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We study the valuation of the variance swaps under stochastic volatility with delay
and jumps. In our model, the volatility of the underlying stock price process not only
incorporates jumps, which are found to be active empirically, but also exhibits past dependence: the behavior of a stock price right after a given time

Variance swaps are forward contracts on future realized variance, the square of the realized volatility, which provide an easy way for investors to gain exposure to the future realized variance of the asset returns instead of directly exposure to the underlying assets. The market for such derivatives develops quickly after the collapse of LCTM in 1998 when the volatilities increased to an unprecedented high level and many investors are now interested in these derivatives to hedge volatility. Recently, several papers address the valuation of variance swaps or other volatility derivatives (see [

It is known that the assumption of constant volatility in Black-Scholes model [

There are various works showing delayed response is an important factor in stock prices. Some statistical studies of stock prices (see [

Our model of stochastic volatility exhibits past dependence: the behavior of a stock price right after a given time

In addition to delay, jumps are another evidence in the financial market. During the last decade, financial models based on jumps processes have acquired increasing popularity in risk management and option pricing applications. A good reference is [

The literature has mainly focused on two approaches: (1) time-varying volatility models that allow for market extremes to be outcome of normally distributed shocks that have a randomly changing variance and (2) models that incorporate discontinuous jumps in the asset price. Neither stochastic volatility models nor jump models have alone proven entirely empirically successful. For example, in the time-series literature, the models run into problems explaining large price movements such as the October 1987 crash. Hence, a price jump cannot explain the enormous increase in implied volatility following the crash of 1987. In response to these issues, researchers have proposed models that incorporate both stochastic volatility and jumps components.

Eraker et al. [

The key risk factors considered in option pricing models, besides the diffusive price risk of the underlying asset, are stochastic volatility and jumps, both in the asset price and its volatility. Models that include some or all of these factors were developed in [

There is currently fairly compelling evidence for jumps in the level of financial prices. The most convincing evidence comes from recent nonparametric work using high-frequency data as in [

The jumps in stock market volatility are found to be so active that this discredits many recently proposed stochastic volatility models without jumps (see [

Another advantage of our stochastic volatility model with delay and jumps is mean-reversion: the volatility is allowed to mean revert. Such models have shown some success in modeling interest rate (e.g., [

In this paper, we incorporate a jump part in the stochastic volatility model with delay proposed by Swishchuk [

The rest of the paper is organized as follows. In Section

A

The measure of realized variance which will be used is defined at the beginning of the contract. The continuous time realized variance over the life of the contract

The discrete-time variance

Valuing a variance forward contract or swap is no different from valuing any other derivative security. The value of a forward contract on future realized variance with strike price

In this paper, we are interested in valuing variance swaps for security markets when stochastic volatility

In this way,

In this section, we recall the model and approach of pricing variance swaps for stochastic volatility with delay and without jumps presented in the paper of Swishchuk [

In our model, we assume that the price of the underlying asset

Throughout the paper, we denote

Now we can rewrite the equation of

We suppose that the following conditions are satisfied,

Condition (C1) guarantees the existence and uniqueness of a solution of (

To price the variance swaps, we need to calculate

Let

There is a probability measure

The discounted asset price

Therefore, in the risk-neutral world, the underlying asset price

Let us take the expectations under risk-neutral measure

We note that this equation has stationary particular solution and approximate general solution.

In this way, we have obtained the following result in [

If conditions (C1)–(C3) are satisfied, then the price of a variance swap at time

In the following section, we will derive some analytical closed formulae for the expectation of the realized variance for stochastic volatility with delay and jumps. First of all, we need to define the jumps and add them to the stochastic volatility model with delay. In (

We assume that the price of the underlying asset

Recall that our purpose is to calculate

Let

Note that the change of measure does not change the Poisson intensity

In the risk-neutral world, the volatility can be defined as follows:

Now let us take the expectation under risk-neutral probability

Notice that (

Hence, the expectation of the realized variance or say the fair delivery price

There is no way to write a solution in explicit form for arbitrarily given initial data. But we can understand an approximate behavior of solutions of (

The only solution to this equation is

In this way, we have that

Hence, the expectation of the realized variance or say the fair delivery price

Of course, (

Summarizing, we have the following result.

Consider stock price satisfying (

In the section we will consider the jumps represented by a compound Poisson process, and since it allows the jumps size to be a random number but not always one in Poisson process, the model is more realistic. Our approach in the last section can be easily used in compound Poisson process case.

In the risk-neutral world, the volatility can be defined as follows:

Our first step is still taking the expectation under risk-neutral probability

Equation (

In general case, we substitute

Therefore, the only solution to this equation is

Of course, (

Summarizing, we have the following result.

Consider a stock price satisfying (

It is interesting to see that when

In the previous section, we assume that the mean value and variance of the jump size

In the risk-neutral world, the volatility still satisfies the following equation:

Let

In order to compute the two expectations in this equation, we first introduce two lemmas as follows (see [

Define

Define

Therefore,

Taking into account (

To get the expectation of the realized variance in the risk-neutral world

After taking the first derivative of this equation, we obtain

Let us denote the Laplace transform of a function

By change of variable and the property of Laplace transform, (

The characteristic function of (

Hence, the expectation of the realized variance or say the fair delivery price

Of course, (

Summarizing, we have the following result.

Consider a stock price satisfying (

From expression (

Volatility is the standard deviation of the change in value of a financial instrument with specific time horizon. It is often used to quantify the risk of the instrument over that time period. The higher the volatility is, the riskier the security is. The variance is a square of volatility and is also a measure of risk of a financial instrument. In this section we look at the estimation for variance through the delay to minimize the risk.

The compound Poisson case in Section

By Section

This expression contains all the information about our model since it contains all the initial parameters. We note that

If

If

Therefore,

To reduce the risk we need to take into account the following relationship with respect to the delay

In this case, there is a way to control the delay.

In this section, we apply the analytical solutions from Section

In the end of February 2002, we wanted to price the fixed leg of a variance swap based on the

Statistics on log-returns of

Series | Log-returns of |
---|---|

Sample | |

Observations | 1300 |

Mean | 0.000235 |

Median | 0.000593 |

Maximum | 0.051983 |

Minimum | −0.101108 |

Std. Dev. | 0.013567 |

Skewness | −0.665741 |

Kurtosis | 7.787327 |

From the histogram of the

There are several parameters which need to be estimated from the data, the jump intensity

From the data in Table

A GARCH(1,1) regression is applied to the series, and the results are obtained as in Table

Estimation of the GARCH(1,1) process.

Dependent variable: log-returns of | ||||

Method: ML-ARCH | ||||

Included observations: 1300 | ||||

Convergence achieved after 28 observations | ||||

— | Coefficient | Std. error | Prob. | |

C | 0.000617 | 0.000338 | 1.824378 | 0.0681 |

Variance equation | ||||

C | 2.58 | 3.91 | 6.597337 | 0 |

ARCH(1) | 0.060445 | 0.007336 | 8.238968 | 0 |

GARCH(1) | 0.927264 | 0.006554 | 141.4812 | 0 |

R-squared | −0.000791 | Mean dependent var | — | 0.000235 |

Adjusted R-squared | −0.003108 | S.D. dependent var | — | 0.013567 |

S.E. of regression | 0.013588 | Akaike info criterion | — | −5.928474 |

Sum squared resid | 0.239283 | Schwartz criterion | — | −5.912566 |

Log-likelihood | 3857.508 | Durbin-Watson stat | — | 1.886028 |

We use the following relationship:

ARCH(1,1) coefficient

Parameter

Now, applying the analytical solutions (

Dependence of delivery price on maturity (

Dependence of delivery price on delay (

Dependence of delivery price on jump intensity (

Dependence of delivery price on delay and jump intensity (

Dependence of delivery price on delay and maturity (

Dependence of delivery price on jump intensity and maturity (

We note that we provided back testing in [

In this paper we studied stochastic volatility model with delay and jumps to price variance swaps. We applied a general approach to derive the analytical close forms for expectation of the realized continuously sampled variance for stochastic volatility with delay and jumps. The jump part in our model is represented by a general version of compound Poisson processes. The key features of the model are the following: (i) continuous-time analogue of discrete-time GARCH model, (ii) mean reversion, (iii) containing the same source of randomness as stock price, (iv) completeness of the market, (v) incorporating the expectation of log-return, and (vi) incorporating the jumps in volatility. The model is also easy to implement and time saving. Besides, we presented a lower bound for delay as a measure of risk. From the numerical example, we found that after adding jumps in volatility, the expectation of the realized variance is higher than the one without jumps for variance swaps. It is easy to explain it since the existence of jumps means the market is more risky which asks for higher cost of variance swaps.

For further study, we may add jumps in the spot price of the underlying asset which is also an important characteristic found in real markets. Besides, the jump part can be in more complicated form, for example,

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The research is supported by NSERC Grant RPG 312593. The authors would like to thank an anonymous referee very much for valuable comments and suggections that improved the present paper. The authors remain responsible for any errors in this paper.