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Digital option replication

Digital Option Replication- 1063 S 361 379,Static Option Replication (FRM Part 1, Book 3, Financial Markets and Products, Exotic Options)

Model Dependency of the Digital Option Replication Replication under an Incomplete Model TomበTICH¯* 1. Introduction A digital option is a special type of financial derivative with a non-linear discontinuous payoff function. In spite of this, the payoff is simple enough to allow (relatively) easy valuation of these contracts The paper focuses on the replication of digital options under an incomplete model. Digital options are regularly applied in the hedging and static decomposition of many path-dependent What is a digital option? A digital option is a type of option that provides a fixed payout if the underlying market moves beyond the strike price. As long as traders correctly predict the 3. Digital Option Replication Respecting the number of revisions in time we can distinguish dynamic replication and static replication. The main drawback of dynamic replica-tion is 14/07/ · In this paper we focus on replication of European digital options. A digi- tal option is a financial derivative which if exercised pays its owner some fixed amountQor the value ... read more

Stochastic Processes In this section we briefly define all processes applied in the paper. The simplest building blocks are the Poisson process or closely related ones such as a gamma process and the Wiener process, which provides ingredients for construction of almost all processes with a diffusion part. Hence, the Wiener process is a martingale; its expected increment is zero at any time and the variance is closely related to the time change.

We can, besides others, digital option replication , build on the basis of the Wiener process the geometric Brownian motion GBM. It is the process which was supposed to be the one followed by stock prices in Black and Scholes Two key facts are that the financial-assets gain return continuously and that digital option replication prices cannot be negative.

Both ideas are supported by GBM, since the price is given by an exponential formula. Since the volatility of asset returns is very difficult to measure and forecast, some slightly more realistic models suppose its stochastic feature. However, a candidate to model the volatility must respect the empirical fact that it regularly reverts back to its long-run equilibrium.

Besides others, it is the case of the Hull and White HW model wdt 12 Here, a describes the tendency of mean-reversion, b is the long-run mean equilibrium and s is the volatility of the volatility. The Wiener process of HW 12 which drives the volatility is usually supposed to be independent of the one of the GBM These processes are also typical by the stochastic continuity — the probability of digital option replication occurrence for given time t is zero.

The Lévy process can be decomposed into a diffusion part and a jump part. Clearly, not all parts must be present. The modeling of financial prices is usually restricted to exponential Lévy models. In fact, it is equivalent to deducing —. dt in the case 2 — as a mean corof geometric Digital option replication motion, digital option replication.

We can therefore interpret recting parameter to the exponential of the Lévy process Xt. The classic works incorporating jumps in price returns were based on jump-diffusion models such as the Merton model These models are typical by a finite number of jumps in any time interval. However, the modern models of financial returns are of infinite activity — thus, the jumps, although small in scale, occur infinitely many times in any time interval. In fact these models do not need to be constructed of diffusion components, since the infinite activity allows description of the true feature either jumps or skewness and kurtosis in the distribution of returns well enough.

In addition, the terminal price can be produced by simulation within one step. Many Lévy models are regarded as subordinated Brownian motions. Since the random process g t plays the role of digital option replication time in the original model, it must be non-decreasing in time.

Of course, the process still evolves in time t. However, so-called internal time gives us a very nice economic interpretation of subordinated processes Finance a úvûr — Czech Journal of Economics and Finance, 56, digital option replication ,ã. Transforming the original time into the stochastic process we can also model other parameters of the digital option replication and fit the model more closely to the set of real data.

Barndorff-Nielsen, digital option replication , In this paper we apply the variance gamma VG model5 for more details see, e. An exhaustive analysis of dynamic option replication time-frames is beyond the scope of this paper, but we will review a sampling of methods and results. By simplifying quantitative methods to a binary option approach, we discover that options can be replicated in a dynamic method by using quantitative and technical approaches Replication of European Digital Options The digital call can be thought of as a limit of a call spread.

Reasoning: a binary option's payout graph has an infinite slope at the strike price, whereas all vanilla options and Reviews: 2. Post a Comment. Thursday, July 14, Digital option replication. Digital option replication Replication of European Digital Options The digital call can be thought of as a limit of a call spread.

By simplifying quantitative methods to a binary option approach, we discover that options can be replicated in a dynamic method by using quantitative and technical approaches Pricing a Digital Option This document was uploaded by user and they confirmed that they have the permission to share it.

Static Option Replication FRM Part 1, Book 3, Financial Markets and Products, Exotic Options , time: at July 14, Email This BlogThis! The counterexample is that although the reverse can happen there is no such line close to minus one. A more justifiable idea is that it is natural implication of skewness. Indeed, if we examine the strategy for various levels of skewness, we could see that the distribution of the terminal error is symmetric only if the underlying distribution is also symmetric.

As the skewness of the underlying distribution increases decreases , and under the no-arbitrage condition, the median becomes negative and a distinct line in the positive area appears. The last example of dynamic strategy concerns the stochastic environment given by the time changed VG model VGSE ; see equations 15 and The parameters to put into the CIR model must be such that the condition 2.

Since we have chosen relatively low volatility and high speed of mean-reversion, the results are not very different from the VG model and therefore are not presented here. To briefly name the big difference — the deviation is somewhat lower and the mean is slightly positive around 0. The stochastic environment given by the time-fluctuation implies the change of all distribution: the volatility, skewness and kurtosis. The maximal potential error is one — ST is close to Note also that the difference between the values of these two assets H and f increases with the approaching maturity time.

Since the portfolio is superreplicating, we must proceed as follows. At the beginning, the financial institution gets the yield from the sold option. The capital which is to be spent to long the spread is, however, higher. The difference which is nonzero and negative is borrowed at the riskless rate up to maturity. Furthermore, some estimates may violate the non-negative condition.

For more results, see Carr et al. All the positions are left intact up to maturity. exp r. Since the portfolio is left intact up to maturity, the number of rebalancing intervals is irrelevant. Therefore, only one result will be provided for each method. The presentation of results confirms the theoretical bounds of the error. The total error is either —0. It significantly influences the median, skewness and kurtosis.

The mean value of the replication error is zero, which confirms the no-arbitrage opportunity. Note again that if the mean value differs from zero, one of positions is preferred either f or H and the relative prices change. The SV model, VG model and VGSE model are applied by virtue of the same principle. We suppose that actual market prices correspond to the relevant model of the underlying asset price evolution.

On the basis of market prices market prices are everything we need to know to replicate the option statically the initial difference is calculated.

The results are clear from Figure 6 and Table 4. Apparently, these results are very similar for all models. None of the strategies allows an arbitrage opportunity; the mean is zero and the standard deviation is very low. It seems that the SV model is close to the BS model, and the VG model is close to the VGSE model, as should be supposed.

The dynamic strategy is based on more or less frequent trading with the underlying asset. Without doubt, as we increase the frequency of trading the total costs we incur will also rise. By contrast, the static strategy requires trading only at the beginning, when the portfolio is set up. Although the transaction costs are commonly higher on the financial derivatives market than on the spot market, the difference is usually not so high as to make the dynamic strategy more favorable.

Another problem arises when the market is not sufficiently liquid or constraints on portfolio positions must be respected. This issue concerns mainly trading with options. As we bring the error bounds closer to zero, the number of options to be purchased and sold sharply rises. The requirement need not be met with the true market characteris8 Of course, also large short positions in the underlying can be prohibited. However, the delta of digital options is low, so it should not significantly affect the dynamic replication.

It can be either impossible to purchase such a high number of options or this act would be associated with unfavorable transaction costs. Conclusions Digital options can be regarded as a derivative at a half-step between plain vanilla options and complicated exotic options. On one hand the theoretical pricing is relatively simple; on the other the discontinuity in the payoff function can cause a serious problem in replication and hedging.

The task of this paper was to examine the relationship of the replication error on the completeness of the model. More particularly, we have studied the dynamic replication and the static replication within four distinct models. Three of them were supposed to be incomplete. We showed how replication methods work if the underlying process is not known or cannot be utilized when the replication portfolio is constructed.

The dynamic method performs relatively well only in a complete setting and with frequent rebalancing of the replication portfolio.

Under incomplete models, the frequency of BS-rebalancing does not play such a significant role SV model or it is almost insignificant VG model. By contrast, the static replication performed well also if the underlying process was not known by the subject. However, the inevitable assumption is that the market price must correspond to the true evolution, otherwise, the results may be poor.

The strength of the static replication is that it does not require trading up to the maturity and it allows us to manage the theoretical bounds of the replication error. Of course, if the market significantly changes its view about the underlying price process, the positions can be rebalanced to minimize the expected error. Working paper, Department of Statistics and Applied Probability University of California, ANDERSEN, L. The Journal of Computational Finance, vol.

BARNDORFF-NIELSEN, O. Research report No. Finance and Stochastics, vol. Quantitative Finance, vol. Journal of the Royal Statistical Society, Series B, vol. Journal of Econometrics, March BLACK, F.

Journal of Political Economy, vol. BOCHNER, S. Proceedings of the National Academy of Science of the United States of America, vol. CARR, P. Morgan Stanley and MIT Computer Science, Working Paper, The Journal of Finance, vol. Mathematical Finance, vol. CLARK, P. A subordinated stochastic process model with fixed variance for speculative prices. Econometrica, vol. CONT, R. COX, J. DERMAN, E. The Journal of Derivatives, vol.

FAMA, E. Journal of Business, vol. FUJIWARA, T. GIOVANNI, D. de — ORTOBELLI, S. Technical Report, University of Karlsruhe, Springer-Verlag, HESTON, S. Review of Financial Studies, vol. HULL, J. Prentice Hall, Journal of Finance, vol. MACBETH, J. MADAN, D.

European Finance Review, vol. MERTON, R. Journal of Financial Economics, vol. MUSIELA, M. You can then hedge a digital call as a call spread. The gearing of the call spread used to over-replicate the digital depends on the strike width of the call spread. The wider the call spread, the lesser the gearing and the more conservative the price. Above the barrier level, the call spread has the same payoff as the digital call. Below the barrier level, the digital call has a zero payoff but the call spread has a non-zero payoff between its lower strike and its upper strike located at the barrier level.

Therefore, we say that the call spread over-replicates the digital call because its payoff and therefore its premium is always greater or equal to the digital call's payoff. As a trader, I sell you this digital call on ABI stock. How much will I sell it to you? Well, I will replicate the digital using a geared call spread.

You can think of different scenarii and see that this call spread over-replicates the digital call. By doing so, I have therefore priced the digital conservatively. For a digital option, Gamma can be quite large and tricky near the barrier at maturity. If ABI stock goes up to As a trader, this would be extremely difficult to hedge.

As a trader, the call spread gives me a cushion against this risk. Using a call spread allows to smooth the Greeks. The smaller the call spread, the larger Gamma and Vega can get near the barrier. In fact, around the barrier level, they shoot up and then shoot down while changing sign.

We will analyze Call Spreads in more details in the next chapter. You will see that its gamma is smoother than that of a digital call. The larger the strike width, the more this is true. You shoud have understood by now that when I sell a digital call, I actually book and trade a call spread in my risk management system.

As the underlying gets closer to the barrier, you still want to be able to manage your delta hedge properly. It is the reason why the liquidity of the underlying is an important variable when selecting the strike width of the replicating call spread. So the width of the option spread is used as a pricing mechanism to go conservative on the price of a digital option over its model price. It is necessary in the pricing mechanism to account for real-world difficulty in executing large deltas at the barrier that the model does not consider.

The optimal width of the call spread depends on several parameters among which the size of the digital, the size of the nominal, the underlying's liquidity, the peak delta around the barrier and the implied volatility around the barrier.

In practice, some traders rather take a constant shift of the barrier. Basically, it allows them to take an additional margin for managing the risks if the underlying was to get close to the barrier. This can be more efficient when risk managing a large book of exotic options. When taking a barrier shift, a trader is pricing a new digital whose replicating centered call spread is the hedge of the actual digital.

We will shortly speak about the greeks of a digital call at initiation. Note that the risks and therefore the greeks are dynamics. For example, the greeks will be quite different if you get closer to maturity. As I cannot describe every scenario, the best way for you to learn this material is to use the pricer , ask yourself many questions and find your answer using the pricer.

For example, what happens to the greeks if just before maturity the spot price is exactly at the barrier level? You open the pricer , you select Digital call in the option type input, you set the stock price at the strike level and you set the maturity close to 0. You will be able to calculate the greeks and see all the related graphics. You will then have to interpret them.

The holder of a digital call is always long the forward price since a higher forward increases the probability of the option finishing in-the-money. I don't think I am making you a favor if I describe all the graphics with precision. The best way to develop yourself is to decipher these plots by yourself. For example, you should be able to understand why does the delta converges to zero and not to 1 as in the case of European calls when the stock price increases well above the barrier level.

Note that the plot of the delta is simply the first derivative of the premium plot with respect to the spot price. Since a digital call has positive delta, the trader selling it will have to buy delta of the underlying. Therefore the trader will be long dividends, short interest rates and long borrow costs of the underlying.

While their magnitudes are quite different, Gamma and Vega behave similarly and depend about the position of the forward price regarding the barrier.

The Gamma plot can be easily deduced from the Delta plot since it is simply the first derivative with respect to the spot price.

Replication of European Digital Options The digital call can be thought of as a limit of a call spread. One can therefore make a good estimate of the price of a digital option by using option spreads.

Reasoning: a binary option's payout graph has an infinite slope at the strike price, whereas all vanilla options and Reviews: 2 An exhaustive analysis of dynamic option replication time-frames is beyond the scope of this paper, but we will review a sampling of methods and results.

By simplifying quantitative methods to a binary option approach, we discover that options can be replicated in a dynamic method by using quantitative and technical approaches.

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Words: 8, Pages: UDC: Introduction A digital option is a special type of financial derivative with a non-linear discontinuous payoff digital option replication. In spite of this, the payoff is simple enough to allow relatively easy valuation of these contracts. It is the reason why digital options can be and regularly are applied to decompose and hedge statically the positions in many options with more complicated and usually discontinuous payoff functions; see for example Andersen — Andreasen — Elizier,Carr — Chou,Carr — Ellis — Gupta, or Derman — Ergener — Kani, It is clear, that in order to ensure efficient risk management of complicated exotic options, the procedures for pricing and hedging digital options must be no less efficient.

However, the valuation of digital options is easy only in the Black and Scholes setting. By contrast, relaxing some Black and Scholes restrictions can cause an incompleteness of the model, digital option replication by stochastic volatility, presence of jumps or non-normally distributed returns non-normality of returns is commonly modeled by suitable Lévy models with an infinite intensity of jumps.

Another problem arises if a trader is not sure about the underlying model. Hence, he or she can only guess and, therefore, probably applies the incorrect one, digital option replication. Obviously, it can also happen that the only model available to apply is the Black and Scholes model.

Although the trader can know the true evolution, it can be comprised of such complicated features that the application can be impossible. This situation can again lead to incorrect results. Several authors have already been concerned with the efficiency of replication methods under misspecified input data.

The majority of them studied the case digital option replication wrong volatility, digital option replication , see e, digital option replication. Ahn et digital option replication. By contrast, in many works on static digital option replication it was argued that to get perfect results, the underlying process can be regarded as irrelevant only if the market prices all assets efficiently. All support is greatly acknowledged. Finance a úvûr — Czech Journal of Economics and Finance, 56,ã, digital option replication.

The task of the paper is to examine the method of dynamic replication in such a case via Monte Carlo simulation and compare the results with the static replication. We also verify the irrelevance of the underlying process on the effect of static replication in the case of market efficiency. We study two basic approaches to option replication static and dynamic within four different settings or different types of scenarios of the underlying asset price evolution.

The only same things valid under each model are that the trader is not able to trade continuously and the only applicable model of digital option replication underlying asset price to execute the replication strategy is digital option replication Black and Scholes model. The considered evolutions of the option underlying asset price are i geometric Brownian motion thus the Black and Scholes model, digital option replication , BS modelii geometric Brownian motion with stochastic volatility following the Hull and White process Hull — White, thus the stochastic volatility model, SV modeliii variance gamma process Madan digital option replication Seneta, regarded as a Brownian motion subordinated by a gamma time process thus a special type of the exponential Lévy model, VG modeland iv variance gamma process with stochastic volatility or more generally the model in stochastic environment, VGSE model driven by additional random time — the Cox-Ingersoll-Ross process The last two examples can digital option replication regarded as special cases of subordinated or time-changed processes, developed digital option replication Bochnerfirst introduced in economics probably by Clarkand tested in econometrics, e.

by Stock The paper proceeds as follows: In the following Section 2 we briefly review the typology of options, digital options and their pricing. Subsequently, we recall the principles of dynamic and static replication of options, see e. Tich˘, Next, we define and briefly describe all stochastic processes applied in this paper. The paper ends with conclusions. Digital Option By an option we generally mean a non-linear financial derivative that gives its owner long position the right to buy call options or the right to sell put options the underlying asset S under predefined conditions.

Simultaneously, the seller of the option has an obligation to respect the right of the owner hence the short position.

The predefined conditions concern, for example, the underlying amount of assets, the maturity time Tthe exercise price K. More advanced texts are e. Mussiela — Rutkowski, or Cont — Tankov, Sometimes also other non-standard conditions are defined average price, barrier level, etc. and such options are digital option replication to as exotic options. Moreover, the option which one can exercise only on the maturity date is called a European option and the option exercisable at any time at or prior to maturity is called an American digital option replication.

In this paper we focus on replication of European digital options. A digital option is a financial derivative which if exercised pays its owner some fixed amount Q or the value of any specified asset. Hence, unlike the plain vanilla call or put, the payoff does not depend on the difference between the spot price and the exercise price.

It indicates that the payoff function is not smooth. Obviously, the payoff conditions of digital options can be further complicated, digital option replication , e. by the existence of a barrier level or a gap in the payoff. The simplest example of a digital option is the cash-or-nothing call put.

The owner of this option receives at option maturity T the specified amount Q if the terminal price of the underlying asset ST is not at least as high dig—cash as the exercise price K, digital option replication. Notice that the payoff of discontinuous path-dependent options is further complicated. For example, for the case of an up-and-out call it is zero up to the exercise price level, then it is linearly increasing up to S barrier.

The pricing of digital options as the one of 1 is significantly dependent on the pricing of plain vanilla options. As we can see at the maturity time see Figure 1the value of option fT is either zero or Q times one. Before the maturity time, the crucial variable to value S Finance a úvûr — Czech Journal of Economics and Finance, 56,ã. Thus, the properties of the digital call value function are close to the delta function of the vanilla call.

Similarly, the digital call delta will be close to digital option replication gamma of the vanilla call. N d— 3 This result is general enough to be valid for various types of underlying processes. Still, the digital option price will be given by the probability of exercising, digital option replication , which should be very close to the vanilla call delta.

Digital Option Replication Respecting the number of revisions in time we can distinguish dynamic replication and static replication. The main drawback of dynamic replication is implied by its definition — the method is based on an ever-changing replicating portfolio which consists of one riskless riskless zero-bond or bank account B and n risky assets where n is the number of underlying independent risk factors.

Hence, it needs to be done continuously, which is obviously impossible. Even if the rebalancing interval is small, but not infinitely, the replication error can be high through the discontinuity in the payoff. Static option replication is based on static decomposition of complicated payoffs into a set of more simple payoffs. The decomposition should be such that it will be sufficient to leave the structure of the portfolio intact up to maturity.

Fortunately, the payoff of a cash-or-nothing call can be easily decomposed into a tight spread of plain vanilla options.

Stochastic Processes In this section we briefly define all processes applied in the paper. The simplest building blocks are the Poisson process or closely related ones such as a gamma process and the Wiener process, which provides ingredients for construction of almost all processes with a diffusion part. Hence, the Wiener process is a martingale; its expected increment is zero at any time and the variance is closely related to the time change.

We can, besides others, digital option replication , build on the basis of the Wiener process the geometric Brownian motion GBM. It is the process which was supposed to be the one followed by stock prices in Black and Scholes Two key facts are that the financial-assets gain return continuously and that digital option replication prices cannot be negative.

Both ideas are supported by GBM, since the price is given by an exponential formula. Since the volatility of asset returns is very difficult to measure and forecast, some slightly more realistic models suppose its stochastic feature.

However, a candidate to model the volatility must respect the empirical fact that it regularly reverts back to its long-run equilibrium. Besides others, it is the case of the Hull and White HW model wdt 12 Here, a describes the tendency of mean-reversion, b is the long-run mean equilibrium and s is the volatility of the volatility.

The Wiener process of HW 12 which drives the volatility is usually supposed to be independent of the one of the GBM These processes are also typical by the stochastic continuity — the probability of digital option replication occurrence for given time t is zero.

The Lévy process can be decomposed into a diffusion part and a jump part. Clearly, not all parts must be present. The modeling of financial prices is usually restricted to exponential Lévy models.

In fact, it is equivalent to deducing —. dt in the case 2 — as a mean corof geometric Digital option replication motion, digital option replication.

We can therefore interpret recting parameter to the exponential of the Lévy process Xt. The classic works incorporating jumps in price returns were based on jump-diffusion models such as the Merton model These models are typical by a finite number of jumps in any time interval.

However, the modern models of financial returns are of infinite activity — thus, the jumps, although small in scale, occur infinitely many times in any time interval. In fact these models do not need to be constructed of diffusion components, since the infinite activity allows description of the true feature either jumps or skewness and kurtosis in the distribution of returns well enough. In addition, the terminal price can be produced by simulation within one step.

Many Lévy models are regarded as subordinated Brownian motions. Since the random process g t plays the role of digital option replication time in the original model, it must be non-decreasing in time.

Of course, the process still evolves in time t. However, so-called internal time gives us a very nice economic interpretation of subordinated processes Finance a úvûr — Czech Journal of Economics and Finance, 56, digital option replication ,ã. Transforming the original time into the stochastic process we can also model other parameters of the digital option replication and fit the model more closely to the set of real data.

Barndorff-Nielsen, digital option replication , In this paper we apply the variance gamma VG model5 for more details see, e. An exhaustive analysis of dynamic option replication time-frames is beyond the scope of this paper, but we will review a sampling of methods and results. By simplifying quantitative methods to a binary option approach, we discover that options can be replicated in a dynamic method by using quantitative and technical approaches Replication of European Digital Options The digital call can be thought of as a limit of a call spread.

Reasoning: a binary option's payout graph has an infinite slope at the strike price, whereas all vanilla options and Reviews: 2. Post a Comment. Thursday, July 14, Digital option replication. Digital option replication Replication of European Digital Options The digital call can be thought of as a limit of a call spread.

By simplifying quantitative methods to a binary option approach, we discover that options can be replicated in a dynamic method by using quantitative and technical approaches Pricing a Digital Option This document was uploaded by user and they confirmed that they have the permission to share it. Static Option Replication FRM Part 1, Book 3, Financial Markets and Products, Exotic Options , time:

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The paper focuses on the replication of digital options under an incomplete model. Digital options are regularly applied in the hedging and static decomposition of many path-dependent 3. Digital Option Replication Respecting the number of revisions in time we can distinguish dynamic replication and static replication. The main drawback of dynamic replica-tion is 15/09/ · Replicating the Digital Option. The trick is to replicate the digital option’s payoff with regular calls. As a starting point, consider buying a call with \(K=\) and selling a call Replication of European Digital Options The digital call can be thought of as a limit of a call spread. One can therefore make a good estimate of the price of a digital option by using 14/07/ · By contrast, in many works on static digital option replication it was argued that to get perfect results, the underlying process can be regarded as irrelevant only if the market 14/07/ · In this paper we focus on replication of European digital options. A digi- tal option is a financial derivative which if exercised pays its owner some fixed amountQor the value ... read more

Putting this another way, the value of an American digital option is twice the value of a European digital option. We will discuss further about barrier shifts in the chapter on barrier options. Since the volatility of asset returns is very difficult to measure and forecast, some slightly more realistic models suppose its stochastic feature. As a trader, the call spread gives me a cushion against this risk. JEL classification: G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing G20 - Financial Economics - - Financial Institutions and Services - - - General D52 - Microeconomics - - General Equilibrium and Disequilibrium - - - Incomplete Markets C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General Statistics Access and download statistics. This situation can again lead to incorrect results.

The holder of a digital call will be long volatility if the forward price is lower than the barrier level since a higher volatility will increase the probability of the spot finishing above the barrier at maturity, digital option replication. By contrast, in many works on static digital option replication it was argued that to get perfect results, the underlying process can be regarded as irrelevant only if the market prices all assets efficiently. Normally, some numerical approximation technique should be used. exp r. Digital Option Replication Respecting the number of revisions in time we can distinguish dynamic replication and static replication. The only same things valid under each digital option replication are that the trader is not able to trade continuously and the only applicable model of the underlying asset price to execute the replication strategy is the Black and Scholes model.

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