Fast Fourier Transform Based Computation of American Options under Economic Recession Induced Volatility Uncertainty

The menace of Economic recession to uncertainty in the payoff of investments and standard of living cannot be over emphasized. This paper presents fast Fourier transform method for the valuation of American style options under the exposure of Economic recession. A multi-factor affine Exponential jump model with Recession induced Stochastic volatility and Intensity, which is a partial Integro-Differential Equation (PIDE) is presented. We show how to determine the characteristic function of the model via generating function. A close form characteristic formula for a financial claim satisfying the PIDE in pricing both European style and American style options in Fourier based transform was done. A numerical based Fourier transform algorithm FFT for European call option valuation was extended to the model under study. The algorithm was further extended to American call options valuation by adding premium price to the European call options price. Numerical result was presented to reflect the effect of economic recession induced volatility on options prices and that of the usual volatility. The result shows some significant vicis-situdes in the options values in the two states of the Economy. The result output indicated that the model is effective and reliable compared to other existing models. The fast Fourier transform (FFT) approach gave better option value and compared to both Black-Scholes Merton (BSM) and American Option solver as shown in the table under numerical result section. We used Nigerian Flourmill Stock (NFS) prices for data calibration and reported the stock performance during the first Nigerian recession and recovery year in the Appendix section.


Introduction
Financial mathematics encompasses many relevant areas in which option valuation is one of them and connects many different fields of study in mathematics.
It relies on the application of various mathematical concepts and tools for survival. Such concepts are Probability theory, optimization theory, numerical analysis, partial differential equations, Ordinary Differential Equations, integral representations, transformation and many more. Among  of recession according to NBER cited in [1] is a significant decline in economic activity spread across the economy, lasting more than a few months, normally visible in production, employment, real income, and other indicators. Some other definitions of economic recession obtainable in financial press emphasised that recession begins with two consecutive quarters of decline in Gross Domestic Product (GDP). NBER ponders on GDP as the single best measure of total economic activity and reflects on the GDP definition to be too narrow in measuring economic activity and to reliably date economic recessions. There exists scenario whereby recession may not include two consecutive quarters of negative growth such as United States recession of 2001 but according to Thomas Hsu [1], declines in GDP are closely correlated to recession periods. Nigerian economic recession outbreak in year 2016 was based on decline in GDP and rise in some other macroeconomic indicators such as high inflation rate and unemployment rate after two consecutive quarters. Some measures were invented by axiomatic methods: for example, as cited in [2]. Probability Measure was invented by A.N.  [2] [3] [4] [5] [6]). Probability theory has been seen as the vehicle for dealing with uncertainty in finance and insurance risks.
Probability theory as a mathematical theory is useful in describing and analysing situations where randomness or uncertainty are present [7]. Definition of uncertainty had been given by different authors in various scenarios. According to from option prices. There is strong tendency that predictions of the volatility of asset prices induced by economic recession factor among other market risk factors may not really be accurate. This becomes a challenge for investors that do not like taking risk, that is, the risk averters. There are some existing methods of calculating the volatility of asset prices in financial market especially while dealing with instantaneous prices (i.e. the real time prices) of assets in the financial market. Among the methods known to us are 1) Standard deviation approach, 2) using Historical value of the volatility and 3) implied volatility. The Standard deviation approach tells us how tightly the stock price is grouped around the mean or moving average. When the prices are spread apart, the implication is that one has a relatively large standard deviation but if the prices are relatively closed to each other or bunched together, this connotes that the standard deviation is negligible.
The causal effect of volatility on stock prices is discussed briefly as follows. The stock market prices rise when volatility decreases and increase in volatility causes fall in stock market value (prices). Increase in volatility leads to increase in market risk but decrease in returns of the market. In the case of options on stocks market, the causal effect of volatility changes on options depends on the type. For example, increase in volatility leads to increase in call options value but for put options, increase in volatility leads to decline in the payoff of the put options vice versa. An intuition behind the introduction of recession induced volatility uncertainty is revealed by huge volatility fluctuations during the period of Economic recession compared to the period of normalcy (recession-free). This in turn affects investors' prediction in the market. Since Economic recession induces a high level of uncertainty on investors activities to include decision making and the payoffs of stocks in general, we then proposes giving a close attention to volatility changes in relation to economic recession and financial models for options valuation. Nigeria economic recession outbreak in 2016 and its effects on the payoffs uncertainty of Nigeria Stocks Exchange (NSE) among other investments is among the motivating factors for proposing economic recession induced volatility formulation in Options pricing. A good knowledge of the behavior or the level of uncertainty in economic recession induced volatility by investor's will help in decision making during recession period.
The rest part of the paper is organized as follows: Preliminaries on financial modeling is discussed in Section 2 in addition to some other subsections to include Fourier transform, uncertainty and uncertain measure. Section 3 deals with accounting for jumps in the asset price linked with recession, Section 4 shows the model formulation. Numerical Fourier based transform of options is presented in Section 5 while Section 6 is conclusion.

Some Literature Review on Uncertainty
The theory of uncertainty in financial market could be traced back to the research work of [2] [3] [4]. Without uncertainty, the probabilities of risky events are known and frictionless markets can precisely price contracts contingent on risky events broadly. The volatility of the stock market or GDP is often used as a measure of uncertainty because when a data series becomes more volatile it is harder to forecast. Other common measures of uncertainty include forecaster disagreement, mentions of "uncertainty" in news, and the dispersion of productivity shocks to firms [9]. It was further stressed by Nicholas Bloom [9] that the volatility of stock markets, bond markets, exchange rates, and GDP growth all rise suddenly during economy recessions. In Philip et al. [10], it was stressed that investment under the exposure of economic recession tends to have negative growth. This in turn poses a lot of challenges on investors in a financial or money market as investors' decision in the market also depends on the daily information on the state of the economy.
In the history of options pricing in financial market, the famous Black -Scholes model [23] with the assumption of constant rate of return and volatility have been criticized by many researchers due to the fact that it does not reflect the stochastic nature of financial markets wholly. As a result of this deficiency in the B-S model, some other realistic models have been formulated and come to stay in the sense that the models shows better random movement of financial  [31].
Geske-Johnson scheme with Richardson Extrapolation was adopted to numerically extend fast Fourier Transform algorithm to American-style security, which features a continum of potential exercise times up to expiration.
Our contribution to the existing literature on options valuation is highlighted in sequel: 1) We incorporate economic recession induces volatility uncertainty into exponential jump model with stochastic volatility and intensity. The economic recession induce volatility is assumed to be an uncertain variable.
2) Derivation of Characteristic function of the affine model with recession uncertainty effect is presented.
3) The Fourier transform of the affine model is performed.
4) The Characteristic function of the affine model is extended to fast Fourier transform algorithm of Carr & Madam [29] in pricing European options. It was extended to pricing American style options by adding time premium such that the limit value tends to zero at the expiry time.

5) Volatility
Surface of the affine model based on the recession induced uncertainty is presented.
consisting of subsets of Γ is called an algebra over Γ if the following three conditions hold: Under the notion of countable union, if 1 2 , , , n ∧ ∧ ∧ ∈   such that the collection  becomes a σalgebra over Γ .
is a Borel set of real numbers.

Fourier Transform
Fourier transform in mathematical finance was firstly used to determine the dis-   : .
Equivalently, ( ) f x is a piecewise integrable real function over the entire real line satisfying the condition For the Fourier transform and inverse Fourier transform of a function to exist, it is necessary that the function is absolutely integrable over R with finite value as defined in (3) above. An absolutely integrable functions on a given interval The space of square integrable functions is represented by where 1 i = − and ω is a parameter.
We can recover x which belongs to either 1 L or 2 L -spaces.

Accounting for Jumps in the Asset Prices Linked with Recession
The major market parameters which most financial modeler will wish to consider while formulating a financial model relies on price, interest rate, dividend rate, volatility and time. In an economy threatened by recession, asset price tends to experience lots of fluctuation. Asset price especially stocks can never be stable. It is very necessary to consider interest rate and volatility to be stochastic in nature. Even if one considers interest rate to be constant, in reality volatility cannot be constant. The Stock volatility is seen as a measure of the uncertainty on the payoff or returns of the stock which investors look up to for decision making and taking. During economic recession, the rate of volatility variation is considered higher than the state of normalcy of an economy. As a result an economy recession factor is invoked in this paper as we shall see later.

Jumps in the Underlying Stock Price
Consider the dynamics of a stock price ( ) S t given by where r is the interest rate, q is the dividend rate, ( ) N t is a Poisson process with stochastic intensity ( ) t λ , m is an average jump amplitude given as where jump size ν is a random variable. The equation consists of diffusion process and jump process. There are two sources of fluctuations in the stock price which we classified as "changes in economic state" and "changes due to supply and demand factors". The inclusion of jump in the model is to cater for the arrival of useful information into the market that will have an abnormal consequence on the stock price. Among the factors that are responsible for jumps in the stock price highlighted by Matthew, S.M in [32], we added economic recession factor. The factors are grouped into:  Firm specific jumps: Caused by news inflow to the market on individual firm's profit/loss report.  Industry/sector specific jumps: Caused by news that can affect specific company or industry such as news on sudden declaration of Holiday.
where λ is the intensity. In line with Matthew [24], if information on economic recession or other source of panic enters the financial market (stock) causing an instantaneous jump in the stock price such that the price jump where t ν is taking as the absolute magnitude jump. Then the relative change in price is given by

The Model Formulation
Let ( ) ξ γ be an economy recession induced parameter variation define on an is an uncertain variable. We have that Example 2: Let the above economic recession induce parameter variation describes the filtration in the market and the economy where market activities takes place, respectively. The (resp. p m R × ) from an uncertainty probability space The expected value of the uncertain random variable ξ is defined as where p E and E  represent the expected values under the uncertainty space and the probability space, respectively.
The above definition connects both the notion of probability and uncertainty such that the random variable is defined from a probability space to uncertainty space. It makes sense to introduce the notion of uncertainty to financial models especially while pricing during economy recession or strong financial crisis in the market. The works of the authors cited in this paper to have contributed to parameter uncertainty in financial models did not consider uncertainty with respect to economic recession. We extended the notion of uncertainty to the term structure of stochastic volatility to include economy recession.
Let an asset X be described as a two-state regime switching process which is free to jump between the states defined as where lm λ is the rate of transition from state l to state m and the times spent in state l before transiting to state m is * Suppose further that the asset X price is defined on a filtered probability space Ω    such that the market filtration is generated by the combination of Wiener process and jump process at a given time, and P is taken as a risk-neutral probability measure. The dynamics of the underlying stock price ( ) S t (not necessarily a stock) is given by where r is the interest rate, q is the dividend rate, with ρ is less than 1. As we take m to be an average jump amplitude since the stock price is expected to jump either upward or downward. Then setting ( ) where p is the probability of upward move jump and q, the downward move jump such that In [16], the mean positive jump and negative jumps were given as

Determination of Characteristics Function
The notion of characteristics function is indispensable in the study of random variables in the context of jump diffusion processes. It is a viable tool in Fourier transformation of options pricing. It is worth noting that the distribution function of jump diffusion processes in closed form may not be readily available or known but the characteristics function is explicitly known. However, the characteristics function of some certain stochastic processes or model may not be readily available in closed form but it can be determined.
Definition 10: The characteristics function of a random variable X is a func- where ( ) X f x is the probability density function of the random variable X.
This definition almost coincide with the definition of inverse Fourier Transform in (6) if the expectation exists. For a continuous probability density function, the moment-generating func- where ( ) f x is the probability distribution function.

The Feynman-Kac Formula Revisited
The formula states that a probabilistic expectation wrt some Ito-diffusion processes can be obtained as a solution of a related partial differential equation.
For an example, for a 1-dimensional stochastic process which is a solution of stochastic differential equation Then the function with final condition ( ) ( ) is the solution to the PDE The Equation (26) is useful as it can be extended to characteristics function as Similarly, Remarks 4:
2) The moment-generating function

Solution to the Proposed Uncertain Affine Exponential Jump Model with Recession Induced Stochastic Volatility and Intensity
Consider the Equation (16) Substituting Equation (32) is the moment-generating function of the jump size distribution and m is an average jump amplitude given earlier by Equating the coefficients of the term structures (the stochastic volatility and the intensity) in Equation (35) to zero, the following ordinary differential equations were obtained: The above systems of solutions were solved as follows.
The result is finally written as Similarly, integrating the ODE in the Equation (49), one will have Writing an explicit solution for Corollary 6 : (Characteristic formula for early exercise Option) For an American option, consider a range of time t T τ ≤ ≤ , where t is the initial (starting) time, τ is an early exercise time and T is the expiry time.
Suppose one is able to determine an optimal payoff time * T τ < in a stopping region, then the early exercise payoff of the claim

Numerical Fourier Based Transform of
Then the call option price function is written as By rationalising the base yields From difference of two's squares, the denominator reduces to ( ) ( Since 2 1 i = − , then negative sign in the denominator changed to positive, The size of the consistent spacing ζ between the N values of k has the fol- The relation is observable if we make comparison between Equations (95) and (97).
The smaller the values assumed by η , the better the fineness of the integration grid and vice versa.

Numerical Experiment
Consider an American Stock with initial price 0 100 S = , Strike Price 80 K = ,  Figure 1 shows the true representation of the call options value obtained during recession free period in Table 1 above while Figure 2 shows the payoff of the   American style call options during recession period. The inference we can draw from both the Table 1 and the two Figure 1 and Figure 2 is that during a very short maturity period of a month in the options life span, the options value in both economic states (recession and recession-free) coincides while as the maturity period increases beyond a month, the upward volatility change induced by recession causes slight increase in the call options prices. If further experiment is carried out on put options, reverse will be the case in terms of volatility change effect on the put options payoff.

Conclusions
The notion of economic recession and its effect on volatility uncertainty on the payoff of European and American Options based on some certain assumptions was presented in this paper. Economic recession is becoming a global issue and not too far to be recognized as a re-occuring incident. In the history of economic recession, US is at the forefront as the continent has experienced several eco- An investor's acquaintance with volatility gives a better comprehension of why option prices behave in certain ways and this will guide them in decision mak- We also reported Nigerian Flourmill stock performance, during recession and recovery year in this study. The Stock volatility is seen as a measure of the uncertainty on the payoff or returns of the stock which may require an estimation.
The two common methods in practice for an estimation of volatility of stocks or assets generally are known to be an estimation of historical volatility or implied volatility on that stock. Estimation of Recession induced volatility may be difficult to be determined accurately. Nevertheless, the fact we try to establish here is that the stocks price experiences a high level of uncertainty during the period of recession. The volatility of stocks tends to increase during recession compared to the period of normalcy. One major point is that volatility is never constant in an ideal real life situation. The Flourmill stock prices data used for calibration purpose revealed that during Nigerian recession, the stock prices became more volatile compared to other periods. According to the Assumption 1 above in this paper, we suggest the use of historical data prior to recession period to determine the level of uncertainty posed by Economic recession which we refer to as Economic recession induced volatility. We hope the figures we generated using MATLAB was presented in the Appendix section. Figures A1-A5