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.
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 [
tional call price by integrating using an approximate probability density function
ders stochastic forward rate processes which are lognormally distributed conditional on the volatility state variables. See also [
Some of the well-known numerical procedures for deriving option pricing that are tree-based binomial or tree- based trinomial are available in [
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.
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, [
where
Note that it can be shown, applying the Ito formula, that the variance rate
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.
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
In (4),
In (5), we have a log normal model for the asset price
At this point, we introduce a second factor
(6) can be transformed to
The dynamic processes (8)-(9) below are defined with respect to the martingale probability measure Q, where
An equivalent Two-factor Black-Derman-Toy model [
The
As mentioned previously, in (4),
As stated previously, in Equation (7), we define the volatility
We assume
Then it follows (see [
Alternatively,
where
Assumption 1: The Brownian motion processes
where
Also, the Brownian motion processes
See [
From (6) and (10), it is clear that
Equation (11) follows because from [
where
Note that
For
Define
Then the average variance is:
and where
Proof: See Appendix A
(a)
(b)
Remark 2:
Some of the limitations of the model can be described as follows:
a) Since we can verify that
b) We have assumed that the error terms
where the expectation is obtained using the risk neutral distribution of
Remark 3:
Proposition 2 restates the result that the risk neutral property of
We can evaluate any security that is a derivative of
Then the price
In the next sections, we will derive a simple Black-Sholes type expression for the call option price
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
are
Assumption 3:
Assumption (3) ensures that the roots are real and are well defined.
Let
Then
where
Define
and also suppose Assumption (2) holds. Note that the functions
Remark 4:
If
Similarly if
Proposition 3:
Suppose
If Assumption (3) holds then the conditional risk-neutral distribution of
If
In other words,
Next we consider the case of
Conditional Risk-neutral Distribution function of
Suppose
If Assumption (3) holds then the conditional risk-neutral distribution of
Example 1:
Let
In the next section we consider the evaluation of price of a security that is derivative of stock price
Example 2:
Let
CDF of lnX(s), m(s) > 0
Assumption 4:
We will utilize the Assumption (4) later for deriving the price of any derivative security.
Conditional Call Option Price
Assumption 4
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.
Proposition 4:
Given
where
Remark 5:
To simplify the presentation of the results, we have suppressed usually the dependence of
Proof:
We prove Proposition 4 below using the risk-neutral distribution results (Proposition 3) of lnX(s) for
Case 1:
Here, we make use of risk-neutral distribution of lnX(s) results for
where
Case 2:
Since
Then,
This completes the proof.
Proposition 5:
Suppose
Case 1:
Then given that
So in this case
Case 2:
Remark 6:
We define
(i) Hedge ratio =
Then, given that
(ii) Since
(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
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
(ii) The hedge ratio expressions are similarly derived for the case of
Conditional Put-Call Parity
Consider a non-dividend paying European put option with strike price K and exercise date s. Then the price
Unconditional Call Option Price
where
One could evaluate the option price (26) numerically as follows:
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).
Figures 4-6 represent respectively, conditional call option price given h = −0.5146, 0, 0.5146.
Call option price functional values for the Equation (26) for m = 1, as the time to maturity
For m = 1, (26) reduces to (28):
The unconditional cost of call option as a weighted average of the cost of call option, as approximated for m = 1, can be represented by
Implied Volatility Functions
By definition, an implied volatility function is the function
In other words, we find a suitable value for implied volatility
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.
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 volatility3
Let
In this section, we show that the implied volatility skewness property of negative correlation-
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 [
Raj Jagannathan, (2016) A Linear Regression Approach for Determining Explicit Expressions for Option Prices for Equity Option Pricing Models with Dependent Volatility and Return Processes. Journal of Mathematical Finance,06,303-323. doi: 10.4236/jmf.2016.62026
Some preliminary results are stated below prior to the proof of Proposition 1.
Application of Least Squares Linear Regression (see [
where
The regression equation obtained is:
and where
is the regression coefficient
2) Regress the function
Note that (see [
We can show that (see [
Proof:
Using Ito’s Lemma, we have
This completes the proof.
is the regression coefficient
Then the regression equation is
Assumption:
Note that
Assumption:
Proof of Proposition 1:
1)
2)
where
where
Proof of Proposition 3:
Now we assume
If
In other words,
The equations defined in (12) hold under the Assumption (2) so that the roots of the quadratic Equation (13) are well defined.
Substituting for
In other words
An explicit expression for
Then,