{V }/8E{V } 3/2 is the convexity adjustment.As we can see from 7.5 , to calculate volatility swaps we need both E{V } and Var{V }.We have already calculated E{σ 2 R S } E V see 6.10 .Let us calculate Var V : E V 2 − E V 2 .In this way, we need E V 2 Eσ 4 R S .Taking into account the expression for V σ 2 R S we have: S t , x t , γ t σ 2 S s , x s , γ s dt ds .7.6 In this way, the variance of V , Var V , is Eσ 2 S t , x t , γ t σ 2 S s , x s , γ s dt ds − 1 T T 0 e tQ σ 2 s, x, 0 dt