The Exact Smile of certain Local Volatility Models
We introduce a new class of local volatility models. Within this framework, we obtain expressions for both (i) the price of any European option and (ii) the induced implied volatility smile. As an illustration of our framework, we perform specific pricing and implied volatility computations for a CEV-like example. Numerical examples are provided.
keywords: CEV, local volatility, stochastic volatility, implied volatility.
Local volatility models are a class of equity models in which the volatility of an asset is described by a function of time and the present level of . That is, . Among local volatility models, perhaps the most well-known is the constant elasticity of variance (CEV) model of Cox (1975). An extension of the CEV model to defaultable assets (the Jump-to-Default CEV or JDCEV model) is derived in Carr and Linetsky (2006). One advantage of these two local volatility models is that they admit closed-form pricing formulas for European options, written as infinite series of special functions. Another advantage of local volatility models is that, for models whose transition density is not available in closed form, accurate density approximations are often available. See, for example, Pagliarani and Pascucci (2011).
In this paper, we introduce a new class of local volatility models which, like the CEV and JDCEV models, allow for European option prices to be expressed in closed form as an infinite series. Additionally, we derive an expression for the exact implied volatility surface induced by our class of models. Previous studies of the implied volatility surface induced by local volatility models focused on heat-kernel expansions to derive asymptotic approximations of the volatility smile (see e.g., Gatheral et al. (2010); Henry-Labordère (2005) and references therein). It is worth mentioning that Dupire (1994) solves the inverse problem of finding a formula for the local volatility function the produces a given observed implied volatility surface exactly.
Essential for our mathematical presentation, is the use of spectral theory. The spectral representation theorem has been widely applied in mathematical finance. An exhaustive review would be prohibitive. However, we mention the seminal work of Davydov and Linetsky (2003), who lay the groundwork for option-pricing with eigenfunctions in a scalar diffusion setting. For applications of eigenfunction methods in a stochastic volatility (i.e., multivariate) setting, we refer the reader to Fouque et al. (2011); Lorig (2012a). While previous spectral-related work has focused exclusively on eigenfunction expansions for self-adjoint operators in Hilbert space, here, we focus on generalized eigenfunction expansions for normal operators. To our knowledge, this is the first time the spectral theory of normal operators has been used in a financial setting.
The rest of this paper proceeds as follows: in section 2 we present our class of models and describe our assumptions about the market. In section 3 we derive a formula for the price of a European option, written in a general form which is valid for any model within our framework. In section 4 we provide an formula for the implied volatility smile induced by our class models. In section 5, as an example of our framework, we perform explicit pricing and implied volatility computations for a CEV-like example. Numerical results are provided at the conclusion of the text. An appendix with some mathematical background is also provided. Concluding remarks can be found in section 6.
2 Model and assumptions
We assume a frictionless market, no arbitrage and take an equivalent martingale measure chosen by the market on a complete filtered probability space . The filtration represents the history of the market. All processes defined below live on this space. For simplicity we assume zero interest rates and no dividends so that all assets are martingales. We consider an asset whose dynamics are given by
where, , , the function is an element of (the Schwartz space of rapidly decreasing functions on ; see equation (50) for a definition) and is a Brownian motion. The restriction is needed to prove Theorem 4. We assume that , the initial value of is known. Note that has local volatility . Obviously, if then is a geometric Brownian motion. This will be key for the implied volatility analysis in section 4. Observe that both zero and infinity are natural boundaries according to Feller’s boundary classification for one-dimensional diffusions (see Borodin and Salminen (2002) pp. 14-15). That is, both zero and infinity are unattainable.
In what follows it will be convenient to introduce . A simple application of Itô’s formula shows that satisfies
3 Option pricing
We wish to find the time-zero value of a European-style option with payoff at time . Using risk-neutral pricing we express the initial value of the option as the risk-neutral expectation of the option payoff
where the notation indicates -expectation starting from . Suppose (compactly supported functions with continuous derivatives up to order 2). Then, the function satisfies the Kolmogorov backward equation
where is the generator of the process . The domain of is defined as the set of for which the limit exists in the strong sense. For any the generator has the explicit representation
where , without the subscript , indicates differentiation with respect to .
It is possible to extend our results to payoff functions that are continuous and have linear growth in (e.g. Call options). However, rigorous justification for this it outside the scope of this paper. Numerical tests are provided to support this claim.
The operators and are normal operators in the Hilbert space and satisfy the following (improper) eigenvalue equations (neither nor have any proper eigenvalues)
Note that, as shown by Dirac (1927), for any analytic , we have
Thus, the generalized eigenfunctions satisfy the orthogonality relation
See also, Friedman (1956), equation (4.35). Note also that Borel-measurable functions of normal operators (e.g., ) are well-defined by the spectral theorem for normal operators, as explained in Appendix A.
We seek a solution to Cauchy problem (4) of the form
The solution to the above equations is
After a bit of algebra, we find an explicit representation for
See Appendix B. ∎
4 Implied volatility
In this section we fix and a call option payoff . Note that
The following definitions will be useful:
The Black-Scholes Price is defined as
For each fixed spot , time to maturity , and strike price , the Implied Volatility is defined implicitly as the unique number such that
where is as given in Theorem 4.
Note that . As shown in Lorig (2012b), when can be expanded as an analytic series whose first term corresponds to , one can obtain the exact implied volatility corresponding to .
For the existence and uniqueness of the implied volatility can be deduced by using the general arbitrage bounds for call prices and the monotonicity of . See Fouque et al. (2011), Section 2.1, Remark (i).
Observe that, for any and , the function is given by its Taylor series:
Observe also that, by monotonicity of we have for all . Therefore, is an invertible analytic function. By the Lagrange inversion theorem, the inverse function is also analytic.
Clearly, is an analytic function of (we derived its power series expansion). It is a useful fact that the composition of two analytic functions is also analytic (see Brown and Churchill (1996), section 24, p. 74). Thus, in light of Remark 10, we deduce that is an analytic function and therefore has a power series expansion in . We write this expansion as follows
Taylor expanding about the point we have
Solving the above equations for we find
The right hand side of (33) involves only for . Thus, the can be found recursively.
Explicitly, up to we have
We summarize our implied volatility result in the following theorem:
Everything we have done so far is exact. The accuracy of the implied volatility expansion (23) is limited only by the number of terms one wishes to compute.
5 CEV-like example
In the constant elasticity of variance (CEV) model of Cox (1975) the dynamics of are assumed to be of the form . A key feature of the CEV model is that, when , the local volatility function increases as , which (i) is consistent with the leverage effect and (ii) results in a negative implied volatility skew. However, values of also cause the volatility to drop unrealistically close to zero as increases. If we choose then from (1) the dynamics of become
Note that the local volatility function behaves asymptotically like as and behaves asymptotically like a constant as .
Because is unbounded as (recall ), the function . However, we can modify the domain of to be where and is arbitrary. The operators and would then be defined on and the domain of these operators would include an absorbing boundary condition at (signifying default of the first time reaches the level ). Note that . In the analysis that follows, it will simplify computations considerably if we continue to work on as working on would require modifying the eigenfunctions from complex exponentials to sines . However, the simplification comes at a cost; in light of the conditions of theorem (4) our results may not be valid for values of .
Thus, the -fold integral (17) collapses into a single integral
Although we have written the option price as an infinite series (10), from a practical standpoint, one is only able to compute for some finite . For any finite we may pass the sum through the integral appearing in (41). Thus, for the purposes of computation, the most convenient way express the approximate option price is
Note, to obtain the approximate value of , only a single integration is required. This makes our pricing formula as efficient as other models in which option prices are expressed as a Fourier-type integral (e.g. exponential Lévy processes, Heston model, etc.).
Monte Carlo Test
To text the accuracy of approximation (42), we compute the price of a series of European call options using approximation (42) with . We then compute the price of the same series of call options by means of a Monte Carlo simulation using a standard Euler scheme with a time step of years and sample paths. The largest relative error obtained in the Monte Carlo simulations (i.e., standard error divided by price) was . Finally, we convert call prices to implied volatilities by inverting Black-Scholes numerically. The results of this procedure are plotted in figure 1. The implied volatilities resulting from the two methods of computation are indistinguishable.
Convergence of Transition Density
Define the transition density and the approximation of the transition density , which are obtained by setting the payoff function equal to a Dirac delta function . Explicitly
In order to test the rate of convergence of to , in figure 2, we plot the approximate transition density for different values of . For we see virtually no difference between and .
Convergence of Implied Volatility
Finally, to see how well the implied volatility expansion of section 4 performs, we define the approximation of the implied volatility
where the are given by (33). In figure 3 we provide a numerical example illustrating convergence of to . We compute using a two-step procedure. First, we approximate using (42) with . In light of the Monte Carlo simulation above, this should introduce almost no error. Then, to find , we solve numerically. Implied volatility is plotted as a function of the -moneyness to maturity ratio, . Convergence is fastest for values of near and slows as moves away from in the negative direction.
In this paper we introduce a class of local stochastic volatility models. Within our modeling framework, we obtain a formula (written as an infinite series) for the price of any European option. Additionally, we obtain an explicit expression for the implied volatility smile induced by our class of models. As an example of our framework, we introduce a CEV-like model, which corrects one possible short-coming of the CEV model; namely, our choice of local volatility function does not drop to zero as the value of the underlying increases. Finally, in the CEV-like example, we show that option prices can be computed with the same level of efficiency as other models in which option prices are computed as Fourier-type integrals.
The author would like to thank Bjorn Birnir and two anonymous reviewers for their helpful comments.
Appendix A Spectral theory of normal operators in a Hilbert space
In this appendix we briefly summarize the theory of normal operators acting on a Hilbert space. A detailed exposition on this topic (including proofs) can be found in Reed and Simon (1980) and Rudin (1973).
Let be a Hilbert space with inner product . The adjoint of an operator acting in is an operator such that . Here, for simplicity, we have assumed . An operator is said to be normal in if it is closed, densely defined and commutes with its adjoint: .
Suppose is a normal operator acting on the Hilbert space . For any Borel measurable function , the operator can be constructed as follows. First, one solves the proper and improper
where and denote the discrete and continuous spectrum of , respectively. For the improper eigenvalue problem one extends the domain of to include all functions for which makes sense and for which the following boundedness conditions are satisfied
After normalizing, the proper and improper eigenfunctions of satisfy the following orthogonality relations
The operator is then defined as follows (see Hanson and Yakovlev (2002), section 5.3.2)
Appendix B Proof of Theorem 4
Our strategy is to show that generates a semigroup . This will guarantee that is an analytic function of , which in turn, justifies the use of expansion (10). Throughout this section we will work on the Hilbert space . We let , the Schwartz space of rapidly decreasing functions on :
We note that is a dense subset of . Thus, has a unique extension with domain . Our analysis begins with a Theorem from Chernoff (1972):
Let be the generator of a contraction semigroup on a Banach space. Let be a dissipative operator with a densely defined adjoint. Assume that the inequality
holds for some and (i.e., the operator is bounded relative to with relative bound ). Then the closure of generates a contraction semigroup .
Recall, an operator is dissipative if for all .
The operator is the generator of a contraction semigroup on . Thus, we must (i) show that has a densely defined adjoint, (ii) show that is dissipative and (iii) derive conditions under which is bounded relative to with relative bound less than or equal to one.
To show (i) we note that the adjoint of , given by , has domain . As mentioned above, is densely defined in . To show (ii), we note that, if an operator satisfies the positive maximum principle
Let be a linear operator with domain . Then satisfies the positive maximum principle if and only if
for some , , , and satisfying
Operators of the form (52) are called Lévy-type operators.
See Theorem 2.12 of Hoh (1998). ∎
The operator is clearly of the form (52). Hence, satisfies the positive maximum principle and is therefore dissipative. Finally, for part (iii), the following Theorem gives conditions under which is bounded relative to with relative bound one.
Suppose (which is the condition given in Theorem 4). Then is bounded relative to with relative bound less than or equal to one.
Clearly, for any we have
The proof of Theorem 4 is complete.
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