Research Article | Open Access
Option Price Decomposition in Spot-Dependent Volatility Models and Some Applications
We obtain a Hull and White type option price decomposition for a general local volatility model. We apply the obtained formula to CEV model. As an application we give an approximated closed formula for the call option price under a CEV model and an approximated short term implied volatility surface. These approximated formulas are used to estimate model parameters. Numerical comparison is performed for our new method with exact and approximated formulas existing in the literature.
In , a decomposition of the price of a plain vanilla call under the Heston model is obtained using Itô calculus. Recently, in , the decomposition obtained in  has been used to infer a closed form approximation formula for a plain vanilla call price in the Heston case, and on the basis of this approximated price, a method to calibrate model parameters has been developed and successfully applied. In this paper, we use the ideas presented in  to obtain a closed form approximation to plain vanilla call option price under a spot-dependent volatility model.
The model presented here assumes the volatility is a deterministic function of the underlying stock price, and therefore, there is only one source of randomness in the model. These models are sometimes called local volatility models in the industry and GARCH-type volatility models in financial econometrics. Recall that these models are different from the so-called stochastic volatility models, like Heston model, where the volatility process is driven by an additional source of randomness, not perfectly correlated with the stock price innovations.
As an application, for the particular case of CEV model, we obtain an approximation of the at-the-money (ATM) implied volatility curve as a function of time and an approximation of the implied volatility smile as a function of the log-moneyness, close to the expiry date. We use these approximations to calibrate the CEV model parameters.
2. Preliminaries and Notations
Let be a positive price process under a market chosen risk neutral probability that follows the modelwhere is a standard Brownian motion, is the constant interest rate, and is a function of such that is a square integrable random variable that satisfies enough conditions to ensure the existence and uniqueness of a solution of (1).
The following notation will be used in all the paper:(i)We define the Black-Scholes function as a function of and such that where denotes the cumulative probability function of the standard normal law, and are strictly positive constants, and Note that the price of a plain vanilla European call under the classical Black-Scholes theory is where is the price of the underlying process at , is the constant volatility, is the strike price, and is the expiry date.(ii)We will denote frequently by the time to maturity.(iii)We use in all the paper the notation , where is the completed natural filtration of (iv)In our setting, the call option price is given by (v)Recall that from the Feynman-Kac formula, the operator satisfies .(vi)We define the operators , , and In particular, we have that
Lemma 1. Then, for any , and for any positive quantities , , , and , one haswhere is a constant that depends on , , and
Proof. For any we have where is a polynomial of order and the exponential decreasing on of the Gaussian kernel compensates the possible increasing of and
3. A General Decomposition Formula
Here we obtain a general abstract decomposition formula for a certain family of functionals of that will be the basis of all later computations.
Assume we have a functional of the form where is a function of and is a function of
Then we have the following lemma.
Lemma 2 (generic decomposition formula). For all , one has
Proof. Applying the Itô formula to process we obtainNow, applying Feynman-Kac formula for , multiplying by , and taking conditional expectations, we obtainOn the other hand, using Itô calculus rules, it is easy to see thatFinally, substituting this expression in (12) we finish the proof.
For the Black-Scholes function previous lemma reduces to the following corollary.
Corollary 3 (BS decomposition formula). For all , one has
Proof. Applying Lemma 2 to and , and using equalities the corollary follows straightforward. Note that to apply Itô formula to Black-Scholes function, because the derivatives of this function are not bounded, we have to use an approximation to the identity and the dominated convergence theorem as it is done, for example, in . For simplicity we skip this mollifying argument across the paper.
Remark 4. For clarity, in the following we will refer to terms of the previous decomposition as
Remark 5. In , an alternative formula that can be used for local volatility models is proved. The formula presented in  uses, as a base function, function , but this formula is numerically worse than the new formula presented here that uses as a base function . This happens because in the formula presented in  the volatility is put into the approximated term, instead of keeping it on the Black-Scholes term as we do here. It is precisely because volatility is a deterministic function of the underlying asset price that we can do that.
4. Approximation Formula
In this section we obtain an approximation formula to plain vanilla call price by approximating terms (I)–(IV). The main idea is to use again Lemma 2 to estimate the errors.
Theorem 6 (BS decomposition formula with error term). For all , one haswhere is an error. Terms of are written in Appendix A.
Proof. We apply Lemma 2 to terms (I)–(IV). Concretely, functions and in every case are
5. CEV Model
The constant elasticity of variance (CEV) model is a diffusion process that solves the stochastic differential equationNote that, writing , CEV model can be seen as a local volatility model. This model, introduced in , is one of the first alternatives to Black-Scholes point of view that appeared in the literature. The parameter is called the elasticity of the volatility and is a scale parameter. Note that for , the model reduces to Osborne-Samuelson model, for , the model reduces to Bachelier model, and for , the model reduces to Cox-Ingersoll-Ross model. Parameter controls the steepness of the skew exhibited by the implied volatility.
5.1. Approximation of the CEV Model
Applying Corollary 3 to CEV model, we obtain the following.
Corollary 7 (CEV exact formula). For all , one has
We will write
The exact formula can be difficult to use in practice, so we will use the following approximation.
Corollary 8 (CEV approximation formula). For all , one haswhere is an error. Terms of are written in Appendix B. We have that and is an increasing function on every parameter.
5.2. Numerical Analysis of the Approximation for the CEV Case
In this section, we compare our numerically approximated price of a CEV call option with the following different pricing methods:(i)The exact formula, see [5, 6, 8]. The Matlab code is available in .(ii)The Singular Perturbation Technique, see .
The results for a call option with parameters , , , , and are presented in Table 1.
The results in the case that are presented in Table 2.
The results in the case that are presented in Table 3.
Finally, the results in the case that are presented in Table 4.
Note that the new approximation is more accurate than the approximation obtained in .
In Figure 1, we plot the surface of errors between the exact formula and our approximation.
We calculate also the speed time of execution (in seconds) of every method running the function timeit of Matlab 1.000 times. The computer used is an Intel Core i7 CPU Q740 @1.73 GHz 1.73 GHz with 4 GB of RAM with a Windows 10 (×64). The results are presented in Table 5.
We observe that singular perturbation method is the fastest method to calculate the price of CEV call option. The method developed in this work is a little more expensive in computation time. But to compute the exact price is much more expensive than any of the other two methods. Note that, in our method, we also are able to calculate at the same time the price and the Gamma of the log-normal price.
6. The Approximated Implied Volatility Surface under CEV Model
In the above section we have computed a bound for the error between the exact and the approximated pricing formulas for the CEV model. Now, we are going to derive an approximated implied volatility surface of second order in the log-moneyness. This approximated implied volatility surface can help us to understand better the volatility dynamics. Moreover we obtain an approximation of the ATM implied volatility dynamics.
6.1. Deriving an Approximated Implied Volatility Surface for the CEV Model
In this section, for simplicity and without losing generality, we assume So denotes time to maturity. The price of an European call option with strike and maturity is an observable quantity which will be referred to as . The implied volatility is defined as the value that makes
Using the results from the previous section, we are going to derive an approximation to the implied volatility as in . We use the idea to expand the function with respect to an asymptotic sequence converging to Thus, we can writeand assuming we can choose Then, we can expand with respect to this scale as and write
Let Write as a shorthand for We can rewrite Corollary 8 as
On the other hand we can consider the Taylor expansion of around We have that and this expression can be rewritten as
Then, equating this expression to we have
Note that is linear with respect to the log-moneyness, while is quadratic.
Remark 9. Note that the pricing formula has an error of as we have proved in Corollary 8, and this is translated into an error of into our approximation of the implied volatility. The quadratic term of the volatility shape is not accurate.
We calculate now the short time behavior of the approximated implied volatility . We write the approximated equations in terms of , because the case is the most interesting, and in terms of the log-moneyness
Lemma 10. For close to one has
Proof. Note that
Remark 11. Note that (34) is a parabolic equation in the log-moneyness. Also, from the above expression it is easy to see that the slope with respect to is negative when and positive when , showing that the implied volatility for short times to maturity is smile-shaped. This is consistent with the result in . Furthermore, there is a minimum of the implied volatility with respect to attained at
Remark 12. Note that, in stochastic volatility models, the implied volatility depends homogeneously on the pair , and in fact it is a function of the log-moneyness As extensively discussed in  and exemplified for GARCH option pricing in , this homogeneity property is at odds with any type of GARCH option pricing. We found also this phenomenon in the quadratic expansion (34).
The behavior of the approximated implied volatility when the option is ATM is easy to obtain:
6.2. Numerical Analysis of the Approximation of the Implied Volatility for the CEV Case
In this section, we compare numerically our approximated implied volatilities with implied volatility computed from call option prices calculated with the exact formula and with the ones obtained using the following formula obtained in :where and is the forward price.
In Figure 2, we can see that the implied volatility dynamics behaves well for long dated maturities and short dated maturities when is close to 1. When this is not the case, the formula behaves well at-the-money but the error increases far from the ATM value. This behavior is a consequence of the quadratic error of our approximation.
Comparing the ATM volatility structure, we have the following graphics.
In Figure 3, we observe that, for ATM options, the approximated implied volatility surface fits really well the real implied volatility structure.
Now, we put the implied volatility approximation found in (34) into Black-Scholes formula and compare the obtained results with Hagan and Woodward results. The results for a call option with parameters , , , , and are presented in Table 6.
The results in the case that are presented in Table 7.