# Option Prices

## Top PDF Option Prices:

### Singular Fourier Padé Series Expansion of European Option Prices

In this section, we demonstrate the performance of the SFP method through various numerical tests. The purpose of this section is first to test whether the error convergence analysis presented in Section 7 is in line with the numerical findings in this section. Second, we test the ability of the SFP method to price any European-style option that are deep in/out of the money and have long/short maturities. Third, we analyse whether the SFP method can provide consistent accuracy when approximating small or large values of option prices. Finally, in a major development of the method, we test whether the SFP method can retain global spectral convergence even when the PDF is piecewise continuous. A number of popular numerical methods are implemented to test the SFP method in terms of the error convergence, convergence rate and computational time. These methods include the COS method (a Fourier COS series method, Fang and Oosterlee 2009a), the filter-COS method (a COS method with an exponential filter to resolve the Gibbs phenomenon; see Ruijter et al. 2013), the CONV method (an FFT method, Lord et al. 2008), the Lewis-FRFT method (a fractional FFT method, Lewis 2001, Chourdakis 2004), and the B-spline, Haar wavelet and the SWIFT methods (a wavelet-based method; see Ortiz-Gracia and Oosterlee 2013, 2016). When we implement the CONV and Lewis-FRFT methods, we use Simpson’s rule for the Fourier integrals to achieve fourth-order accuracy. In the filter-COS method, we use an exponential filter and set the accuracy parameter to 10 as Ruijter et al. (2013) report that this filter provides better algebraic convergence than the other options. We also set the damping factors of the CONV and Lewis-FRFT to 0 and any value greater than zero, respectively.

### Closed Form Approximations for Spread Option Prices and Greeks

Our closed-form approximation formula for spread option prices can offer insights on the designing and analyzing of real options embedded in financial and real contracts. Spread options with zero spread, a.k.a. exchange options, have been employed extensively by researchers to model real options, partly because of the availability of the Margrabe formula. For example, McDonald and Siegel (1985) use the Margrabe formula to study the investment and valuation of firms when there is an option to shut down. Shevlin (1991) investigates the valuation of R&D firms with R&D limited partnerships. Albizzati and Geman (1994) value the surrender option in life insurance polices by extending the Margrabe formula to a Heath-Jarrow-Morton stochastic interest rate framework. Grinblatt and Titman (1989), and Johnson and Tian (2000) apply the Margrabe formula to study the design and effectiveness of performance-based contracts and executive stock options. However, in many of these applications, it is more natural to assume that we have a spread option instead of an exchange option. The spread K may correspond to the cost or salvage value of shutting down a firm, the cost or salvage value of terminating an R&D partnership, the monetary penalty of surrendering the life insurance policy prematurely, a minimal level (K > 0) of performance difference that a manager has to achieve over a benchmark, or a cushion (K < 0) to insure the manager that he will not be unfairly penalized because of pure bad luck. The extra degree of freedom arising from a nonzero K could be extremely important in designing and analyzing these real options.

### Thinking by analogy, systematic risk, and option prices

Although, Black and Scholes (1973) formula is the most widely used formula in financial markets, a number of empirical facts have been discovered which cast doubt on its validity: 1) The Black-Scholes implied volatility is not a constant, rather it exhibits a skew/smile when plotted against the striking price. See Rubinstein (1985, 1994), and Dumas, Fleming, and Whaley (1998) among others. 2) In contradiction with Black Scholes formula, systematic risk of the underlying has been found to affect option prices. In fact, both the level of implied volatility as well as the steepness of implied volatility curve increases with systematic risk. See Duan and Wei (2009). 3) Investor sentiment affects option prices. Specifically, implied volatility curve steepens in a bearish market. See Han(2008).4) In contrast with the

## How Duration Between Trades of Underlying Securities Affects Option Prices

number of transactions. In this article, we concentrate on the question of how derivatives prices are calculated when durations possess a distribution function that better reflects the observed empirical behavior. Our contribution is threefold. Firstly, we propose a general model that explicitly incorporates waiting times as one of the building blocks of stock price dynamics under the physical measure. In particular, one of the key elements of our approach is to exploit the idea of time-changes by using transaction times, given by the high-frequency time-stamps of trades, to describe the “business time”. Secondly, we show how option prices are calculated by choosing a risk-adjusted measure. Thirdly, based on empirical waiting-time data from blue-chip companies, we investigate a particular distribution for duration and we employ it to calibrate risk-neutral parameters to IBM options data.

## Canonical Representation Of Option Prices and Greeks with Implications for Market Timing

We introduce a canonical representation of call options, and propose a solution to two open prob- lems in option pricing theory. The first problem was posed by (Kassouf, 1969, pg. 694) seeking “theoretical substantiation” for his robust option pricing power law which eschewed assumptions about risk attitudes, rejected risk neutrality, and made no assumptions about stock price distribu- tion. The second problem was posed by (Scott, 1987, pp. 423-424) who could not find a unique solution to the call option price in his option pricing model with stochastic volatility–without ap- pealing to an equilibrium asset pricing model by Hull and White (1987), and concluded: “[w]e cannot determine the price of a call option without knowing the price of another call on the same stock”. First, we show that under certain conditions derivative assets are superstructures of the underlying. Hence any option pricing or derivative pricing model in a given number field, based on an anticipating variable in an extended field, with coefficients in a subfield containing the un- derlying, is admissible for market timing. For the anticipating variable is an algebraic number that generates the subfield in which it is the root of an equation. Accordingly, any polynomial which satisfies those criteria is admissible for price discovery and or market timing. Therefore, at least for empirical purposes, elaborate models of mathematical physics or otherwise are unnecessary for pricing derivatives because much simpler adaptive polynomials in suitable algebraic numbers are functionally equivalent. Second, we prove, analytically, that Kassouf (1969) power law specifica- tion for option pricing is functionally equivalent to Black and Scholes (1973); Merton (1973) in an algebraic number field containing the underlying. In fact, we introduce a canonical polynomial representation theory of call option pricing convex in time to maturity, and algebraic number of the underlying–with coefficients based on observables in a subfield. Thus, paving the way for Wold decomposition of option prices, and subsequently laying a theoretical foundation for a GARCH option pricing model. Third, our canonical representation theory has an inherent regenerative mul- tifactor decomposition of call option price that (1) induces a duality theorem for call option prices, and (2) permits estimation of risk factor exposure for Greeks by standard [polynomial] regression procedures. Thereby providing a theoretical (a) basis for option pricing of Greeks, and (b) solving Scott’s dual call option problem a fortiori with our duality theory in tandem with Riesz represen- tation theory. Fourth, when the Wold decomposition procedure is applied we are able to construct an empirical pricing kernel for call option based on residuals from a model of risk exposure to persistent and transient risk factors.

## Density forecast comparisons for stock prices, obtained from high frequency returns and daily option prices

We compare density forecasts derived from option prices using the Black-Scholes (1973) and Heston (1993) models with forecasts obtained from historical time series using the Corsi (2009) heterogeneous autoregressive model of realized variance (HAR-RV). However, the risk-neutral density is a suboptimal forecast of the future distribution of the asset price as there is no risk premium in the risk neutral world, while in reality investors are risk-averse. Hence we need to use economic models and/or econometric methods to transform risk-neutral densities into real-world 3 densities. Economic theory motivates pricing kernel

## Recovering Probabilities and Risk Aversion from Option Prices and Realized Returns

We might hope to recover even more information from option prices, in particular, the stochastic process followed by the underlying index price. Unfortunately, given the recovered risk-neutral probability distribution for a given expiration date, there are an infinite number of possible stochastic processes which are consistent with these prices. To sort through these processes, we need to make additional assumptions. We design a model that is as close as possible to the standard binomial option pricing model while allowing options to be valued under an arbitrary prespecified risk- neutral probability distribution applying to a single given expiration date which corresponds to the ending nodes of a recombining binomial tree. We call the resulting stochastic process an implied binomial tree [Rubinstein 1994]. The strongest assumption we make initially (also a property of standard binomial trees) is that all paths leading to the same ending node have the same risk-neutral probability. Applying this model to post-crash option prices produces a tree of local (or one move) volatilities with the following general features:

### ESTIMATION OF STOCK OPTION PRICES USING BLACK-SCHOLES MODEL

This paper attempted to predict the accuracy of the Black-Scholes option pricing model in pricing the stock option contracts for the selected 8 companies. The study uses the Black-Scholes model along with its parameters to estimate the stock option contracts prices. This helps in finding whether the stock options are rightly priced or not. The study finally attempts to identify the pricing errors between the Market price of the option contracts and the calculated option prices. This is done with the help of Mean Absolute Percentage Error and Mean Absolute Deviation tools. The results of the study indicate that there were only a small difference between the calculated prices and the market price of the option contracts.

### Analytic Approximation of Finite Maturity Timer Option Prices

In 2007, Soci´et´e G´en´erale Corporate and Investment Banking introduced a new variance deriva- tive to the market called “timer options” (See Sawyer (2007)). A timer option is similar to a plain-vanilla option, except that the expiry is not deterministic. Rather it is specified as the first time when the accumulated realized variance exceeds a given budget. See Hawkins and Krol (2008), Carr and Lee (2010) and Lee (2012) for some insights on why such a product can be attractive to investors. Interestingly, timer options were studied many years ago by Neuberger (1990) and Bick (1995) when such security did not even exist in the market place. Sawyer (2008) also reported that Soci´et´e G´en´erale Corporate and Investment Banking started to sell other timer-style options such as “timer out-performance options” and “timer swaps”. The pricing of timer options is usually done through Monte Carlo simulation. A naive implementation can take tens of hours to get a reasonably accurate estimate of the option price. See C.X. Li (2010, 2013) and Bernard and Cui (2011) for techniques to speed up the simulation. Another approach is to compute the timer option price exactly through multidimensional numerical integration. See Liang, Lemmens and Tempere (2011) where the authors provide such an integration for the Heston model and the 3/2 model. One shortcoming besides being complex and slow is that this method works for a limited set of models. A third approach is to approximate timer option prices through methods such as perturbation. Saunders (2010) considers an asymptotic expansion for stochastic volatility models under fast mean-reversion. Another paper in this category which is very closely related to the current one is Li and Mercurio (2013). The method in Li and Mercurio (2013) is extremely fast and very accurate, with percentage errors in the order of 0.01% for the Heston model for the parameters they used.

### Impact on option prices of divergent consumer confidence

This paper investigates the impact on option prices of divergent con- sumer conÞdence. To model this, we assume that consumers disagree on the expected growth rate of aggregate consumption. With other condi- tions unchanged in the discrete-time Black-Scholes option-pricing model, we show that the representative consumer will have declining relative risk aversion instead of the assumed constant relative risk aversion. In this case all options will be underpriced by the Black-Scholes model under the assumption of bivariate lognormality. We also extend Benninga and Mayshar’s (2000) results about impact on option prices of heterogeneous beliefs and preferences to an N-agent economy.

### Anchoring Heuristic in Option Prices

An anchoring-adjusted option pricing model is developed in which the expected return of the underlying stock is used as a starting point that gets adjusted upwards to form expectations about corresponding call option returns. Anchoring bias implies that such adjustments are insufficient. In continuous time, the anchoring price always lies within the bounds implied by expected utility maximization when there are proportional transaction costs. Hence, an expected utility maximizer may not gain utility by trading against the anchoring prone investors. The anchoring model is consistent with key features in option prices such as implied volatility skew, superior historical performance of covered call writing, inferior performance of zero-beta straddles, smaller than expected call option returns, and large magnitude negative put returns. The model is also consistent with the puzzling patterns in leverage adjusted option returns, and extends easily to jump-diffusion and stochastic-volatility approaches.

### A Linear Regression Approach for Determining Explicit Expressions for Option Prices for Equity Option Pricing Models with Dependent Volatility and Return Processes

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 [21]-[23]). Analyzing the proposed model is computationally simpler than it is for the other affine jump process models. The results of this paper can also be applied to bond option, foreign currency option and futures option models and to more complex derivative securities including various types of mortgage-backed securities.

### Other positions of parameters in the asymptotic expansions of Knock-in barrier option prices

Abstract. Joshi’s general method for computing the asymptotic expansions of european options was conceived by placing the strike price at the center of the binomial tree [6]. This set-up showed that the errors in approximating the continuous pricing models by discrete ones through asymptotic expansion is of order 1 n . In this paper, we find other positions, aside from the center, for the parameters K (strike price) and B (barrier level) in the asymptotic expansions of Knock-in barrier option prices under Joshi’s general method. This has been shown to be possible for the case of an Up-and-In Put (UIP) barrier option in the paper done by Llemit and Escaner [1].

### 2SABR Implied Volatility and Option Prices

The SABR model is used to model a forward Libor rate, a forward swap rate, a forward index price, or any other forward rate. It is an extension of Black’s model and of the CEV model. The model is not a pure option pricing model— it is a stochastic volatility model. But unlike other stochastic volatility models such as the Heston model, the model does not produce option prices directly. Rather, it produces an estimate of the implied volatility curve, which is subsequently used as an input in Black’s model to price swaptions, caps, and other interest rate derivatives.

## The relationship between conditional value at risk and option prices with a closed-form solution

In this paper we derived a simple, closed form and analytic relationship between CVaR and European options. We discussed the significance of the equation and relationship, in particular with respect to implied volatility and risk management. We showed that we can account for implied volatility smile effects when we measure risk in terms of CVaR. We have demonstrated that CVaR cannot be calculated independently from options, otherwise there may exist arbitrage opportunities. We showed CVaR and VaR surfaces should be modelled like option price surfaces and contrary to common practice, are not predictable over short time periods.

## Estimating Realistic Implied Correlation Matrix from Option Prices

Correlation measures how two securities move in relative to each other. The correlation coefficient can range from −1 to 1. A perfect positive correlation coefficient of 1 means the two security moves in sync with each other. On the other hand, a perfect negative correlation coeffi- cient of −1 means the securities will move in an opposite direction. Correlation is useful in portfolio management because it can be used to reduce risk. In general, correla- tion can be computed from historical data of asset prices. However, the derivation of implied correlation index is something completely different; it is derived from the option premiums of index options and single-stock op- tions that are index constituents. Since the option pre- mium represents the risk associated with the underlying assets, the implied correlation index can be extracted by comparing the option premiums.

## Analogy Making, Option Prices, and Implied Volatility

All the parameters and variables in the Black Scholes formula are directly observable except for the standard deviation of the underlying’s returns. So, by plugging in the values of observables, the value of standard deviation can be inferred from market prices. This is called implied volatility. If the Black Scholes formula is correct, then the implied volatility values from options that are equivalent except for the strike prices should be equal. However, in practice, a skew is observed in which in- the-money call options’ (out-of-the money puts) implied volatilities are higher than the implied volatilities from at-the-money and out-of-the-money call options (in-the-money puts).

## Fractional diffusion models of option prices in markets with jumps

Examples where it has been assumed that share prices follow jump processes include: the early work of Press Ref. [28]; Merton’s Jump Diffusion model Ref. [23]; and the work of Mantegna and Stanley, see Refs. [20, 21, 22], which builds on the work of Mandelbrot Ref. [19] to show that Truncated L´evy Flights can be very successful at capturing the high-frequency empirical probability distribution of the S&P 500 index. Based on this work, Koponen Ref. [15] and Boyarchenko and Levendorskiˇi Ref. [3] proposed the use of modified LS (also known as L´evy-α-stable) processes to model the dynamics of securities. This modification introduced a damping effect in the tails of the LS distribution to ensure finite moments and gain mathematical tractability; these models are known in the mathematical finance literature as KoBoL processes. Finally, motivated by the most important properties of the dynamics of share prices including size and frequency of both positive and negative jumps in the stock price movements, Carr, Geman, Madan and Yor proposed the CGMY process Ref. [6]. This is essentially the same model as the KoBoL, and has quickly become one of the most widely used models for equity prices.