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

Abstract

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.

1. 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 as commodities. A stock's volatility is the simplest measure of its riskiness 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. Why trade volatility or variance? As mentioned in [1], “just as stock investors think they know something about the direction of the stock market so we may think we have insight into the level of future volatility. If we think current volatility is low, for the right price we might want to take a position that profits if volatility increases”.

The Black-Scholes model for option pricing 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 and financial engineering 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 this paper, we show how to model and price variance and volatility swaps for two different types of stochastic volatility, local current and local semi-Markov volatilities.

The paper is organized as follows. Section 2 is devoted to the description of different types of volatilities and swaps. Martingale characterization of semi-Markov processes is considered in Section 3. Section 4 summaries relevant results from [3]. Minimal martingale measure for stock price with local semi-Markov volatility is constructed in Section 5. Section 6 contains pricing of variance swap for local semi-Markov stochastic volatility (Theorem 6.1). Section 7 is devoted to the pricing of volatility swap for local semi-Markov stochastic volatility (Theorem 7.1).

2. 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, 𝜎∶=1𝑛−1^𝑛_𝑖=1𝑅_𝑖−𝑅², (2.1) where 𝑅_𝑖∶=ln(𝑆_𝑡𝑖/𝑆_𝑡𝑖−1), 𝑅∶=(1/𝑛)∑^𝑛_𝑖=1ln(𝑆_𝑡𝑖/𝑆_𝑡𝑖−1), and 𝑆_𝑡𝑖 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 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).

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.

The above issues have been addressed and studied in several ways, such as(i)volatility is assumed to be a deterministic function of the time: 𝜎≡𝜎(𝑡) (see Wilmott et al. [8], Merton [9]); extended the term structure of volatility to 𝜎∶=𝜎_𝑡 (deterministic function of time), with the implied volatility for an option of maturity 𝑇 given by 𝜎²_𝑇=(1/𝑇)∫^𝑇₀𝜎²_𝑢𝑑𝑢,(ii)volatility is assumed to be a function of the time and the current level of the stock price 𝑆(𝑡): 𝜎≡𝜎(𝑡,𝑆(𝑡)) (see Dupire [6], Hull [2]); the dynamics of the stock price satisfies the following stochastic differential equation: 𝑑𝑆(𝑡)=𝜇𝑆(𝑡)𝑑𝑡+𝜎(𝑡,𝑆(𝑡))𝑆(𝑡)𝑑𝑊₁(𝑡), (2.2) where 𝑊₁(𝑡) is a standard Wiener process,(iii)the time variation of the volatility involves an additional source of randomness, besides 𝑊₁(𝑡), represented by 𝑊₂(𝑡), and is given by 𝑑𝜎(𝑡)=𝑎(𝑡,𝜎(𝑡))𝑑𝑡+𝑏(𝑡,𝜎(𝑡))𝑑𝑊₂(𝑡), (2.3) where 𝑊₂(𝑡) and 𝑊₁(𝑡) (the initial Wiener process that governs the price process) may be correlated (see Hull and White [10], Heston [11]),(iv)the volatility depends on a random parameter 𝑥 such as 𝜎(𝑡)≡𝜎(𝑥(𝑡)), where 𝑥(𝑡) is some random process (see Elliott and Swishchuk [12], Swishchuk [3, 13]),(v)another approach is connected with stochastic volatility, namely, uncertain volatility scenario. This approach is based on the uncertain volatility model developed in Avellaneda et al. [14], where a concrete volatility surface is selected among a candidate set of volatility surfaces. This approach addresses the sensitivity question by computing an upper bound for the value of the portfolio under arbitrary candidate volatility, and this is achieved by choosing the local volatility 𝜎(𝑡,𝑆(𝑡)) among two extreme values 𝜎_min and 𝜎_max such that the value of the portfolio is maximized locally,(vi)the volatility 𝜎(𝑡,𝑆_𝑡) depends on 𝑆_𝑡∶=𝑆(𝑡+𝜃) for 𝜃∈[−𝜏,0], namely, stochastic volatility with delay (see Kazmerchuk et al. [15]).

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 𝑊₁(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: 𝑑𝜎_𝑡=𝜅𝜃−𝜎_𝑡𝑑𝑡+𝛾𝜎_𝑡𝑑𝐵_𝑡. (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.

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

3.1. 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⊗ℬ₊)-valued stochastic process (𝑥_𝑛,𝜏_𝑛; 𝑛≥0), with 𝜏₀≤𝜏₁≤⋯≤𝜏_𝑛≤𝜏_𝑛+1≤⋯, (see [21, 22]).

Definition 3.1. A Markov renewal process is a two-component Markov chain, (𝑥_𝑛𝜏_𝑛;𝑛≥0), homogeneous with respect to the second component with transition probabilities 𝑃𝑥_𝑛+1∈𝐵,𝜏_𝑛+1−𝜏_𝑛≤𝑡∣ℱ_𝑛=𝑄𝑥_𝑛,𝐵,𝑡=𝑃(𝑥,𝐵)𝐺_𝑥(𝑡)a.s., (3.1) where ℱ_𝑛 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_𝑛𝜏_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].

Remark 3.5. The current life is a Markov process with generator 𝑄𝑓(𝑡)=𝑓^′(𝑡)+𝜆(𝑡)[𝑓(0)−𝑓(𝑡)], where 𝜆(𝑡)=−𝐹^′(𝑡)/𝐹(𝑡), 𝐹(𝑡)=1−𝐹(𝑡), Domain(𝑄)=𝐶¹(𝑅).

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: 𝑄𝑓𝑥₀,𝜏₀=𝑞𝑥₀𝐸𝑓𝑥₁,𝜏₁−𝑓𝑥₀,𝜏₀|ℱ|₀, (3.2) where 𝑞(𝑥)=1/𝑚(𝑥), 𝑚(𝑥)=∫₀^+∞𝐺_𝑥(𝑡)𝑑𝑡, ℱ_𝑡=𝜎{𝑥(𝑠),𝜏_𝜈(𝑠);0≤𝑠≤𝑡}.

Lemma 3.8. The compensating operator of the Markov renewal process can be defined by the relation 𝑄𝑓(𝑥,𝑡)=𝑞(𝑥)₀^+∞𝐺_𝑥(𝑑𝑠)_𝑋𝑃𝑥,𝑑𝑦𝑓𝑦,𝑡+𝑠−𝑓(𝑥,𝑡). (3.3)

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, 𝑚_𝑛∶=𝑓𝑥_𝑛,𝜏_𝑛−^𝑛_𝑖=1𝜏_𝑖−𝜏_𝑖−1𝑄𝑓𝑥_𝑖−1,𝜏_𝑖−1, (3.4) 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 𝐸_𝑥|𝑓(𝑥₁,𝜏₁)|<+∞.

Let 𝑥(𝑡) be a Markov process with infinitesimal generator 𝑄.

Theorem 3.10. The process 𝑚(𝑡)∶=𝑓(𝑥(𝑡))−𝑓(𝑥)−^𝑡₀𝑄𝑓(𝑥(𝑠))𝑑𝑠 (3.5) is an ℱ^𝑥_𝑡=𝜎{𝑥(𝑠);𝑠≤𝑡}-martingale (see [12, 22]).

This statement follows from the Dynkin formula (see [23]).

Theorem 3.11. The quadratic variation of the martingale 𝑚(𝑡) is the process ⟨𝑚(𝑡)⟩=^𝑡₀𝑄𝑓²(𝑥(𝑠))−2𝑓(𝑥(𝑠))𝑄𝑓(𝑥(𝑠))𝑑𝑠. (3.6) (See [12]).

3.2. Jump Measure for a Semi-Markov Process

The jump measure for 𝑥(𝑡) is defined in the following way (see [22]): 𝜇([0,𝑡]×𝐴)=_𝑛≥0𝟏𝑥_𝑛∈𝐴,𝜏_𝑛≤𝑡,𝐴∈𝒳,𝑡≥0. (3.7) It is known (see [12]) that the predictable projection (compensator) for 𝜇 has the following form: 𝜈𝑑𝑡,𝑑𝑦=_𝑛≥0𝟏𝜏_𝑛<𝑡≤𝜏_𝑛+1𝑃𝑥_𝑛,𝑑𝑦𝑔_𝑥𝑛(𝑡)𝐺_𝑥𝑛(𝑡)𝑑𝑡, (3.8) where 𝐺_𝑥(𝑡)=1−𝐺_𝑥(𝑡), 𝑔_𝑥(𝑡)=𝑑𝐺_𝑥(𝑡)/𝑑𝑡, for all 𝑥∈𝑋.

3.3. Martingale Characterization of Semi-Markov Processes

Let 𝑥_𝑡 be a semi-Markov process with semi-Markov kernel 𝑄(𝑥,𝐵,𝑡)=𝑃(𝑥,𝐵)𝐺_𝑥(𝑡) and 𝛾(𝑡) be the current life.

Lemma 3.12. The process 𝑚^𝑓_𝑡∶=𝑓𝑥_𝑡,𝛾(𝑡)−^𝑡₀𝑄𝑓𝑥_𝑠,𝛾(𝑠)𝑑𝑠 (3.9) is a martingale with respect to the filtration ℱ^∗_𝑡∶=𝜎{𝑥_𝑠,𝜏_𝜈(𝑠);0≤𝑠≤𝑡}, where 𝑄 is the infinitesimal operator of Markov process (𝑥_𝑡,𝛾(𝑡)): ()=()𝑄𝑓𝑥,𝑡𝑑𝑓𝑥,𝑡+𝑔𝑑𝑡^𝑥(𝑡)𝐺_𝑥(𝑡)^𝑋𝑃𝑥,𝑑𝑦𝑓𝑦,𝑡−𝑓(𝑥,𝑡). (3.10)

This statement follows from Theorem 3.10 and the fact that (𝑥(𝑡),𝛾(𝑡)) is a Markov process (see Remark 3.6).

Let us calculate the quadratic variation of the martingale 𝑚^𝑓_𝑡.

Lemma 3.13. Let 𝑄 in (3.10) be such that if 𝑓∈Domain(𝑄), then 𝑓²∈Domain(𝑄). The quadratic variation ⟨𝑚^𝑓_𝑡⟩ of the martingale 𝑚^𝑓_𝑡 in (3.9) is equal to 𝑚^𝑓_𝑡=^𝑡₀𝑄𝑓²𝑥_𝑠,𝛾(𝑠)−2𝑓𝑥_𝑠,𝛾(𝑠)𝑄𝑓𝑥_𝑠,𝛾(𝑠)𝑑𝑠. (3.11)

This statement follows from Theorem 3.11 and the fact that (𝑥(𝑡),𝛾(𝑡)) is a Markov process (see Remark 3.6).

Lemma 3.14. Let the following condition (Novikov's condition) is satisfied 𝐸^𝑃1exp2^𝑡₀𝑄𝑓²𝑥_𝑠,𝛾(𝑠)−2𝑓𝑥_𝑠,𝛾(𝑠)𝑄𝑓𝑥_𝑠,𝛾(𝑠)𝑑𝑠<+∞,∀𝑓²∈Domain(𝑄). (3.12) Then 𝐸^𝑃𝑒^𝑓_𝑡=1, where 𝑒^𝑓_𝑡∶=𝑒^{𝑚𝑓𝑡−(1/2)⟨𝑚𝑓𝑡⟩}, (3.13) and 𝑒^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).

4. 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”). Assume the stock price 𝑆_𝑡 satisfies the stochastic differential eqaution 𝑑𝑆_𝑡=𝑆_𝑡𝜇𝑑𝑡+𝜎𝑥_𝑡,𝛾(𝑡)𝑑𝑤_𝑡, (4.1) where 𝜇∈𝑅 is the appreciation rate and 𝜎(𝑥_𝑡,𝛾(𝑡)) is the semi-Markov volatility, and the bond price 𝐵(𝑡) is 𝐵(𝑡)=𝐵₀𝑒^𝑟𝑡, (4.2) where 𝑟>0 is the risk-free rate of return (interest rate). Then the folllowing two theorems are valid.

Theorem 4.1 (Price of Variance Swap [3]). The value of a variance swap for semi-Markov stochastic volatility 𝜎(𝑥_𝑡,𝛾(𝑡)) equals to 𝑃_𝑣𝑎𝑟(𝑥)=𝑒^−𝑟𝑇1𝑇^𝑇₀𝑒^𝑡𝑄𝜎²(𝑥,0)𝑑𝑡−𝐾_𝑣𝑎𝑟. (4.3)

Theorem 4.2 (Price of Volatility Swap [3]). The value of a volatility swap for semi-Markov stochastic volatility 𝜎(𝑥_𝑡,𝛾(𝑡)) is 𝑃_𝑣𝑜𝑙(𝑥)≈𝑒^−𝑟𝑇1𝑇^𝑇₀𝑒^𝑡𝑄𝜎²()−𝑥,0𝑑𝑡1/𝑇²∫^𝑇₀∫^𝑇₀𝐸𝜎²𝑥_𝑡,𝛾(𝑡)𝜎²𝑥_𝑠,𝛾(𝑠)𝑑𝑡𝑑𝑠−(1/𝑇)∫^𝑇₀𝑒^𝑡𝑄𝜎(𝑥,0)𝑑𝑡²81𝑇∫^𝑇₀𝑒^𝑡𝑄𝜎²(𝑥,0)𝑑𝑡^3/2−𝐾_𝑣𝑜𝑙. (4.4) Here 𝑄 is defined as ()=()𝑄𝑓𝑥,𝑡𝑑𝑓𝑥,𝑡+𝑔𝑑𝑡^𝑥(𝑡)𝐺_𝑥(𝑡)^𝑋𝑃𝑥,𝑑𝑦𝑓𝑦,𝑡−𝑓(𝑥,𝑡). (4.5)

In this paper, we generalize these results for local semi-Markov volatilities.

5. 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 𝑑𝑆_𝑡=𝑆_𝑡𝑟𝑑𝑡+𝜎_loc𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)𝑑𝑤_𝑡 (5.1) with the volatility 𝜎∶=𝜎_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.

Remark 5.1. We note that process (𝑆_𝑡,𝑥_𝑡,𝛾_𝑡) is a Markov process on (𝑅₊,𝑋,𝑅₊) with infinitesimal operator ()=()𝑄𝑓𝑠,𝑥,𝑡𝜕𝑓𝑠,𝑥,𝑡+𝑔𝜕𝑡^𝑥(𝑡)𝐺_𝑥(𝑡)^𝑋𝑃𝑓(()𝑥,𝑑𝑦𝑠,𝑦,𝑡−𝑓𝑠,𝑥,𝑡)+𝑟𝑆𝜕𝑓𝑠,𝑥,𝑡+1𝜕𝑠2𝜎²(𝑠,𝑥,0)𝑆²𝜕²𝑓(𝑠,𝑥,𝑡)𝜕𝑠². (5.2)

Remark 5.2 (Local Semi-Markov Volatility). Let's suppose that the stock price 𝑆_𝑡 satisfies the following stochastic differential equation 𝑑𝑆_𝑡=𝑆_𝑡𝑟𝑑𝑡+𝜎𝑆_𝑡,𝑥_𝑡,𝑡𝑑𝑤_𝑡 (5.3) 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-Markov volatility to the previous one with local current semi-Markov volatility by the following expansion: 𝜎𝑆_𝑡,𝑥_𝑡,𝑡=𝜎𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)+𝜏_𝜈(𝑡)𝑑𝜎𝑆_𝑡,𝑥_𝑡,𝑡𝜏𝑑𝑡+𝑜_𝜈(𝑡). (5.4) The error of estimation will be 𝐸𝜎𝑆_𝑡,𝑥_𝑡,𝑡−𝜎𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)²≤𝐸𝜏_𝜈(𝑡)²×𝐶, (5.5) where 𝐶=max_{0≤𝑡≤𝑇}𝐸[𝑑𝜎(𝑆_𝑡,𝑥_𝑡,𝑡)/𝑑𝑡]².

In this way, it is enough to consider the case with the local current stochastic volatility in (5.1).

5.1. Minimal Martingale Measure

We consider the following (𝐵,𝑆) security market. Let the stock price 𝑆_𝑡 satisfies the following equation: 𝑑𝑆_𝑡=𝑆_𝑡𝜇𝑑𝑡+𝜎𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)𝑑𝑤_𝑡, (5.6) 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 𝐵(𝑡)=𝐵₀𝑒^𝑟𝑡, (5.7) 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).

Lemma 5.3. Under the assumption ∫^𝑇₀((𝑟−𝜇)/𝜎(𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)))²𝑑𝑡<+∞, a.s., the following holds:(1)There is a probability measure 𝑃^∗ equivalent to 𝑃 such that 𝑑𝑃^∗𝑑𝑃=exp^𝑇₀𝑟−𝜇𝜎𝑆_𝑡,𝑥_𝑡(𝑡)(𝑡)−1,𝛾𝑑𝑤2^𝑇₀𝑟−𝜇𝜎𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)²𝑑𝑡 (5.8) is its Radon-Nikodym density.(2)The discounted stock price 𝑍(𝑡)=𝑆_𝑡/𝐵(𝑡) is a positive local martingale with respect to 𝑃^∗ and is given by 𝑍(𝑡)−1=exp2^𝑡₀𝜎²𝑆_𝑡,𝑥_𝑠,𝛾(𝑠)𝑑𝑠+^𝑡₀𝜎𝑆_𝑡,𝑥_𝑠,𝛾(𝑠)𝑑𝑤^∗(𝑠), (5.9) where 𝑤^∗(𝑡)=∫^𝑡₀((𝑟−𝜇)/𝜎(𝑆_𝑡,𝑥_𝑠,𝛾(𝑠)))𝑑𝑠+𝑤(𝑡) is a standard Brownian motion with respect to 𝑃^∗.

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: 𝑑𝑆_𝑡=𝑆_𝑡𝑟𝑑𝑡+𝜎𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)𝑑𝑤^∗(𝑡), (5.10) and discounted process 𝑍(𝑡) has the presentation 𝑑𝑍(𝑡)=𝑍(𝑡)𝜎𝑆_𝑡,𝑥_𝑠,𝛾(𝑠)𝑑𝑤^∗(𝑡). (5.11)

Remark 5.6. A sufficient condition (Novikov's condition) for the right-hand side of (5.8) to be a martingale is 1𝐸exp2^𝑇₀𝑟−𝜇𝜎𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)²𝑑𝑡<+∞. (5.12)

6. 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 to𝑁𝜎²_𝑅(𝑥)−𝐾_𝑣𝑎𝑟, (6.1) where 𝜎²_𝑅(𝑥) is the realized stock variance (quoted in annual terms) over the life of the contract,𝜎²_𝑅(𝑥)1∶=𝑇^𝑇₀𝜎²𝑆_𝑡,𝑥_𝑠,𝛾(𝑠)𝑑𝑠, (6.2)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 𝜎²_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: 𝑃_𝑣𝑎𝑟(𝑥)=𝐸𝑒^−𝑟𝑇𝜎²_𝑅(𝑥)−𝐾_𝑣𝑎𝑟, (6.3) 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 𝜎²_𝑅(𝑥) is defined in (6.2).

Let us show how we can calculate 𝐸𝑉(𝑥), where 𝑉(𝑥)∶=𝜎²_𝑅(𝑥). For that we need to calculate 𝐸𝜎²(𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)).

We note (see Section 3.3, Lemma 3.12) that for 𝜎(𝑥)∈Domain(𝑄) the following process 𝑚^𝜎_𝑡∶=𝜎𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)−𝜎(𝑥,0)−^𝑡₀𝑄𝜎𝑆_𝑠,𝑥_𝑠,𝛾(𝑠)𝑑𝑠 (6.4) is a zero-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).

The quadratic variation of the martingale 𝑚^𝑓_𝑡 by Lemma 3.13 is equal to 𝑚^𝜎_𝑡=^𝑡₀𝑄𝜎²𝑆_𝑠,𝑥_𝑠,𝛾(𝑡)−2𝜎𝑆_𝑠,𝑥_𝑠,𝛾(𝑡)𝑄𝜎𝑆_𝑠,𝑥_𝑠(𝑡)𝜎,𝛾𝑑𝑠,₂(𝑠,𝑥,𝑡)∈Domain(𝑄). (6.5)

Since 𝜎(𝑆_𝑠,𝑥_𝑠,𝛾(𝑠)) satisfies the following stochastic differential equation: 𝑑𝜎𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)=𝑄𝜎𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)𝑑𝑡+𝑑𝑚^𝜎_𝑡 (6.6) then, we obtain from Itô formula (see [22]) that 𝜎²(𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)) satisfies the following stochastic differential equation: 𝑑𝜎²𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)=2𝜎𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)𝑑𝑚^𝜎_𝑡+2𝜎𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)𝑄𝜎𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)𝑑𝑡+𝑑𝑚^𝜎_𝑡, (6.7) where ⟨𝑚^𝜎_𝑡⟩ is defined in (6.5).

Substituting (6.3) into (6.5) and taking the expectation of both parts of (6.7), we obtain that 𝐸𝜎²𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)=𝜎²(𝑠,𝑥,0)+^𝑡₀𝑄𝐸𝜎²𝑆_𝑠,𝑥_𝑠,𝛾(𝑠)𝑑𝑠,𝑠=𝑆₀, (6.8) and solving this equation we have 𝐸𝜎²𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)=𝑒^𝑡𝑄𝜎²(𝑠,𝑥,0). (6.9) Finally, we obtain that (𝑥)=1𝐸𝑉𝑇^𝑇₀𝑒^𝑡𝑄𝜎²(𝑠,𝑥,0)𝑑𝑡. (6.10) In this way, we have obtained the following result (see (6.3)–(6.10)).

Theorem 6.1. The value of a variance swap for local current semi-Markov stochastic volatility 𝜎(𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)) is 𝑃_𝑣𝑎𝑟(𝑠,𝑥)=𝑒^−𝑟𝑇1𝑇^𝑇₀𝑒^𝑡𝑄𝜎²(𝑠,𝑥,0)𝑑𝑡−𝐾_𝑣𝑎𝑟, (6.11) where 𝑄 is defined in (5.2).

7. 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 𝑁𝜎_𝑅(𝑆)−𝐾_𝑣𝑜𝑙, (7.1) where 𝜎_𝑅(𝑆) is the realized stock volatility (quoted in annual terms) over the life of contract, 𝜎_𝑅(𝑆)∶=1𝑇^𝑇₀𝜎²_𝑠𝑑𝑠, (7.2)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: 𝑃_𝑣𝑜𝑙(𝑥)=𝐸𝑒^−𝑟𝑇𝜎_𝑅(𝑥)−𝐾_𝑣𝑜𝑙, (7.3) 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{𝜎²_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]): 𝐸𝜎²_𝑅(𝑆)≈𝐸{𝑉}−Var{𝑉}8𝐸{𝑉}^3/2, (7.4) where 𝑉∶=𝜎²_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}.

7.1. Pricing of Volatility Swap

From Brockhaus and Long approximation [20] we have (see also [19, page 16]): 𝐸𝜎²_𝑅(𝑆)≈𝐸{𝑉}−Var{𝑉}8𝐸{𝑉}^3/2, (7.5) where 𝑉∶=𝜎²_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{𝜎²_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=𝐸𝜎⁴_𝑅(𝑆). Taking into account the expression for 𝑉=𝜎²_𝑅(𝑆) we have: 𝐸[𝑉]²=1𝑇^2𝑇₀𝐸𝜎²𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)𝜎²𝑆_𝑠,𝑥_𝑠,𝛾(𝑠)𝑑𝑡𝑑𝑠. (7.6)

In this way, the variance of 𝑉, Var[𝑉], is Var[𝑉]=𝐸𝑉²−(𝐸[𝑉])²=1𝑇^2𝑇₀𝐸𝜎²𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)𝜎²𝑆_𝑠,𝑥_𝑠(𝑠)1,𝛾𝑑𝑡𝑑𝑠−𝑇^𝑇₀𝑒^𝑡𝑄𝜎²(𝑠,𝑥,0)𝑑𝑡². (7.7) Taking into account (7.3)–(7.7), we obtain:𝑃_𝑣𝑜𝑙(𝑠,𝑥)=𝑒^−𝑟𝑇𝐸𝜎_𝑅(𝑆)−𝐾_𝑣𝑜𝑙=𝑒^−𝑟𝑇𝐸1𝑇^𝑇₀𝜎²𝑆_𝑠,𝑥_𝑠,𝛾(𝑠)𝑑𝑡−𝐾_𝑣𝑜𝑙≈𝑒^−𝑟𝑇√(𝑉)𝐸𝑉−Var8(𝐸𝑉)^3/2−𝐾_𝑣𝑜𝑙=𝑒^−𝑟𝑇1𝑇^𝑇₀𝑒^𝑡𝑄𝜎²()−𝑠,𝑥,0𝑑𝑡1/𝑇²∫^𝑇₀∫^𝑇₀𝐸𝜎²𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)𝜎²𝑆_𝑠,𝑥_𝑠,𝛾(𝑠)𝑑𝑡𝑑𝑠−(1/𝑇)∫^𝑇₀𝑒^𝑡𝑄𝜎²(𝑠,𝑥,0)𝑑𝑡²8(1/𝑇)∫^𝑇₀𝑒^𝑡𝑄𝜎²(𝑠,𝑥,0)𝑑𝑡^3/2−𝐾_𝑣𝑜𝑙. (7.8) Summarizing (7.3)–(7.8), we have the following.

Theorem 7.1. The value of volatility swap for local current semi-Markov stochastic volatility 𝜎(𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)) is 𝑃_𝑣𝑜𝑙(𝑠,𝑥)≈𝑒^−𝑟𝑇1𝑇^𝑇₀𝑒^𝑡𝑄𝜎²()−𝑠,𝑥,0𝑑𝑡1/𝑇²∫^𝑇₀∫^𝑇₀𝐸𝜎²𝑆_𝑡,𝑥_𝑡,𝛾(𝑡)𝜎²𝑆_𝑠,𝑥_𝑠,𝛾(𝑠)𝑑𝑡𝑑𝑠−(1/𝑇)∫^𝑇₀𝑒^𝑡𝑄𝜎(𝑥,0)𝑑𝑡²8(1/𝑇)∫^𝑇₀𝑒^𝑡𝑄𝜎²(𝑠,𝑥,0)𝑑𝑡^3/2−𝐾_𝑣𝑜𝑙, (7.9) where 𝑄 is defined in (5.2).

Acknowledgments

The first author would like to thank the Department of Mathematics, Faculty of Economics, University of Rome 1 “La Sapienza”, Rome, Italy, very much for their hospitality and friendly environment during his June 2009 visit. The research of The first author is supported by NSERC.