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.

Show more
35 Read more

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.

Show more
40 Read more

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

Show more
33 Read more

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.

Show more
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.

Show more
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

Show more
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:

Show more
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.

Show more
15 Read more

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.

Show more
38 Read more

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.

Show more
20 Read more

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.

Show more
37 Read more

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.

21 Read more

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].

Show more
11 Read more

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.

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.

Show more
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.

Show more
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).

Show more
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**.

Show more
In this paper we derive N th order stochastic dominance **option** bounds from concurrently expiring options. We show that given the **prices** of a unit bond, underlying stock, and n **option** **prices**, the kth order stochastic dominance op- tion bounds are given by a pricing kernel the (N − 2)th derivative of which is (n/2)-segmented and piecewise constant if n is even or ((n+1)/2)-segmented and piecewise constant if n is odd. Since stochastic dominance rules are generally accepted, the derived **option** bounds in this paper are practically meaningful.

A er the implementation of the predictive model into gamma hedging strategy, signiﬁ cant improvement in its performance is observed. During the observed period of 23 months 62 portfolios were created. Positions were closed next day to minimize time impact on the estimated changes in **option** **prices**. Initial capital was increased by 133.24%, with average proﬁ t 1.44% per realized portfolio. It means that the average proﬁ t was eight times higher than the average proﬁ t of growth portfolio and 5 times higher than decline portfolios. All three strategies are relatively similarly risky (measured by standard deviation of returns). If we compare strategies by the maximal drawdown, strategy with predictive model is deﬁ nitely less risky than other two strategies. Detail comparison can be seen in Tab. II.

Show more