Modeling of Price. Ximing Wu Texas A&M University

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1 Modeling of Price Ximing Wu Texas A&M University As revenue is given by price times yield, farmers income risk comes from risk in yield and output price. Their net profit also depends on input price, but output price consists of the major price risk faced by farmers. Incorporation of price risk into decision making process is fundamentally important. Both historical price information and projected future price come into play. Due to the usual lengthy gap between planning and harvest, expected price is the dominant factor. Given that many agricultural commodities are represented in the future and option markets, it is a common practice to infer the collective belief of future price movement from option pricing. An European call option, the right (but not obligation) to buy at a future maturity time τ at a (strike) price X, is fairly priced according to c(x; τ) = e rτ (S X)f(S)dS, (1) X where r is the intertemporal discount rate, S is the price at maturity, and f(s) is the predictive density of price at maturity. Intuitively, the price of this call option equals the expected profit under its predictive density (multiplied by a time discount factor). The discount rate is often proxied by a risk free interest rate. The predictive density of future price is termed the implied Risk Neutral Density (RND) of the underlying asset since option prices are risk neutral with respect to the underlying risky asset. In reality, agents may be risk averse. To factor in risk aversion, one needs to model the aggregate market utility function and the corresponding coefficient aversion. Nonetheless, changes in RND is expected to mostly reflect changes in agents belief about future outcomes under the assumption that agents preference is relatively stable. Bahra (1997) provides an excellent review of the RND. 1

2 Intuitively, one can infer the RND of a financial asset from its option prices over a range of call and put (right to sell) prices with the same maturity date. Breeden and Litzenberger (1978) show that differentiating (1) twice gives 2 c(x; τ) X 2 = e rτ f(x). (2) This identity provides an alternative route to estimate the RND; that is, one first comes up with a model for c(x; τ) then differentiates it twice to calculate its implied RND. Note that for the imputed RND to be positive, one needs to impose positivity and concavity in the modeling of c(x; τ). Call options with strike price less than the market price are termed in-the-money (ITM), higher than the market price are termed out-of-the-money (OTM), and equal to the market price termed at-the-money (ATM). Since options are priced usually within a limited range at either sid eof ATM strike, estimation of the RND essentially amounts to interpolating between observed option prices and extrapolating outside ot their range for tail probabilities. Fitting an assumed parametric RND to observed option prices. Note that a put option is priced according to X p(x; τ) = e rτ (X S)f(S)dS. (3) 0 Consider a set of strike prices X 1,..., X n and their corresponding call and put prices c(x i ; τ) and p(x i ; τ). Suppose the RND is given by parametric density f( ; θ), where θ is a vector of unknown parameters with a finite dimension. Denote c i (θ) p i (θ) = e rτ X (S X)f(S; θ)ds (4) = e rτ X 0 (X S)f(S; θ)ds (5) The RND can then be estimated by the method of least squares: min θ {c(x i ; τ) c i (θ)} 2 + {p(x i ; τ) p i (θ)} 2. In addition, under the no arbitrage condition, the mean of the RND should equal the forward 2

3 price of the underlying asset. Define S(θ) = 0 Sf(S; θ)ds. This no arbitrage condition can be incorporated into the estimation via min θ {c(x i ; τ) c i (θ)} 2 + {p(x i ; τ) p i (θ)} 2 + {S(θ) e rτ S} 2. According to Black-Scholes model, RND is log-normal, or RND return distribution is normal. The log normal distribution (associated with a normal distribution with mean α and standard deviation β) has a density f(x) = 1 xβ (ln x α) 2 2π e 2β 2. Suppose that log S T φ(log S + (µ 1/2σ 2 )τ, σ τ), we then have c(x; τ) = SΦ(d 1 ) exp( rτ)xφ(d 2 ), where Φ is the standard normal CDF and d 1 = log(s/x) + (r + 1/2σ2 )τ σ τ d 2 = d 1 σ τ. Note here S/X reflects the moneyness of the strike price relative to current price of the underlying asset. Thus Black-Scholes model assumes homogeneous of degree one with respect to strike and price. One can calculate the implied volatility σ, using the B-S formula, based on observed strike, asset price, option price, interest rate and maturity date. Under B-S model, the implied volatility is a constant. However, empirical evidences tend to suggest a U shape relationship between strike and implied volatility, indicating deviation from the log-normality suggested by the B-S model. In practice, since option prices are noisy, the fitting of the option pricing function is often 3

4 undertaken with a transformation to the implied volatility, which is a less variable function of the strike. That is, the option prices are first converted to their implied volatility, which is then fitted as a function of contributing factors. The fitted implied volatility is then converted back to option price, whose second derivative is taken to calculate the RND. This conversion and back-conversion is conducted using the B-S model. However, this practice does not imply, nor does it require, that the B-S model is correct. It is a merely a non-linear monotone transformation to facilitate the fitting. For flexibility, one can use mixture of log-normal distribution to account for possible non- Gaussianality of the underlying return distribution. Nonparametric estimations provide a viable alternative to estimate the RND. There are three commonly used methods: kernel regression, spline regression and maximum entropy method. The local polynomial estimator is a kernel regression technique that estimates not only the conditional mean, but its derivatives locally. It is based on the idea of low order local Taylor expansion via weighted regression: min β Y i p β j (z)(z Z i ) j j=0 2 K h (z Z i ), where K h is a kernel with bandwidth h. The coefficient β j (z), j = 0,..., P reflects the local j-th derivative. Since the RND is the second derivative of the option pricing function, we need p 2. In practice, p is usually set to be an odd number; p = 3 is often used for the RND estimation. The local polynomial estimation is highly flexible; on the other hand, the suggested RND is not necessarily positive since this estimator does not impose restrictions on the sign of its coefficients. A second possibility is to use spline estimator. Desired shape restrictions are relatively easy to impose in spline estimation. This approach also estimates the option pricing function first, then calculate its second derivative to recover the RND. Maximum entropy approach. Background: Suppose we know a few moments about an unknown distribution, is it possible to construct a distribution based on the given moment information? This answer is positive, but there might be an infinite number of distributions 4

5 that satisfy those moment conditions. The method of maximum entropy provides an approach for such a construction that enjoys some appealing theoretical properties. For a random variable x with a density function f, its entropy is given by H(f) = f(x) log f(x)dx. Jaynes (1957) advocated the method of maximum entropy: max H(f), f s.t. f(x)dx = 1, m j (x)f(x)dx = µ j (x), j = 1,..., J, where µ j are functions defined on the support of x and µ j are their expectations. It turns out the entropy maximization problem has the following nice solution: where a 0 f J (x) = exp( J a j m j (x) a 0 ), j=1 = log{ exp( J j=1 a jm j (x))dx} such that f J integrates to one. Note that the coefficients a j s are Lagrangian multipliers associated with the moment conditions, reflecting their boundedness. This maximum entropy density satisfies the given moment condition and maintains the highest degree of uncertainty regarding the underlying distribution. In this sense, it is the least-biased or least committed estimator of the underlying distribution. It nests many commonly used distributions. For instance, if x is positive and m 1 = x, it leads to the exponential distribution; if x is defined on the real line and m 1 = x, m 2 = x 2, it leads to the normal distribution. It turns out that option price can be viewed a moment condition of the RND. Let m j (X j, S t ) = (S T X j ) + = max(s T X j, 0), where x j is a strike price. This is indeed the payoff function of a call option given terminal price S T. Thus the expected payoff of this option is m j (X j, S T )f(s T )ds T, which is also the price of this call option. Thus given a set of options, one can estimate the 5

6 RND f(s T ) using the maximum entropy approach subject to moment conditions m j, j = 1,..., J. The resultant maxent density takes the form f J (x) = exp{ J a j (x X j ) + a 0 }. j=1 This is, as a matter of fact, a linear log-spline density. One can further smooth this curve to improve differentiability. Pricing kernel. Let p(s T ) be the (physical) density of an asset at time T and f(s T ) be the corresponding RND. The pricing of a call option is given by its expected payoff C(X; τ) = exp( rτ) (S T X) + f(s T )ds T By a change of measure, we have C(X; τ) = exp( rτ) exp( rτ) (S T X) + f(s T ) p(s T ) p(s T )ds T (S T X) + K(S T )p(s T )ds T The ratio between the RND and the price density is called the pricing kernel, or stochastic discount factor (SDF). It summarizes the risk preference of a representative agent. For instance, SDF (for a call option) is hypothesized to be monotone decreasing w.r.t. price as people s marginal value to payoff decreases with their wealth. The pricing kernel provides yet another approach to estimate the RND. It can be readily approximated by a series estimation. Suppose K(S T ) a l g l (S T ), 6

7 where g l s are series basis functions. We then have C(X i ; τ) exp( rτ) = a l exp( rτ) a l Ψ(X i ) (S T X i ) + ( a l g l (S T ))p(s T )ds T (S T X i ) + ( g l (S T ))p(s T )ds T Although p(s T ) is not observed, we can estimate it using historical stock returns. Denote the estimate by ˆp(S T ), we can calculate ˆΨ(X i ), replacing p with ˆp. We then estimate the series coefficients by the OLS min a 1,...,a L (C(X i ; τ) a l ˆΨ(Xi )) 2. Denote the estimated SDF by ˆK(S T ) = L âl ˆΨ(S T ), we then estimate the RND by ˆf(S T ) = ˆK(S T )ˆp(S T ). Risk of revenue. Revenue=Price Yield. To calculate the expectated revenue, one needs to know the joint distribution of price and yield. Again, there are two general approaches. The parametric approach assumes a parametric funcitonal form. The multivariate Gaussian is commonly used, which requires only the estimation of mean and variance matrix. However, Gaussian distributions are sometimes restrictive: symmetric and zero tail dependence. Nonparametric multivairate density estimation. Let {X i, Y i } n estimate the joint density using the Kernel density estimator be an i.i.d. sample. We can ˆf(x, y) = 1 nh x h y ( ) x Xi K K h x ( y Yi where h x, h y are bandwidths. Note that nonparametric multivariate density estimation suffers the so-called curse of dimensionality in that the convergence rate declines with the dimension. For details, see Li and Racine (2007). Copula approach. Let F (x, y) be the joint distribution of (x, y) and F x, F y the marginal distributions; let the corresponding densities be f(x, y), f x, f y. Sklar s Theorem states that, 7 h y ),

8 via changes of variables, F (x, y) = C(F x (x), F y (y)), where C : [0, 1] 2 [0, 1] is called the copula function that links two marginal distributions to their joint distributions. Assuming differentiability, taking derivatives with respect to x and y on both sides of the equation gives f(x, y) = f x (x)f y (y)c(f x (x), F y (y)), where c : [0, 1] 2 R + is the copula density function. The copula distribution/density is distribution/density function defined on the unit cube with uniform marginal distributions. It completely summarizes the dependence between x and y and is invariant to positive monotone transformation of the marginals. It offers the advantage of separation between the marginal distributions and their dependence. For general overview of copulas, see, e.g., Nelsen (2007). Sklar s Theorem suggests one can estimate the marginal and copula functions separately. Estimation of the individual marginal densities is generally easier due to its low dimension. If the copula function is also easy to estimate, then it can be advantageous to estimate a joint density via the copula approach. Parametric copula. Popular parametric copulas include the Gaussian, T, Frank, Gumbel, Clayton and so on. Except for Gaussian copula, almost all parametric copulas have only one parameter and are only defined for bivariate distributions. Hence they can be restrictive. Copula density can be estimated nonparametrically as well. Like the usual densities, it suffers the curse of dimensionality. In addition, since copulas are defined on a bounded support, it is also plagued by the boundary bias. This is particularly severe for copula densities, which often tend to infinities at the corners. Boundary bias corrected kernel estimations are called. Alternatively, one can use bounded kernel in the estimation. See, e.g., Chen (1999). Alternatively, one can first transform the domain of the density from [0, 1] 2 to R 2, on which the density can be estimated by the kernel method free of boundary biases, then transform the resultant density back to its original support. 8

9 References Bahra, B. Implied risk-neutral probability density functions from option prices: theory and application. Grith, M., hardle, W., Schienle, M. Nonparametric estimation of risk-neural densities. Lai, W. Comparison of methods to estimate option implied risk-neutral dnesities, Quantitative Finance, 2014, 14: Buchen, P. and Kelly, M. The maximum entropy distribution of an asset inferred from option prices, Journal of Financial and Quantitative Analysis, 1996, 31: Li, Q. and Racine, J. Nonparametric Econometrics, 2004, Princeton University Press. Nelsen, R.B. An Introduction to Copulas, 2007, Springer-Verlag. Chen, S. X. (1999). Beta kernel estimators for density functions. Computational Statistics and Data Analysis, 31(2):