# Continuous-Time Limit of Binomial Option Pricing

### Introduction

In the context of the contemporary business environment, the economic theory is worked on different models related to continuous-time security trading. As per the change in the market trends, it has addressed that these models are appropriate for managing the insofar various characteristics of the model wherein trades occur very discreetly over a time period. It seems a pervasive element for checking that the limit of discrete-time market models in terms of the period ranging from trades shrink to zero. It helps in determining the impact of continuous time trading (Duffie and Protter, 1992). That is termed the primary objective of the present study. If {(Sn, θn)} is being termed as an appropriate approach of security price processes and trading tactics converging in distribution within specific pairs (S, θ). In addition to that, there are some additional conditions considered under the sequence {ʃ θtn dStn} of stochastic integrals that are defining the gains from trade converges in distribution to the stochastic integral {ʃ θt dSt}.

These conditions recently developed and improved in an appropriate manner for secure applications in finance. In this regards, several examples have determined for elaborating the situation. Furthermore, the present study provides parallel conditions for managing the convergence of gains in probability. Therefore, it can be stated that this paper provides significant support in the form of "user's guide” rather than by providing a set of new convergence results (Dolinsky, Nutz and Soner, 2012). In this context, the option pricing formula is being termed an essential limit of a discrete-time binomial option pricing formula because the number of periods per unit of real-time goes to infinity

Further assessment of new formula has determined that this approach establishes a connection between discrete- and continuous-time financial models concerning a standard technique that is used for estimating continuous-time derivative asset prices with the help of different numerical methods based on discrete-time reasoning. For examples: just assume that {Sn} is termed as a sequence of security price processes converging in terms of distribution while managing the distribution to the geometric Brownian motion price process S under the Black-Scholes model in which {Sn} satisfying basic technical norms (Mele and Fornari, 2012).For this, it has been indicated that there is an appropriate cumulative return process Rn for Sn converges in distribution to the Brownian motion cumulative return process R underlying S, plus at the similar technical condition on the sequence “{Rn}”.

In this regards, it has been addressed that if an investor ignores the distinction between Sn and S is facing some loss then an attempt to replicate a call option payoff concerning Black-Scholes stock hedging Cx(Stn, T - t ) for managing the loss. Therefore, an investor’s strategy will become successful in the limit (in the sense that the final payoff of the hedging strategy converges in distribution to the option payoff as n -» ∞ ). Further investigation has derived that this kind of stability result is to be expected by applying the natural verifiable mathematical conditions that are sufficient for managing different kinds of convergence result (Embrechts, Klüppelberg and Mikosch, 2013). In the context, further assessment is performed through counterexamples concerning certain conditions that are not pathological under which convergence fails. Therefore, the primary task of the present study is to provide a useful set of tools. This way an individual or an organisation can explore the boundaries between discrete- and continuous-time financial models and it also provides stability within the financial gain process dS as per the simultaneous perturbations of the price process S as well as a trading strategy.

As per the requirement of the present study, Preliminaries are playing a critical role in handling different tasks and strategies. This section sets out some of the basic definitions and notation in which the element Dd has denoted the space of Rd-valued cadlag sample paths on specific fixed time interval T = [O, T]. There are natural extensions of results addressed in each case to 3 = 10, m) (Ziegler, 2012). The Skorohod topology on Dd is adopted throughout unless otherwise noted. A cadlag process is being termed as random variable S and worked on some probability space that is valued in Dd. A sequence {Sn} of cadlag processes that could be different probability spaces is converging the cadlag process in distribution that is denoted as S, denoted Sn =» S, if E[ h ( Sn)] -» E[h(S) ]f or any bounded continuous real-valued function h on Dd.

For analysing the situation, Donsker's theorem is being termed as a practical example which normalised "coin toss" in the form of random walk converges during the distribution to Brownian motion. For example: { Yk } is identified as a sequence of independent random variables that are come with the equally likely outcomes + 1 and - 1, and let Rtn = (Y + ... + Y[nt])/√n for any time t, where [ t ] is being considered as smallest integer that could be less than or equal to t. Then Rn-» B in which B is identified as a standard Brownian motion (Dolinsky and Soner, 2013). Donsker's theorem plays a critical role in the general forms of the random walk.

In financial models, the importance of { Yk } in the form of a discrete-time return process is significantly enhanced in which Rn is the normalised the cumulative return process. Furthermore, the corresponding price process Sn is being termed by Stn: = SO Ֆ( Rn), for some initial price where So > 0 and stochastic exponential Ֆ( Rn)of Rn concerning the case.

The general definition of the stochastic exponential has been introduced into this economic context in which it is well known that Sn = S, where St = SoeBt,-'/2. That is linked with the returns generated by a coin toss random walk. Therefore, the asset price process converges in distribution to the solution of the stochastic differential equation dSt = St dBt. This is the typical Black-Scholes example (Bingham and Kiesel, 2013). By extending this example, it shows that the Black-Scholes formula can be found at the limit of discrete-time models in the context of the general class of cumulative return processes in which Rn converging in distribution to Brownian motion.

However, A process X is the form of semimartingale if there exists a decomposition X = M + A, in which M is a denoted as local martingale and A can be termed as an adapted cadlag process with the process and approach of finite variation on the short time intervals. Seniimartingales are termed as the most general processes that contain "stochastic differentials”. For determining the treatment of stochastic integration and stochastic differential equations, there are two counterexamples are evaluated (Liang, Phillips, Wang and Wang, 2016). In each case, there is significant different addressed even though a trading strategy θn converges for managing with a trading strategy θ. In addition to that, a price process Sn converges in distribution to a price process S so as it is not appropriate for the financial gain procedure ʃ θn dSn converges in the distribution for managing the financial gain process ʃ θ dS.

The first example is being addressed a very deterministic and well known. In this regards, firstly there is d = 1 security and it is considering the trading strategies θn = θ = 1(T/2,T], all of which hold one unit of the security after time T/2. Let Sn = l[TI2+ t/n,T] for n > 2/T, and let S = 1[T/2,T]. Therefore, θn => θ and Sn => S, it is not the case that (θn, Sn) => (θ, S) in a sense. Apart from that ʃƅ θn dSn = 1 for all n > 2/T and all t > T/2 + l/n, while ʃƅ θ dS = 0 for all t. The further assessment has derived that failure of weak convergence occurs for a rather obvious reason that will be excluded by initial convergence conditions.

The second example is more subtle. Let B is denoted as standard Brownian motion, and let R = σ B that present the "ideal" cumulative return on a particular investment for some constant σ. Suppose that returns are only credited with a lag as per the moving average basis that is denoted as Rnt = n ʃtt - i/n R(s) ds, so that we are dealing instead with the "stale" returns. Suppose an investor considers they invest total wealth Xt at time t by placing a fraction g(X,) that is termed as a risky investment concerning specific remainder that has been invested recklessly (Mishura, 2015). The approach of continuity has been assumed where g is bounded with a bounded derivative. In the ideal case, the evaluation of the wealth process is mentioned below:

In the above equation, x is termed as an initial wealth. With stale returns, likewise, the wealth process Xn is measured by below formula:

It can be determined that that the "stale" cumulative return process Rn converges in distribution to R.

Literature review

Introduction

It is termed as a most critical section of each systematic study in which a researcher has performed a theoretical analysis of various facts and figures that have been acquired through past studies, views of different authors and online articles. In this process, the investigator has considered a range of secondary sources that include the book, journals and online articles.

Inappropriate results for stochastic integrals

As per the study of Biagini and Pinar (2013), it has found that weak convergence of stochastic integrals is being termed as an important form that is simplified different aspects for applications in financial, economic models. There are some fixed elements addressed so as each n, there is a probability space available (Ωn, Fn Pn) and a filtration { Fnt; r Ɛ T} of sub-u-fields of Fn (satisfying the usual conditions). In this regards, Xn and Hn are cAdlAg adapted processes valued respectively in Rm and Mkm (the space of k x rn matrices). It is assumed that En is denoted as an expectation concerning (Ωn, Fn Pn). Furthermore, there is also a probability space along with filtration on which there are certain corresponding properties hold for X and H (Duffie and Protter, 1992). Moreover, (Hn, Xn) => (H,X ). For "( Hn, Xn) => (H,X),". In addition to that there is significant emphasis on certain factor and that the definition requires at least one (and not two) sequence A. For managing the time changes such that λn(s) converges to s uniformly, and (Hλnn(s), Xλnn(s)) converges in law uniformly in s to (H, X ). X" is assumed as semimartingale in the whole process and for each n, which implies the existence of ʃ Hsn- dXsn.

The research of Mele and Fornari (2012) is focused on the Good Sequences of Semimartingales. In this regards, the primary element is {Xn} that could be termed as the key to a different goal. It is defined that a sequence of {Xn} of semimartingales is appropriate if, for any {Hn}, the convergence of { Hn, Xn) to (H, X ) in distribution based on the implies that X is a semimartingale and also implements the convergence of (Hn, Xn, ʃ Hn_ d Xn ) to (H, X, ʃ H _ d X) in distribution. In this context, a proposition is examined in different conditions. If (Xn) is good and (Hn, Xn) converges in distribution, then the value of (ʃ Hn_ d Xn) is also found appropriate (Embrechts, Klüppelberg and Mikosch, 2013). In this context, it remains to determine some appropriate conditions for a sequence {X"} of semimartingales to be right. The whole process is initiated with the pure, simple condition for "goodness," and then it is extended in a general manner. Before starting the condition, it is essential in order to evaluate each semimartingale in which X is defined by the fact that it can be written in the form of sum M + A of a local martingale M with M, = 0 and an adapted process A of finite variation. Therefore, the total variation of A at time t is denoted [A] along with the quadratic variation of M is termed as [M, M ] (Dolinsky and Soner, 2013)

Analysis of options and other derivatives

As per the study of Sundaresan (2000), it has found that establishment of an isomorphic relationship between dynamic stochastic optimal control problems along with the static state space representation frameworks have found very effective for determining markets martingale representation theory that can be used for reducing the dynamic intertemporal problems into a static problem in order to complete markets setting. This approach has been found very effective in solving intertemporal portfolio selection problems along with the asset pricing problems wherein the investors are mainly focused on certain constraints. Brunnermeier and Sannikov (2016) argued that this approach provides significant for assessing explicit solutions with reference to the consumption and portfolio rules within certain constraints. Further investigation of Moreno-Bromberg and Rochet (2018) has focused on determining the efficiency of results in continuous trading within a few securities. This approach has found very effective within the financial service market and showed that continuous trading provides a basis for the implementation of Arrow–Debreu equilibrium within far fewer securities as compared to a full complement of securities. This study has found it very effective to address the issue of welfare results of continuous-trading opportunities that are mainly focused on the long-lived securities. As per the study of Kijima (2016), it has found that the contribution of different securities can be formalized the importance of dynamic trading opportunities in the context of welfare perspective. This approach has found very effective for analysing different trends and market opportunities as per the contemporary market trends.

The study of Bou-Rabee and Vanden-Eijnden (2018) has focused on different models of the valuation of complex derivative securities. The analysis has been performed on the securities that can be termed as mortgage-backed securities. It is termed as collateralized mortgage obligations through exotic options, lookback options, Asian options, passport options, volatility swaps, shout options, options on swaps and others. Further investigation has found that there is a very large fixed-income derivative market so as important to develop and implement models has enhanced significantly that can be used for valuing as well as hedging of complex derivative instruments along with the business transactions (Chakkour and Frénod, 2016). On the other hand, the study of Hirsa (2016) has to evaluate the Numerical and computational advances that are considered for implementation of those models which are no closed-form solutions. This type of problems has been emerged in American options pricing problems along with performing a systematic valuation of tranches of CMOs. Swishchuk (2016) argued that the problem of managing along with measuring the risk factors within large portfolios had gained significant popularity because it has a direct impact on the overall portfolio efficiency and management decisions. Models which are focused on measuring market risk along with the credit risk, as well as different possible interactions among different variables, have gained significant popularity within the last decades.

The research of Lindström, Madsen and Nielsen (2015) has stated that the risk management within large portfolios has been addressed as another important area where numerical as well as computational procedures have become indispensable and. The development of numerical procedures has been linked with the development of the theory as a result of the free-boundary nature of some derivatives that include American style options. However, some derivatives have been focused on the payoffs that are path dependent, which is also termed as look-back options. The study of Sundaresan (2000) has focused on the development of models that are motivated by stylized facts. Further investigation has found that these models are not easily evaluated by Black–Scholes models. For example This option includes the presence of an implied volatility smile or skews within the options data. The analysis of different option has determined that the term structure of volatility smiles has been identified within options data. In addition to that, it has found that the volatility smile effect identifies to depend in a systematic way on the maturity structure of options (Brunnermeier and Sannikov, 2016). This kind of situation is also emerged as to be a lot stronger in short-term options and less so in long-term options in many markets. Therefore, this is little kind of misspecification occurred within the existing models of options pricing because there is certain possibility of skewness is presented within the process of the conditional distribution of returns. However, the study of Moreno-Bromberg and Rochet (2018) has uncovered that equity returns are predictable. This can be termed as the most important implication for derivatives research. Further investigation has focused for reconciling different observations in which options theorists have a tendency in order to pay significant attention to the two fronts in which the first front takes into account jumps in the underlying state variables, and another mode provides an opportunity for volatility in order to be state dependent or stochastic (Kijima, 2016). The results have been mixed and determined that modeling of jumps risks along with stochastic volatility has found very effective for improving the availability of the fit the options data. However, the study of Duarte (2018) has determined that the term structure of implied volatilities still appears in the form of different patterns which cannot be so easily reconciled. The development of options pricing models has focused on a wide range of equity returns, which are predictable without any kind of big impact.

The research Swishchuk (2016) has evaluated Continuous-Time Methods in Finance in which the author has examined that effect of constraints on trading and transactions costs along with implications on derivatives based on the hedging and pricing. The study has found increased importance of constraints along with with the transactions costs within the process of pricing and hedging. Further investigation has determined that both utility-based approaches and no-arbitrage approaches have been identified very useful bounds in order to evaluate the options prices and implied volatility so as an investigator is able to cover different factors such risk factors and time duration of (Remillard, 2016).

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### References

• Biagini, S., and Pinar, M. Ç. (2013). The best gain-loss ratio is a poor performance measure. SIAM Journal on Financial Mathematics, 4(1), 228-242.
• Bingham, N. H., and Kiesel, R. (2013). Risk-neutral valuation: Pricing and hedging of financial derivatives. Springer Science and Business Media.
• Dolinsky, Y., and Soner, H. M. (2013). Duality and convergence for binomial markets with friction. Finance and Stochastics, 17(3), 447-475.
• Dolinsky, Y., Nutz, M., and Soner, H. M. (2012). Weak approximation of G-expectations. Stochastic Processes and their Applications, 122(2), 664-675.
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• Liang, H., Phillips, P. C., Wang, H., and Wang, Q. (2016). Weak convergence to stochastic integrals for econometric applications. Econometric Theory, 32(6), 1349-1375.
• Mele, A., and Fornari, F. (2012). Stochastic Volatility in Financial Markets: crossing the bridge to continuous time (Vol. 3). Springer Science and Business Media.
• Mishura, Y. (2015). The rate of convergence of option prices on the asset following a geometric Ornstein–Uhlenbeck process. Lithuanian Mathematical Journal, 55(1), 134-149.
• Ziegler, A. C. (2012). A Game Theory Analysis of Options: Corporate Finance and Financial Intermediation in Continuous Time. Springer Science and Business Media.
• Sundaresan, M. (2000). [Online]. Continuous-Time Methods in Finance: A Review and an Assessment. Accessed Through:< https://www0.gsb.columbia.edu/mygsb/faculty/research/pubfiles/458/continuous_time_reviewpaper_jf2000.pdf>. [Accessed On 7th June, 2019]
• Brunnermeier, M. K., and Sannikov, Y. (2016). Macro, money, and finance: A continuous-time approach. In Handbook of Macroeconomics (Vol. 2, pp. 1497-1545). Elsevier.
• Moreno-Bromberg, S., and Rochet, J. C. (2018). Continuous-Time Models in Corporate Finance, Banking, and Insurance: A User's Guide. Princeton University Press.
• Kijima, M. (2016). Stochastic processes with applications to finance. Chapman and Hall/CRC.
• Duarte, V. (2018). Machine Learning for Continuous-Time Economics.
• Remillard, B. (2016). Statistical methods for financial engineering. Chapman and Hall/CRC.
• Meyer, G. H. (2015). The time-discrete Method of Lines for options and bonds: A PDE approach.
• Bou-Rabee, N., and Vanden-Eijnden, E. (2018). Continuous-time random walks for the numerical solution of stochastic differential equations (Vol. 256, No. 1228). American Mathematical Society.
• Chakkour, T., and Frénod, E. (2016). Inverse problem and concentration method of a continuous-in-time financial model. International Journal of Financial Engineering, 3(02), 1650016.
• Hirsa, A. (2016). Computational methods in finance. CRC Press.
• Swishchuk, A. (2016). Change of time methods in quantitative finance. Springer International Publishing.
• Lindström, E., Madsen, H., and Nielsen, J. N. (2015). Statistics for Finance. Chapman and Hall/CRC.

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