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The Heston model is one of the most popular stochastic volatility models for option pricing to measure the volatility of different parameters in the financial market. In this work, we study the statistical analysis of Heston Model by partial differential equations. The model proposed by Heston takes into account non-lognormal distribution of the assets returns, leverage effect and the important mean reverting property of volatility. We have assayed on the return distribution on the basis of different values of correlation parameter and volatility, then we measure the effects of parameters
*ρ* (correlation coefficient) an
*σ* (standard deviation) for different situation such as
*ρ* > 0,
*σ* > 0,
* ρ* = 0,
*σ* = 0,
*ρ* < 0,
*σ* < 0 etc. On return distribution of Heston Model which indicates market condition for buyers and sellers to buy and sell options. All solvers used in this analysis are implemented using MATLAB codes and the simulation results are presented graphically.

In 1970, Fischer Black, Myron Scholes and Robert Merton derived the “Black- Scholes model” (sometimes known as “Black-Scholes-Merton”) which changed the way and impact the world of pricing derivatives using stocks as the underlying asset [

Another great obtainment in the financial market is Heston’s stochastic volatility model, which helps to resolve a shortcoming of the BS model. More precisely, we can say that models based on BS assume that the underlying volatility is constant over the life of the derivative and unaffected by the changes in the price level of the underlying security. Howsoever, these models cannot explain long-observed features of the implied volatility surface like as volatility smile and skew, which indicate that implied volatility does tend to vary with respect to strike price and expiry. By assuming that the volatility of the underlying price is a stochastic process rather than a constant, it becomes possible to model derivatives more accurately.

Now-a-days, Heston model is considered as one of the most popular stochastic volatility option pricing models, which is motivated by the widespread evidence that volatility is stochastic and that the distribution of risky asset returns has tail (s) longer than that of a normal distribution [

The real issue was whether the ideal behavior was defensible in the derivation of this formula, Heston made several simplifying assumptions [

The main goal of this paper is to measure true volatility and experimental volatility as well as compare them to examine the present situation of the share market. We want to measure the effect of different parameters of Heston model on return distribution and effect on implied volatility surface with strike and maturity as well as discuss the pricing through the Partial Differential Equation. This work is actually a statistical analysis of option price Heston model and a calculation of different error measurement to test the consistency of different functions.

Black and Scholes [

A1. The stock price follows the stochastic process d s = μ S d t + σ S d W , with fixed μ and σ ;

A2. Unrestricted short-selling of stock, with full use of short-sale proceeds;

A3. No transactions costs and taxes;

A4. No dividends are paid during the life of the option;

A5. There are no riskless arbitrage opportunities:

A6. It is based on European options;

A7. The risk-free rate of interest r is constant and same for all maturities;

A8. Continuous trading;

In order to make a price for a call option on a non-dividend paying stock with the BS Equation, we need to know current stock price, strike price, risk-free interest rate, volatility and time to maturity. It is easy to get all above inputs variables in the market except the volatility. For the price of a non-dividend paying call option, the BS equation is described as:

C ( S , t ) = S N ( d 1 ) − K e − r ( T − t ) N ( d 2 )

where, d 1 = ln ( s k ) + ( r + σ 2 2 ) ( T − t ) σ ( T − t ) and d 2 = d 1 − σ ( T − t )

Here, S is the stock price at time t, T is the maturity date, K is the strike price, N ( d 2 ) is the cumulative normal distribution, σ is the volatility. Although Black-Scholes equation is still widespread used in the market, much evidence has shown that the assumption of fixed volatility is not suitable for actual data. Consequently, in this dissertation, we consider the volatility following a stochastic process rather than a constant during the life of a call option.

The crude assumption of constant volatility in the Black-Scholes formula causes problem. One model where the volatility is a stochastic process is the Heston Stochastic Volatility Model [

d S t = μ S t d t + V t S t d W t 1 (1)

d V t = κ ( θ − V t ) d t + σ V t d W t 2 (2)

and where W t 1 and W t 2 are correlated Wiener processes with ρ, i.e.

d W t 1 d W t 2 = ρ d t (3)

where, μ is the drift coefficient of the stock price, θ is the long term mean of variance, κ is the rate of mean reversion, σ is the volatility of volatility, s t and v t are the price and volatility process respectively, where { v t } t ≥ 0 is a square root mean reverting process, first used by with long run mean θ , and rate of reversion κ . To take into account the leverage effect, stock returns and implied volatility are negatively correlated, w t 1 and w t 2 are correlated wiener process and the correlation coefficients is ρ. All the parameters μ , κ , θ , σ , ρ are the time and state homogeneous.

For some stochastic volatility models, one can find a partial differential equation (PDE), the value of any option must be satisfied by such a PDE. For Heston’s Stochastic Volatility model, a PDE exists, but calculation is quite complicated due to the difficult estimation of the market price of volatility risk. In order to price options in the SV model, we can apply no-arbitrage arguments, or use the risk-neutral valuation method. First we discuss the no-arbitrage method. The riskless portfolio is constructed as in the Black-Scholes model. But the construction method is different. In the SV option pricing model, there is only one traded risky asset S but two random sources d W t 1 and d W t 2 . So the market is incomplete. We cannot perfectly replicate the option solely with the underlying stock. No-arbitrage arguments are not enough to give the option price. We need additional assumptions. In the following derivation, equilibrium arguments are also employed. We know that the market can be completed by adding any option written on stock S. Simply, the market is complete when we have two traded assets, the underlying asset S and a benchmark option V 1 . Then all other options can be replicated by these two traded assets.

To proceed, consider a self-financing/risk-less portfolio with value ∏ consisting of an option with value V ( S , v , t ) which we want to price, − Δ units of the underlying asset S and, in order to hedge the risk associated with the random volatility, − Δ 1 units of benchmark option with value V 1 ( S , v , t ) .

Hence,

∏ = V − Δ S − Δ 1 V 1 (4)

The portfolio is self-financing i.e. for risk-less portfolio, so that

d ∏ = d V − Δ d S − Δ 1 d V 1 (5)

By applying two dimensional form of Ito’s formula, we have

d ∏ = { ∂ V ∂ t + 1 2 v S 2 ∂ 2 V ∂ S 2 + ρ σ S v ∂ 2 V ∂ S ∂ v + 1 2 v σ 2 ∂ 2 V ∂ v 2 } d t + ∂ V ∂ S d S + ∂ V ∂ v d v − Δ 1 { ∂ V 1 ∂ t + 1 2 v S 2 ∂ 2 V 1 ∂ S 2 + ρ σ S v ∂ 2 V 1 ∂ S ∂ v + 1 2 v σ 2 ∂ 2 V 1 ∂ v 2 } − Δ 1 ∂ V 1 ∂ S d S − Δ 1 ∂ V 1 ∂ v d v − Δ d S (6)

Now, we can rewrite it by collecting the terms of dS, dt and dv

d ∏ = { ∂ V ∂ t + 1 2 v S 2 ∂ 2 V ∂ S 2 + ρ σ S v ∂ 2 V ∂ S ∂ v + 1 2 v σ 2 ∂ 2 V ∂ v 2 } d t + ( ∂ V ∂ v − Δ 1 ∂ V 1 ∂ v ) d v − Δ 1 { ∂ V 1 ∂ t + 1 2 v S 2 ∂ 2 V 1 ∂ S 2 + ρ σ S v ∂ 2 V 1 ∂ S ∂ v + 1 2 v σ 2 ∂ 2 V 1 ∂ v 2 } d t + ( ∂ V ∂ S − Δ 1 ∂ V 1 ∂ S − Δ ) d S (7)

To make the portfolio riskless, we choose

∂ V ∂ v − Δ 1 ∂ V 1 ∂ v = 0 (8)

∂ V ∂ S − Δ 1 ∂ V 1 ∂ S − Δ = 0 (9)

To eliminate the terms dS and dv, we solve the Equation (8) and (9) as

Δ 1 = ∂ V ∂ v ∂ V 1 ∂ v (10)

Δ = ∂ V ∂ S − Δ 1 ∂ V 1 ∂ S = ∂ V ∂ S − ∂ V ∂ v ∂ V 1 ∂ v ∂ V 1 ∂ S (11)

The portfolio is risk free if we rebalance the Equation (7) according to (10) and (11). On the other hand, the riskless portfolio must earn a risk free rate, i.e. the return of this risk-free portfolio must equal the (deterministic) risk-free rate of return. Otherwise, there would be an arbitrage opportunity.

d ∏ = { ∂ V ∂ t + 1 2 v S 2 ∂ 2 V ∂ S 2 + ρ σ S v ∂ 2 V ∂ S ∂ v + 1 2 v σ 2 ∂ 2 V ∂ v 2 } d t − Δ 1 { ∂ V 1 ∂ t + 1 2 v S 2 ∂ 2 V 1 ∂ S 2 + ρ σ S v ∂ 2 V 1 ∂ S ∂ v + 1 2 v σ 2 ∂ 2 V 1 ∂ v 2 } d t

= r ∏ d t = r ( V − Δ S − Δ 1 V 1 ) d t (by using Equation (7))

∴ d ∏ = r ( V − Δ S − Δ 1 V 1 ) d t (12)

Using above two equations we have,

∂ V ∂ t + 1 2 v S 2 ∂ 2 V ∂ S 2 + ρ σ S v ∂ 2 V ∂ S ∂ v + 1 2 v σ 2 ∂ 2 V ∂ v 2 + r S ∂ V ∂ S − r V ∂ V ∂ v = ∂ V 1 ∂ t + 1 2 v S 2 ∂ 2 V 1 ∂ S 2 + ρ σ S v ∂ 2 V 1 ∂ S ∂ v + 1 2 v σ 2 ∂ 2 V 1 ∂ v 2 + r S ∂ V 1 ∂ S − r V 1 ∂ V 1 ∂ v (13)

Notice that the left-hand side is a function of V only and the right-hand side is a function of V_{1} only. The only way that this equation holds is that both sides are equal to some function, i.e. g only depends on the independent variables S , v and t. Setting g = ( κ ( θ − V ) − Λ ( S , v , t ) σ V ) yields a special case of a so- called affine diffusion process. For this class of processes, the pricing PDE is tractable analytically. In this case we have,

∂ V ∂ t + 1 2 v S 2 ∂ 2 V ∂ S 2 + ρ σ S v ∂ 2 V ∂ S ∂ v + 1 2 v σ 2 ∂ 2 V ∂ v 2 + r S ∂ V ∂ S − r V = ( κ ( θ − V ) − Λ ( S , v , t ) σ V ) ∂ V ∂ v (14)

Now replacing V by U ( S , V , t , T ) and v by V, we have

1 2 V S 2 ∂ 2 U ∂ S 2 + ρ σ S V ∂ 2 U ∂ S ∂ V + 1 2 V σ 2 ∂ 2 U ∂ V 2 + r S ∂ U ∂ S − r U + ∂ U ∂ t + ( κ ( θ − V ) − Λ ( S , v , t ) σ V ) ∂ U ∂ V = 0 (15)

Λ ( S , v , t ) is called the market price of volatility risk. According to Heston’s assumption, the market price of volatility risk is proportional to volatility i.e. to the square root of the variance.

Λ ( S , v , t ) α V ⇒ Λ ( S , v , t ) = k V , (16)

where k is the proportional constant.

Multiplying both sides of Equation (16) by σ V , then we have,

Λ ( S , v , t ) σ V = k σ V

Thus Equation (16) becomes,

1 2 V S 2 ∂ 2 U ∂ S 2 + ρ σ S V ∂ 2 U ∂ S ∂ V + 1 2 V σ 2 ∂ 2 U ∂ V 2 + r S ∂ U ∂ S − r U + ∂ U ∂ t + ( κ ( θ − V ) − λ V ) ∂ U ∂ V = 0 (17)

Therefore λ ( S , V , t ) represents the market price of volatility risk. The price of volatility risk λ ( S , V , t ) is independent of particular asset. It can be obtained theoretically from any asset depending on volatility risk. Assume the strike price to be K and expiring time T. The price is considered in rectangular area of [ 0 , ∞ ] × [ 0 , ∞ ] and on horizontal time [ 0 , T ] . Then For European call option the option price obeys Equation (4.14) with boundary,

U ( S , v , t ) = max ( 0 , S − k ) U ( 0 , v , t ) = 0 ∂ U ∂ S ( ∞ , v , t ) = 1 r S ∂ U ∂ S ( S , 0 , t ) + κ θ ∂ U ∂ V ( S , 0 , t ) − r U ( S , 0 , t ) + ∂ U ∂ t ( S , 0 , t ) = 0 U ( S , ∞ , t ) = S } (18)

This choice of market price of volatility risk gives us analytical advantages. The drift term of the specified process (4) is an affine function of the state variable itself. The affinity makes the model easier to solve. Since the diffusion of the variance process is also proportional to the square root of the variance, the product of the market price of risk and the diffusion is proportional to variance itself. As a result, the drift term will remain affine under the Equivalent Martingale Measure (EMM). This particular market price of volatility risk helps the model to have a closed-form solution. We can also apply the risk-neutral valuation method to the SV model. The market is incomplete. But it is still free of arbitrage. The equivalent martingale measure is not unique. We have to choose one of all these measures to price the options. So the price of the option is also not unique. It will depend on which equivalent martingale measure we use.

There are many economic, empirical, and mathematical reasons for choosing a model with such a form for a detailed statistical/ empirical analysis).Empirical studies have shown that an asset’s log-return distribution is non-Gaussian. It is characterized by heavy tails and high peaks (leptokurtic). There is also empirical evidence and economic arguments that suggest that equity returns and implied volatility are negatively correlated (also termed ‘the leverage effect’). This departure from normality plagues the Black-Scholes-Merton model with many problems. In this work, we will show effect of effects of ρ and σ on return distribution.

Consider that ρ denotes the correlating factor between the sources of randomness for the underlying and the volatility. ρ can be interpreted as the correlation between the returns and the volatility of the asset. Therefore it captures the leverage effect, affecting the heaviness of the tails, thus the skewness of the return distribution. Intuitively, if ρ < 0, then volatility will increase as the asset price return decreases, this will spread the left tail and squeeze the right tail of the distribution creating a fat left-tailed distribution. Conversely, if ρ > 0, then volatility will increase as the asset price/return increases. This will spread the right tail and squeeze the left tail of the distribution creating a fat right-tailed distribution and if ρ = 0 the skewness is close to zero. As a result ρ, affects the skewness of the distribution. Figures 1-5 show the effect ρ for different values. However, Figures 4-12 are similar to those presented in [

The σ affects to the kurtosis (peak) of the distribution. When σ is zero the volatility is deterministic, because the diffusion process in dV_{t} will be dropped and hence the returns will be normally distributed as in the BSM-model. Increasing σ will increase the peak (kurtosis), creating heavy tails on both sides, i.e. the increase in σ represents the market volatility is more volatile and higher σ shows higher peaks than less one.

In Figures 8-10, we investigate the effect of ρ on the implied volatility surface generated under Heston’s model. Here Maturity (years), Strike and Implied Volatility are denoted by x-label, y-label and z-label respectively. In

In

In

Stochastic volatility models tackle one of the most restrictive hypotheses of the Black-Scholes model framework, which assumes that volatility remains constant during the option’s life. However, by observing financial markets it becomes apparent that volatility may change dramatically in short time periods and its behavior is clearly not deterministic. Among stochastic volatility models, the Heston model presents two main advantages. First, it models an evolution of the underlying asset which can take into account the asymmetry and excess kurtosis that are typically observed (and expected) in financial asset returns. Second, it provides closed form solutions for the pricing of European options.

The study made in this paper demonstrated a technique for constructing smile and skew consistent prices by violating one of the crude assumptions in the BS model, constant volatility. The result shows that the Heston approximation works really well and only face big problems when options with high time to maturity are to be priced. Another problem is that the approximation gives us incorrect prices when the moneyless is below one. To reduce this problem further studies of the volatility smile could be done and were the skew of options that are not in the money could be compare to options that are in the money and trying to repair this. As one could observe from the results above is that the Heston approximation loses its accuracy as the time to maturity increases, but Black and Scholes is also facing the same type of problem. Since the Heston model was not built on the assumption on non-constant volatility, it showed an improvement of modeling stocks and receiving smile consistent option prices.

The authors would like to express sincere thanks to anonymous reviewers for helpful comments on an earlier version of this paper, and they are also grateful to an author assistant of Scientific Research Publishing and an editorial assistant of Journal of Mathematical Finance.

Mondal, M.K., Alim, M.A., Rahman, M.F. and Biswas, M.H.A. (2017) Mathematical Analysis of Financial Model on Market Price with Stochastic Volatility. Journal of Mathematical Finance, 7, 351-365. https://doi.org/10.4236/jmf.2017.72019