A Linear Regression Approach for Determining Explicit Expressions for Option Prices for Equity Option Pricing Models with Dependent Volatility and Return Processes

We consider a risk-neutral stock-price model where the volatility and the return processes are assumed to be dependent. The market is complete and arbitrage-free. Using a linear regression approach, explicit functions of risk-neutral density functions of stock return functions are obtained and closed form solutions of the corresponding Black-Scholes-type option pricing results are derived. Implied volatility skewness properties are illustrated.


Introduction
Stochastic volatility (SV) modeling is the subject of several papers in the option price literature.By assuming that the volatility and the return processes of a stock price model are correlated, one can explain better the skewness of the implied volatility curve.Apart from the single-factor CEV model [1], the models proposed are mostly variations of 2-factor affine-jump diffusion models, [2]- [4], with one of the factors being stock volatility.The 2-factor affine model [2] assumes correlated volatility and asset return processes.In [2], however, one has to numerically integrate conditional characteristic functions obtained as solutions of nonlinear pdf to derive the call option prices.The case of the two factors, namely the asset price and volatility being uncorrelated, is considered in the paper [5], which obtains Call Option Price Conditional on the variance rate 2  V and derives the unconditional call price by integrating using an approximate probability density function ( ) g V .The paper [6] consi- ders stochastic forward rate processes which are lognormally distributed conditional on the volatility state variables.See also [7] pp 182-183, for other numerical approximation methods.Some of the well-known numerical procedures for deriving option pricing that are tree-based binomial or treebased trinomial are available in [8] and [9].GARCH based heteroscedacity models are discussed in [10]- [13] where empirical versions of SV models in discrete time are approached.
In the next section, the proposed two-factor stock price model that allows the volatility factor and the Brownian motion return processes to be dependent and a linear regression approach that derives explicit expressions for the distribution functions of log return of a stock or stock index are used.
In the subsequent section, we obtain a closed form formula for the call option price that has an algebraic expression that is similar to that of a Black-Scholes model, making it much easier to compute its value.
In the following section, we define an implied volatility function and derive its skewness property.Finally, we provide concluding remarks and suggestions for future direction.

Heston's Stochastic Volatility Model
It is known that under a Black-Scholes model formulation the implied volatility function must remain constant for different values of the strike price when the other parameters of the option pricing model are kept constant.However, skewness in implied volatility curves is observed in actual market data for European options.To explain the skewness property of implied volatility functions, [2] considers the following model (1)-( 3) with the condition that the (2) asset price and volatility are correlated: ( ) ( ) ( ) ( ) ( ) ( ) ( ) where , 1, 2 j z j = are Brownian processes.Note that it can be shown, applying the Ito formula, that the variance rate t ν has a square root process model (see [2]).Computation of option price in the case of the above correlated model as described in using a pdf is fairly complicated.To obtain a closed form solution for the option price one has to invert two conditional characteristic functions to compute the difference between two probability functions as the required solution of the pdf.

A Two-Factor Stochastic Volatility Model
Here, we will explicitly specify the sde of the asset price and volatility processes.In this paper, we consider a risk-adjusted diffusion process (4) for spot asset price ( ),0 defined with respect to a probability space ( ) , with the data-gathering measure P ( ) ( ) ( ) In (4), R υ ∈ is called the instantaneous diffusion rate and µ is called the instantaneous drift rate of the diffusion process.
In (5), we have a log normal model for the asset price ( ),0 .

X u u s ≤ ≤
At this point, we introduce a second factor ( ) H u , which is a mean-reverting process, in Equation (7), and corresponds to the volatility (6) can be transformed to

Formulation of a Risk-Neutral Model
The dynamic processes (8)-( 9) below are defined with respect to the martingale probability measure Q, where ( ) B u and ( ) Y u are Brownian motions under Q, where we assume the corresponding Novikov's condition is satisfied.
As mentioned previously, in (4), is the instantaneous diffusion rate and µ is called the instantaneous drift rate of the diffusion process.
As stated previously, in Equation (7), we define the volatility as a mean reverting Gaussian process with θ as its long-term mean 1 .
We assume

( )
H u to be correlated with ( ) B u as in the Equation (8) and that ( ) Y u is a standard Brow- nian motion process.

( )
H u may be expressed as ( ) ( ) ( ) where ( ) See [15] for a similar assumption.See also [3] and [4].From (6) and (10), it is clear that  11) follows because from [16] we know that the Gaussian random variable ( ) H u may be ex- pressed as , where e as the average standard deviation in the case of uncorrelated Brownian motion process Then the average variance is: and where , , s e s e s ε = + where ( ) ( ( ) , 0 Cov e s e s = ) s e s e s ε = + is also normally distributed.
Proposition 2: where the expectation is obtained using the risk neutral distribution of ( ), X s as defined in (6).

Remark 3:
Proposition 2 restates the result that the risk neutral property of ( ), 0.

X t t ≥
holds; the normalized process is a martingale with respect to Q and the market

Derivative Securities
We can evaluate any security that is a derivative of ( ) X s using the risk neutral probability distribution of ( ) X s .In particular, consider a non-dividend paying European call option with strike price K and maturity dates 2 .Then the price ( )  ( ) , where the expectation is obtained using the risk neutral distribution of ( ) X s .Similarly the put option is defined as ( ) , , , , , e . Then, using Put Callparity formula and the Equation (18) we have In the next sections, we will derive a simple Black-Sholes type expression for the call option price and derive its properties.

Implied Risk-neutral distribution of lnX(s)
For easier reference we present below the explicit expressions for the vector where the conditional risk-neutral distribution function of Next we determine an explicit expression for the conditional distribution function So given ( ) Then the roots of the equation Assumption 3: Assumption (3) ensures that the roots are real and are well defined.Let , , ln | ,

s n s m s p s h z z h z h m s n s n s m s p s h
and also suppose Assumption (2) holds.Note that the functions ( ) m s and ( ) Z s and a minimum of ( ) ( ) ln X s as a function of ( ) Similarly if is a concave function of ( ) Z s and a maxi- mum of ( ) ( ) ln X s as a function of ( ) Z s exists.
If Assumption (3) holds then the conditional risk-neutral distribution of )

X s z h z h h F h P X s s h h z h z h h F h P X s s h h n s n s m s p s h z z h z h m s n s h h p s m s
then the roots of the equation defined in (18) are equal so that , n s z z z m s , then there exists a value ( ) In other words, ( ) * , , h ω ρ ξ is the lowest value the conditional random variable assume in this case.
Next we consider the case of , , ,

X s X s z h z h h F h P X s s h V s h z h z h h F h h n s n s m s p s h z z h z h m s
, , , , , , 0.06, 0.08, 0.20, 0.5, 0.8, 0.1 r υ α η θ ρ κ = − . Then in Figure 1 depicts the conditional risk-neutral distribution of In the next section we consider the evaluation of price of a security that is derivative of stock price ( ) X s .
We need the following Assumption (4) to ensure that the call option price is well defined.
Figure 2 shows the conditional risk-neutral distribution of   We will utilize the Assumption (4) later for deriving the price of any derivative security.

Conditional Call Option Price
Next we obtain an explicit closed form expression for the conditional call option price that is similar to the corresponding B-S expression and hence is easier to compute.

s n s m s p s h z z h z h m s n s n s m s p s h h h p s m s
is the cdf of the standard normal variable Z.

Remark 5:
To simplify the presentation of the results, we have suppressed usually the dependence of ( ) exp , e , for ln , , 2 1 1 We prove Proposition 4 below using the risk-neutral distribution results (Proposition 3) of lnX(s) for 0 ρ > .

K C x K r h x z h m s n s z h m s n s m s m s m s n s p s h
Figure 3 shows the unconditional hedge ratio as derived using (28).(ii) Since 0 10, x = we have, if 10, K < the option is said to be in-the-money; if 10 K = , the option is atthe-money and if 10, K > then the option is out-of-the money.(iii) Subject to the condition (22), it can be verified that the call option price function increases (i) as time to maturity s increases and (ii) as ρ increases.

Delta-Neutral Portfolio
Consider the following portfolio that includes a short position of one European call with a long position delta units of the stock.
(i) The portfolio of delta-neutral positions is defined as where the expectation is obtained using the risk neutral distribution of ( ) X s .Here the investor can exercise the option at time s if ( ) . However we have the relationship in terms of conditional distribution of ( ) ln X s given ( ) rs rs rs Unconditional Call Option Price

Put-Call Parity
The Put option price is obtained using Put-Call parity: Again, we can apply the discrete approximation numerical method as in (26) in evaluating (27).
In other words, we find a suitable value for implied volatility σ * so that call option price values both under the new model with parameter values ( ) With a view to explaining this anomaly, several different models have been proposed in the option-price literature.These models are mostly variations of 2-factor affine-jump diffusion models, one of the factors being stock volatility 3Let ( ) ( ) 0 , , , , , , , 10, 0.06,1, 0.6, 0.2, 0.05, 0.08, 0.8 x r α η κ θ υ ρ = − In this section, we show that the implied volatility skewness property of negative correlation-0 ρ < model.The "implied volatility smile curves are rotated clock wise into smirks", which is known as "Volatility asymmetry".See [4], p. 350.The implied volatility can be easily computed and is an increasing function of the time to maturity s-(see Figure 8).

Conclusion
In this paper, we formulate a two-factor model of a stock index, where we assume the volatility process and the Brownian motion process of the model are dependent and use a novel linear regression approach to obtain call option price expressions for the proposed model.We have obtained closed form Black-Scholes type expressions  for option prices under the assumption of constant interest rate.We can also show stochastic interest rate and random economic shocks can also be incorporated in the model (see [21]- [23]).Analyzing the proposed model is computationally simpler than it is for the other affine jump process models.The results of this paper can also be applied to bond option, foreign currency option and futures option models and to more complex derivative securities including various types of mortgage-backed securities.

1 B
u are related as follows:

2 , 0 ; 2 :=
Cov e s e s = (17) Proof: See Appendix A Assumption Some of the limitations of the model can be described as follows: a) Since we can verify that ( , we have only the necessary condition for independence between ( ) s ε and ( ) , U s ξ is satisfied.b) We have assumed that the error terms ( ) the linear regressions are normally distributed and that ( ) ( ) ( ) 1 2 the call option is the present value of the expected terminal value,

If Assumption ( 3 )
holds then the conditional risk-neutral distribution of
s ≠ and ( ) , p s h are as defined in (
The hedge ratio expressions are similarly derived for the case of 0 ρ > using results in Proposition 4.Conditional Put-Call ParityConsider a non-dividend paying European put option with strike price K and exercise date s.Then the price the put option is the present value of the expected terminal value, the option price (26) numerically as follows:
are equal.Implied volatility is a popular estimate of future stock price volatility, obtained from option price data.It is known that under a Black-Scholes model formulation the implied volatility function must remain constant for different values of the strike price when the other parameters of the option pricing model are kept constant.However, skewness in implied volatility curves is observed in actual market data for European options.