Local Volatility Function Models under a Benchmark Approach

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(1)�� . QUANTITATIVE FINANCE RESEARCH CENTRE. QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 124. April 2004. Local Volatility Function Models under a Benchmark Approach David Heath and Eckhard Platen ISSN 1441-8010. www.qfrc.uts.edu.au.

(2) Local Volatility Function Models under a Benchmark Approach. David Heath1 and Eckhard Platen1 April 30, 2004. Abstract. This paper studies a class of one-factor local volatility function models for stock indices under a benchmark approach. It assumes that the dynamics for a large diversified index approximates that of the growth optimal portfolio. The pricing and hedging of derivatives under the benchmark approach does not require the existence of an equivalent risk neutral martingale measure. Fair prices for index derivatives when expressed in units of the index are martingales under the real world probability measure. The real world transition densities for the index and the underlying local volatility function can be determined from a continuum of European call option prices. As specific examples a modification of the constant elasticity of variance model and a version of the minimal market model are discussed together with a smoothed local volatility function that fits a snapshot of S&P500 index options data.. 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30, 62P20. JEL Classification: G10, G13 Key words and phrases: local volatility function, index derivatives, growth optimal portfolio, benchmark approach, fair pricing, modified CEV model, minimal market model. 1. University of Technology Sydney, School of Finance & Economics and Department of Mathematical Sciences, PO Box 123, Broadway, NSW, 2007, Australia.

(3) 1. Introduction. The standard Black-Scholes (1973) model assumes constant or deterministic volatility. However, the existence of implied volatility smiles and skews for options on actual market indices provides strong evidence that volatilities are stochastic. A natural one-factor extension of the Black-Scholes model is obtained by introducing local volatility function (LVF) models. This means that the volatility is allowed to change as a function of the underlying security and time. The resulting LVF models have attracted the interest of many researchers and practitioners. They were pioneered by Dupire (1993, 1994), Derman & Kani (1994) and Rubinstein (1994). Further results on LVF models have been described in Dumas, Fleming & Whaley (1997), Lagnado & Osher (1997), Andersen & Brotherton-Ratcliffe (1998) and Berestycki, Busca & Florent (2002). The constant elasticity of variance (CEV) model is a special LVF model and has been treated, for instance, in Cox & Ross (1976), Beckers (1980), Schroder (1989), Andersen & Andreasen (2000) and Lewis (2000). As explained in Delbaen & Shirakawa (2002), the standard CEV model has the disadvantage that its real world dynamics hits zero with strictly positive probability. As a consequence of this fact certain problems arise with the formulation of a consistent and reasonable pricing system for the CEV model, see Heath & Platen (2002a). Another LVF model is the one-factor minimal market model (MMM) proposed in Platen (2001). Here index dynamics follow a time transformed squared Bessel process, see Platen (2002) and Heath & Platen (2003). The growth optimal portfolio (GOP) of the market, see Kelly (1956) and Long (1990), is the portfolio that maximizes expected log-utility. In this paper it is assumed that the underlying risky asset is a diversified accumulation index that approximates the GOP. This assumption is supported by a result in Platen (2004b), where it is shown that the GOP is approximated by any well-diversified accumulation index. For example, in the case of the world stock market one can use the MSCI accumulation world stock index as proxy for the GOP. In most of the existing literature LVF models are formulated under the assumption that an equivalent risk neutral martingale measure exists. However, the modified CEV model studied in Heath & Platen (2002a) and also the MMM considered in Platen (2001) and Heath & Platen (2003) do not have an equivalent risk neutral martingale measure. Therefore, in this paper we do not assume the existence of such a measure. To obtain under these modeling assumptions a consistent pricing system for derivatives we use the benchmark approach developed in Platen (2002, 2004a, 2004b) where fair prices, when expressed in units of the GOP, are martingales. This paper is organized as follows: In Section 2 the LVF model is introduced. The pricing and hedging of derivatives using the benchmark approach is outlined in Section 3. The real world probability density of the index is inferred in Section 4. 2.

(4) A representation for the LVF that can be constructed from a continuum of call option prices is given in Section 5. The LVF is computed for a snapshot of S&P500 index options data in Section 6.. 2. Local Volatility Model for an Index. For simplicity, we use a deterministic, constant interest rate r with corresponding savings account process B = {Bt , t ∈ [0, T ]} given by Bt = exp{r t}. (2.1). for t ∈ [0, T ]. Let us then consider a financial market with a continuous GOP process D = {Dt , t ∈ [0, T ]}, which we interpret as the accumulation index of the market, see Platen (2004b). At time t ∈ [0, T ] the GOP value is denoted by Dt and is assumed to follow a local volatility function (LVF) model. That is, it is assumed to satisfy a stochastic differential equation (SDE) of the form £¡ ¢ ¤ dDt = Dt r + σ 2 (t, Dt ) dt + σ(t, Dt ) dWt (2.2) for t ∈ [0, T ] with fixed initial value D0 > 0. Here W = {Wt , t ∈ [0, T ]} denotes a standard Wiener process on a filtered probability space (Ω, AT , A, P ), where P is the real world probability measure. The filtration A = (At )t∈[0,T ] , where At describes the information available at time t ∈ [0, T ], is assumed to satisfy the usual conditions, see Karatzas & Shreve (1998). We also assume that the LVF σ : [0, T ] × (0, ∞) → (0, ∞) is such that a unique strong solution of the SDE (2.2) exists. In cases where the GOP process D may hit zero, we choose zero as an absorbing boundary. It is well known, see Karatzas & Shreve (1998), that the GOP volatility equals the market price for risk. The general form of the SDE (2.2) for the GOP, where the risk premium is the square of the volatility, has been pointed out, for instance, in Long (1990). If the resulting volatility process σ = {σ(t, Dt ), t ∈ [0, T ]} is chosen to be deterministic, then a standard lognormal dynamics results for Dt . However, in this paper we consider the case where the index volatility may depend on both the index and time and is therefore in general stochastic. This formulation defines an LVF model. For example, the modified CEV model considered in Heath & Platen (2002a) has an LVF of the form σ(t, Dt ) = (Dt )α−1 ϕ (2.3) for a constant exponent α ∈ (−∞, 1) and scaling parameter ϕ > 0. In Heath & Platen (2002a) it is shown that for the modified CEV model there does not exist an equivalent risk neutral martingale measure. This indicates possible problems when trying to use standard risk neutral pricing for certain LVF models. 3.

(5) Another LVF model is obtained by the minimal market model (MMM), proposed in Platen (2001, 2002, 2004b). Here the LVF is chosen in the form µ σ(t, Dt ) = where. Dt Bt. 1 ¶ 2−ν. ´ √ ³ν γt −1 , 2 ν. (2.4). Dt = (Yt γt ) 2 −1 Bt. (2.5). γt = γ0 exp{η t}. (2.6). and for t ∈ [0, T ] with net growth rate η > 0 and initial parameter γ0 > 0. Furthermore, Y = {Yt , t ∈ [0, T ]} is taken to be a square root process with dimension ν > 2, which satisfies the SDE ³ν ´ p dYt = − η Yt dt + Yt dWt (2.7) 4 for t ∈ [0, T ]. Note that by (2.4) and (2.5) the LVF of the MMM given in (2.4) is also a function of the square root process Y , that is ³ν ´ 1 σ(t, Dt ) = − 1 (Yt )− 2 (2.8) 2 for t ∈ [0, T ]. Consequently, the volatility is proportional to the inverse square root of a square root process. Therefore, the MMM is characterized by a volatility that has a stationary density. This is a particular feature of the MMM, which does not apply for the modified CEV and many other LVF models. For the CEV model the volatility is a fixed function of the index and changes its average value over long periods of time since the index is itself some type of growth process. As is the case for the CEV model, the MMM has no equivalent risk neutral martingale measure, see Platen (2001, 2004b). The following empirical observation indicates that this is likely to be a natural property for a realistic market index model. Consider the Radon-Nikodym derivative process Λ = {Λt , t ∈ [0, T ]} with ½ ¾ Z Z t 1 t 2 Bt D 0 = exp − σ (s, D(s)) ds − σ(s, D(s)) dW (s) (2.9) Λt = Dt 2 0 0 for t ∈ [0, T ]. Obviously, the process Λ is a continuous, nonnegative (A, P )-local martingale and thus a supermartingale, see Karatzas & Shreve (1991). Since we assume that the GOP is approximated by the market index, we can observe the Radon-Nikodym derivative process of the world stock market by interpreting the MSCI world accumulation index as the GOP. The Radon-Nikodym derivative is then by (2.9) simply the savings account, expressed in units of the index. In Figure 2.1 we show the resulting empirical Radon-Nikodym derivative for the period from 1975 until 2003, which declines systematically over time. This is typical for a strict supermartingale and indicates that the (A, P )-local 4.

(6) 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1975. 1985. 1995. 2003. Figure 2.1: Radon-Nikodym derivative for world stock market. martingale Λ may not be an (A, P )-martingale. For an economist or investor the observed systematic decline is not so surprising as it may appear to financial mathematicians. It is simply consistent with the well-known fact that a diversified stock index outperforms in the long term the savings account. However, if the Radon-Nikodym derivative is not an (A, P )-martingale, then the standard risk neutral pricing methodology, see, for instance, Karatzas & Shreve (1998), breaks down. Under standard risk neutral pricing one needs to assume that the process Λ is an (A, P )-martingale, in order to apply the Girsanov theorem, see Karatzas & Shreve (1998). This then ensures that the process W̃ = {W̃t , t ∈ [0, T ]} with Z t W̃t = Wt + σ(s, D(s)) ds (2.10) 0. for t ∈ [0, T ] is a Wiener process with respect to the, so called, equivalent risk neutral martingale measure P̃ , where dP̃ = Λ(T ). dP. (2.11). As mentioned above, LVF models do not always allow for the existence of an equivalent risk neutral martingale measure. We emphasize again that Figure 2.1 indicates that this assumption may not be appropriate when aiming to construct realistic market models.. 5.

(7) 3 3.1. Pricing of European Style Derivatives Fair Pricing and Hedging. Since, in general, an equivalent risk neutral martingale measure may not exist for an LVF model we apply the benchmark approach developed in Platen (2002, 2004b). We call a price that is expressed in units of the GOP a benchmarked price. It follows, under the benchmark approach for the given continuous complete market, that all nonnegative benchmarked portfolios are (A, P )-local martingales, see Platen (2002). For example, this can be easily seen for the benchmarked savings account process B̂ = {B̂t , t ∈ [0, T ]} with B̂t =. Bt , Dt. (3.1). which is proportional to the Radon-Nikodym derivative process Λ. Here one obtains by application of the Itô formula together with (3.1), (2.2) and (2.1) the SDE dB̂t = −B̂t σ(t, Dt ) dWt (3.2) for t ∈ [0, T ]. The process B̂ is clearly an (A, P )-local martingale. It can also be shown that all nonnegative portfolios in the given market are supermartingales as explained in Platen (2002). This result can then be used to show that there is no arbitrage in the sense that it is not possible to generate via a nonnegative portfolio from zero initial capital strictly positive wealth with strictly positive probability, see Platen (2004b). According to Platen (2002) a benchmarked price process is called fair if it is an (A, P )-martingale. It turns out, that for LVF models, for which an equivalent risk neutral martingale measure exists, fair prices coincide with risk neutral prices, see Platen (2002, 2004b). Thus, the concept of fair pricing generalizes that of risk neutral derivative pricing. Let H = H(Dτ ) denote the nonnegative payoff of a contingent claim with maturity date τ ∈ [0, T ], where ¯ ¶ µ H ¯¯ At < ∞ (3.3) E Dτ ¯ for all t ∈ [0, τ ]. Then the fair benchmarked price ûH (t, Dt ) at time t ∈ [0, τ ] of this contingent claim is given by ¯ ¶ µ H(Dτ ) ¯¯ At (3.4) ûH (t, Dt ) = E Dτ ¯ for t ∈ [0, τ ]. Note that this conditional expectation is taken under the real world probability measure P . The corresponding fair price uH (t, Dt ) at time t, expressed in units of the domestic currency, is given by uH (t, Dt ) = Dt ûH (t, Dt ) 6. (3.5).

(8) for t ∈ [0, τ ]. It is shown in Platen (2002), see also Heath & Platen (2002a), that for these fair benchmarked prices a corresponding self-financing hedge portfolio can be constructed, which perfectly replicates the contingent claim. To be more specific, if the function ûH is sufficiently smooth, then we can form a hedge portfolio (0) (1) consisting of δH (t) units of the domestic savings account Bt and δH (t) units of the index Dt at time t with (0). δH (t) = − and. (Dt )2 ∂ ûH (t, Dt ) Bt ∂D. (1). (0). δH (t) = ûH (t, Dt ) − δH (t) B̂t. (3.6). (3.7). for t ∈ [0, τ ]. This self-financing hedge portfolio provides perfect replication in the sense that the corresponding benchmarked profit and loss for maintaining the hedge portfolio remains zero. For a sufficiently smooth function f : (0, τ ) × (0, ∞) → < define the operator L0 by the expression L0 f (t, D) =. ∂f (t, D) ∂f (t, D) + (r + σ 2 (t, D)) D ∂t ∂D 2 1 2 2 ∂ f (t, D) + σ (t, D) D 2 ∂D2. (3.8). for (t, D) ∈ (0, τ ) × (0, ∞). Using (3.4), (2.2) and (2.1) it can be shown by application of the Itô formula that a sufficiently smooth fair benchmarked pricing function ûH (·, ·) satisfies the Kolmogorov backward equation L0 ûH (t, D) = 0. (3.9). for (t, D) ∈ (0, τ ) × (0, ∞) with terminal condition ûH (τ, D) =. H(D) D. (3.10). for D ∈ (0, ∞). It should be emphasized once again that the above methodology does not assume the existence of an equivalent risk neutral martingale measure. The fair benchmarked pricing function ûH (·, ·) is uniquely determined by the expectation (3.4) and satisfies the PDE (3.9) with terminal condition (3.10). However, for certain types of contingent claims the solution to this PDE in its given form may not be unique. For instance, this is the case for European put options under the modified CEV model, as is shown in Heath & Platen (2002a), where multiple solutions to such a PDE exist. This indicates that other perfect hedging strategies are possible. However, it is shown that fair prices are the minimal prices that can be combined with perfect hedging prescriptions for the underlying contingent claim, see Platen (2002). 7.

(9) 3.2. European Call and Put Options. If we denote by K the strike price of a European call option with maturity τ , then, by (3.5), its fair call option price satisfies at time t the relation c(t, Dt , τ, K) = Dt ĉ(t, Dt , τ, K),. (3.11). where its fair benchmarked price is given by the conditional expectation ¯ ¶ µ (Dτ − K)+ ¯¯ ĉ(t, Dt , τ, K) = E ¯ At Dτ Ãµ ¶+ ¯ ! ¯ K ¯ At = E 1− (3.12) ¯ Dτ for t ∈ [0, τ ], see (3.4). For the fair European put option with strike K and maturity τ ∈ [0, T ] the fair price p(t, Dt , τ, K) at time t is given by p(t, Dt , τ, K) = Dt p̂(t, Dt , τ, K), where the corresponding benchmarked price is Ãµ ¶+ K p̂(t, Dt , τ, K) = E −1 Dτ. ¯ ! ¯ ¯ At ¯. (3.13). (3.14). for t ∈ [0, τ ]. A fair zero coupon bond that pays one unit of the domestic currency at the maturity date τ ∈ [0, T ] can be interpreted as an index derivative. Its fair price P (t, Dt , τ ) at time t is given by P (t, Dt , τ ) = Dt P̂ (t, Dt , τ ),. (3.15). ¶ 1 ¯¯ ¯ At Dτ. (3.16). where. µ P̂ (t, Dt , τ ) = E. for t ∈ [0, τ ]. The fair benchmarked pricing functions ĉ(·, ·, τ, K), p̂(·, ·, τ, K) and P̂ (·, ·, τ ) for fixed τ and K all satisfy the PDE (3.9) together with the matching terminal condition (3.10). To calculate, for example, the fair call option price appearing in (3.11) one has to compute the conditional expectation in (3.12). This can be done by various methods, for instance, by Monte Carlo simulation combined with variance reduction techniques, see Heath & Platen (2002b), or numerically solving the PDE 8.

(10) (3.9) - (3.10), see Heath & Platen (2002a). For the modified CEV model and the MMM considered here this conditional expectation can be more easily computed by using the analytic transition densities for the corresponding square root processes involved. Figure 3.1 displays the implied volatility surface for fair European calls as a. 0.22 0.21 0.2 0.19. 10 8 6. 0.6. T. 4. 0.8 1 2. K. 1.2. Figure 3.1: Fair call implied volatility surface for the modified CEV model. function of the strike K and maturity date τ = T , which results for the modified CEV model with α = 21 , r = 0.04, ϕ = 0.2 and D0 = 1. Figure 3.1 shows a negatively skewed surface without any major increases in at-the-money implied volatilities over time.. 0.275 0.25 0.225 0.2. 10 8 6. 0.6 4. 0.8 1 K. T. 2 1.2. Figure 3.2: Fair call implied volatility surface for the MMM. To compare the CEV model with the MMM we display in Figure 3.2 the implied volatility surface for fair European call options under the MMM, where we choose 9.

(11) ν = 4, r = 0.04, η = 0.048, γ = 0.03827, and D0 = 1. In Figure 3.2 we also observe a pronounced negative skew, similar to that appearing in Figure 3.1. However, the term structure of implied volatility shown in Figure 3.2 is characterized by a gradual increase in at-the-money implied volatilities over time. That effect results because the MMM incorporates the average long term growth of the index while maintaining equilibrium dynamics for local volatility in its LVF.. 4. Implied Probability Density. For modeling and calibration it is desirable to estimate the transition probability density of the underlying security, which is here taken to be the index. To achieve this let us denote by pB̂ (t, B̂t , τ, κ) the transition probability density for the benchmarked savings account process B̂ under the real world probability measure P with the deterministic value κ=. Bτ . K. (4.1). Using the benchmarked European call option price with strike K and maturity τ let us define the quantity u(t, B̂t , τ, κ) = κ ĉ(t, Dt , τ, K). By (3.12) together with (3.1) this equation can be rewritten in the form ³ ¯ ´ +¯ u(t, B̂t , τ, κ) = E (κ − B̂τ ) At .. (4.2). (4.3). Using similar arguments to those described in Breeden & Litzenberger (1978), Dupire (1993, 1994) and Derman & Kani (1994) it follows from (4.3) that Z κ ∂ ∂ u(t, B̂t , τ, κ) = (κ − y) pB̂ (t, B̂t , τ, y) dy ∂κ ∂κ 0 Z κ = pB̂ (t, B̂t , τ, y) dy. (4.4) 0. This leads directly to the following result: Proposition 4.1 The transition probability density pB̂ of the benchmarked savings account process B̂ has the form pB̂ (t, B̂t , τ, κ) =. ∂2 u(t, B̂t , τ, κ) ∂κ2. for (t, B̂t , τ, κ) ∈ (0, τ ) × (0, ∞) × (0, T ) × (0, ∞). 10. (4.5).

(12) Therefore, the transition densities with respect to the real world probability measure can be obtained from observed option prices. This contrasts with results described in the above mentioned literature, where transition densities with respect to some hypothetical equivalent risk neutral martingale measure are obtained. Using (4.1) and (4.2), and calculating the partial derivative of u in terms of partial derivatives of c, the transition density pB̂ is given by the expression pB̂ (t, B̂t , τ, κ) =. K 3 ∂2 c(t, Dt , τ, K) Bτ Dt ∂K 2. (4.6). for (t, B̂t , τ, κ) ∈ (0, τ ) × (0, ∞) × (0, T ) × (0, ∞) with κ given by (4.1). Let pD (t, Dt , τ, K) denote the transition density for the GOP process D under the real world probability measure P . Then the following result can be derived by using the transformation (3.1) and formula (4.5). Corollary 4.2. The transition density pD of the index is of the form K ∂2 pD (t, Dt , τ, K) = c(t, Dt , τ, K) Dt ∂K 2. (4.7). for (t, Dt , τ, K) ∈ (0, τ ) × (0, ∞) × (0, T ) × (0, ∞). Similarly, it follows that the transition distribution function FD for the process D is given by the expression Z K FD (t, Dt , τ, K) = pD (t, Dt , τ, y) dy 0. = 1−. K ∂ 1 c(t, Dt , τ, K) + c(t, Dt , τ, K) Dt Dt ∂K. (4.8). for (t, Dt , τ, K) ∈ (0, τ ) × (0, ∞) × (0, T ) × (0, ∞). Consequently, by assuming a continuum of European call option prices we can infer the real world transition density and distribution function for the index. It is important to note that these results do not involve any major modeling assumption, other than the requirement that the GOP is chosen to be the index and represented by a one-dimensional diffusion process with respect to a given LVF. To demonstrate how Corollary 4.2 can be applied, Figure 4.1 displays the real world transition densities of the index as a function of K and τ for the CEV model by using the call option prices that provided the implied volatilities shown in Figure 3.1. Similarly, we show in Figure 4.2 the real world transition densities for the index obtained for the MMM. Note that for later time periods the mean of this density is located at higher values compared to the corresponding densities for the CEV model. 11.

(13) 4 3. 10. 2 1 0 0.5. 8 6 0.75. 4. T. 1 K. 2. 1.25 1.5. Figure 4.1: Implied transition density for the index obtained from fair CEV call option prices.. 5. Representation of the LVF. From the following it follows that for a maturity date τ ∈ [0, T ] and strike K ∈ (0, ∞) the LVF value σ(τ, K) can be recovered from European call option prices. This is similar to results described in Breeden & Litzenberger (1978), Dupire (1993, 1994) and Derman & Kani (1994). However, here it is obtained via the benchmark approach without assuming the existence of an equivalent risk neutral martingale measure. To be able to derive the result conveniently let us assume that 1 ∂ lim u(t, B̂t , τ, κ) = 0 (5.1) κ→0 κ ∂τ and ∂2 lim σ 2 (τ, K) κ 2 u(t, B̂t , τ, κ) = 0. (5.2) κ→0 ∂κ These are reasonable conditions on the asymptotics of u that apply to a wide range of LVF models and lead us to the following statement: Theorem 5.1. For fixed t ∈ [0, τ ] and B̂t > 0 the LVF has the form ! 12 √ Ã ∂ u(t, B̂ , τ, κ) 2 ∂τ t σ(τ, K) = 2 ∂ κ u(t, B̂t , τ, κ) ∂κ2. (5.3). for (τ, K) ∈ (0, T ] × (0, ∞) and t ∈ [0, τ ), again with κ as given in (4.1). Proof: From the SDE (3.2) and relation (3.1) it follows that the transition 12.

(14) 2. 10. 1. 8. 0 0.5. 6 0.75. 4. T. 1 K. 2. 1.25 1.5. Figure 4.2: Implied transition density for the index obtained from fair MMM call option prices. density pB̂ for the process B̂ satisfies the Fokker-Planck equation o ∂ 1 ∂2 n 2 2 p (t, B̂t , τ, κ) − σ (τ, K) κ pB̂ (t, B̂t , τ, κ) = 0 ∂τ B̂ 2 ∂κ2. (5.4). for (τ, κ) ∈ (0, T ) × (0, ∞) with initial condition pB̂ (t, B̂t , t, κ) = δ(B̂t − κ),. (5.5). where δ(·) is the Dirac delta function and t ∈ [0, τ ), B̂t ∈ (0, ∞). It therefore follows by using (4.5) that (5.4) can be rewritten in the form µ 2 ¶ ¾ ½ 2 ∂ ∂ 1 ∂2 2 2 ∂ u(t, B̂t , τ, κ) − u(t, B̂t , τ, κ) = 0 σ (τ, K) κ ∂τ ∂κ2 2 ∂κ2 ∂κ2 and hence ∂2 ∂κ2. ½. ∂ 1 ∂2 u(t, B̂t , τ, κ) − σ 2 (τ, K) κ2 2 u(t, B̂t , τ, κ) ∂τ 2 ∂κ. ¾ = 0.. (5.6). 1 ∂2 ∂ u(t, B̂t , τ, κ) − σ 2 (τ, K) κ2 2 u(t, B̂t , τ, κ) = β0 (τ ) + β1 (τ ) κ. ∂τ 2 ∂κ. (5.7). Then there exist quantities β0 (τ ) and β1 (τ ) such that. From (5.7), (5.1) and (5.2) it follows that β0 (τ ) = 0.. (5.8). β1 (τ ) = 0.. (5.9). and. 13.

(15) Combining (5.7), (5.8) and (5.9) yields (5.3). To express the LVF in terms of call option prices one can use the transformations (4.1) and (4.2) and compute the corresponding partial derivatives. One then obtains the following result, which is equivalent to (5.3): Corollary 5.2. The LVF is given by √ s∂ ∂ c(t, Dt , τ, K) 2 ∂τ c(t, Dt , τ, K) + K r ∂K σ(τ, K) = 2 ∂ K c(t, Dt , τ, K) ∂K 2. (5.10). for (τ, K) ∈ (0, T ] × (0, ∞) with t ∈ [0, τ ) and Dt ∈ (0, ∞). Equation (5.10) coincides with corresponding formulae obtained in Breeden & Litzenberger (1978), Dupire (1993) and Derman & Kani (1994). However, here it is derived without assuming the existence of an equivalent risk neutral martingale measure. Note that the price function c(t, Dt , ·, ·) needs to be such that the expression appearing under the square root in (5.10) is nonnegative. Figure 5.1 shows the LVF computed from (5.10) by using the CEV call option. 0.24 0.22 0.2 0.18 0.16. 10 8 6. 0.8 4. 1. T. 1.2 K. 2. 1.4 1.6. Figure 5.1: LVF implied from CEV call option prices. values that produced the implied volatilities shown in Figure 3.1. These results match the theoretical LVF for the modified CEV model given in (2.3) with the parameter choices α = 12 and ϕ = 0.2. Small differences in values obtained from using (5.10) compared to the theoretical values given in (2.3), which can be noted for the smallest K and T values, are due to numerical truncation errors and the finite difference approximations that are used to compute the partial derivatives appearing in (5.10).. 14.

(16) 0.35 0.3 0.25 0.2 0.15 0.8. 10 8 6 4. 1. T. 1.2 K. 2. 1.4 1.6. Figure 5.2: LVF implied from MMM call option prices. Similarly, we can recover from (5.10) the LVF (2.8) of the MMM. This is shown in Figure 5.2, using the MMM call option data that generated the implied volatilities displayed in Figure 3.2. For fixed K one notes in Figure 5.2 that σ(·, K) gradually increases over time. Note that the MMM implied volatility surface in Figure 3.1 matches quite well the estimated average implied volatilities shown in Cont & da Fonseca (2002). For very short dated options the documented average implied volatilities have more curvature than those obtained for the above standard MMM version. However, such curvature is naturally obtained by extended versions of the MMM that allow for random scaling, as is shown in Heath & Platen (2003). The following example will discuss a case where the observed volatility surface is for longer maturities as flat and negatively skewed as provided by the MMM.. 6. S&P500 Local Volatility Function. To illustrate the above analysis we consider observed index option prices for the S&P500 index. Figure 6.1 shows a fit of the implied volatility surface for S&P500 European call options for 20 April 2004. These implied volatilities were computed using prices obtained from the average of bid and ask prices using the short rate r = 0.03 and the dividend rate d = 0.01. The corresponding closing price for the S&P500 index was 1114. Closing option price data was not used because these prices may lie outside the closing bid and ask range and may not be synchronized with the closing price of the S&P500 index. A total of 83 option prices were used to obtain the displayed fit with a strike range from 1025 to 1200 and maturity dates: 21 May 2004, 18 June 2004, 16 July 2004, 17 September 2004, 17 December 2004, 18 March 2005 and 17 June 2005. 15.

(17) 0.18 0.16. 1. 0.14 0.75. 0.12 1050. 0.5. T. 1100 K. 0.25. 1150 1200. Figure 6.1: Fitted implied volatility surface for the S&P500 index for 20 April 2004. The available strikes for the option price data varied depending on the maturity date. For real markets observed option prices have features that, in general, do not provide a reasonably smooth implied volatility surface. Therefore, we fit the implied volatilities to a sufficiently smooth parametric surface, as is commonly undertaken in practice. The fitting procedure consisted of two basic steps. Firstly, a set of corresponding implied volatilities were computed for each of the 83 option prices. Secondly, a least squares fit for the implied volatility surface was obtained using a set of two-dimensional cubic polynomials. This fitted parametric surface is displayed in Figure 6.1. Using the fitted implied volatility surface a corresponding smoothed option price surface can be deduced. These option prices can then be used to calculate the real world transition densities according to formula (4.7). The resulting transition densities are displayed in Figure 6.2. Note that for the shortest maturity the area under the curve is approximately one as should be expected. A fitting procedure was needed here because equation (4.7) requires a continuum of option prices to be computed for the specified strike and maturity date ranges, whereas observed options data is only available at a fixed number of points within the considered strike and maturity range. The corresponding LVF can be obtained by formula (5.10) and is displayed in Figure 6.3. Because of the form of equation (5.10) and, in particular, the combination of first and second order partial derivatives the shape of this surface turns out to be rather sensitive to the choice of basis functions employed in the fitting procedure. Note however that this LVF returns exactly the implied volatility 16.

(18) 0.008 0.006 0.004 0.002. 1 0.75. 1050. 0.5. T. 1100 K. 0.25. 1150 1200. Figure 6.2: Implied transition densities for the S&P500 index for 20 April 2004. surface displayed in Figure 6.1 and the corresponding smoothed S&P500 option prices.. Conclusion This paper demonstrates that it is possible to back out from derivative prices the real world transition density for a diversified market index modeled by a local volatility function. Furthermore, the corresponding local volatility function can be inferred from a continuum of call option prices. These results are derived without assuming the existence of an equivalent risk neutral martingale measure and are applied directly to S&P500 index options. Future research will focus on extensions of the local volatility function approach to include random scaling effects.. Acknowledgement MSCI data was provided by Thomson Financial. Options data was downloaded from http://finance.yahoo.com.. 17.

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