The Pricing of Double Barriers Knock-in Binary Put Option
Courwork for Derivatives 2
Ying Li (mr_)
Nina Jhatakia (Nina.)
Alina Ma(alinama_)
MSc Mathematical Trading and Finance
Cass Business School
25th, March 2008
Abstract
As a courwork, we are required to price a double barriers knock-in binary put option. We ud finite difference method in 24 ways and multinomial lattice in 12 ways. We also implemented analytic and Markov chain method. At the end, we compared the four methods and Monte Carlo method.
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In this courwork, we discusd the speed, convergence rate and monotonicity of convergence for the methods. We also discusd whether extrapolation improves convergence.
Our Task
The subject is pricing of barrier option in the Black Scholes model. The option is a discretely (daily) monitored European style barrier knock-in option. The initial stock price is S0 = 100: Time to maturity is 40 trading days. The barriers are 105, 95, the binary strike price is 105.
Assume the logarithm of the stock price is a Gaussian process with constant drift and volatility, b = 0.02; c = 0.4 under the risk-neutral measure: Assume time is measured in years and one year has 250 (trading) days. There are no dividends.
The Option Features
1.Sensitive to barriers and strike price
For the option is a knock-in double barriers option, the barriers have big impact to the price. As a binary option, its payoff is not continues, so the strike price is a critical value also.
2.Barriers and strike price are near to the initial price.
For the barriers (105 and 95) and strike (105) are very near to the initial price (100), especially with the high volatility(40%), the possibility of hitting barriers are very high.
3.Discontinuous Payoff拔花生作文
A Binary Option has a discontinuous payoff. That means a continuous underlying price could generate a completely discontinuous option value. That is, in the payoff chart, there is a jump at the strike price. Discontinuous payoffs generate special oscillation problems.
Analytic Solution
Hui(1996) published clod-form formulas for the valuation of one-touch double-barrier binary options. A knock-in one-touch double-barrier pays off a cash amount K at maturity if the ast price touches the lower L or upper U barrier before expiration. The option pays off zero if the barriers are not hit during the lifetime of the option. Similarly, a knock-out pays out a predefined cash amount K at maturity if the lower or upper barriers are not hit during the lifetime of the option. If the ast price touches any of the barriers, the option vanishes. The formula for the knock-out variant is:
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Where
The option we are pricing is just a knock-in one-touch double-barrier with K=1. The following table gives some pricing results related to our option. We u this as a benchmark to judge our pricing results of other methods. The value of a continuous monitoring corresponding option is 0.6024.
Double-Barrier Binary Option Value
(S=100, T, r=
0.7868 0.6629 0.6205 0.6024
Binary put
(K=105)
0.5352 0.0527 0.0010 0.0000
Knock
Out
Knock In 0.2516 0.6102 0.6195 0.6024
Finite Difference solution
高考励志句子
At fist, in order to compare each method thoroughly, we will not only price the discrete monitored option, which is our task, but also we will price the continuous monitored option.
In order to compare the different convergence speed and pricing accuracy, we will u three kinds of finite difference methods: standard Implicit Finite Difference Method (IFD), Standard Crank-Nicosoln Finite Difference Method (CN) and the Improved Crank-Nicosoln Difference Method (Improved). IFD is stable, so it is a good benchmark for us. CN converges quickly. While CN has some stability issues, we can u some techniques to improve it in the Improved. So, we choo the three.
With the three basic methods, we will u different finite grid construction methods. One method puts all the critical values on the nodes as some papers suggested. The other method puts tho just between the nodes as other papers suggested.
特别的近义词和反义词Besides the direct pricing method, we will u indirect pricing method also. It will give us totally different features.
So, we have 3 (CN,IFD and Improved) x 2 (OnNodes and BetweenNodes) x 2
( Directly and Indirectly) x 2 (Continuous and Discrete) = 24 pricing ways. Implementation
1.Continuous and discrete monitor frequency
In order to compare the difference between the continuous and discrete monitoring frequencies, we
calculate the price for both situations. For the former, we u finer and finer grids in both the space and time directions. For the latter, we u finer and finer grids in the space direction only, leaving the time with daily divisions.
2.Direct and indirect pricing
In theory, the price of a double knock-in binary put is that of a normal binary minus that of a double knock out binary put. In short,
DKI_Binary_Put= Binary_Put – DKO_Binary_Put
So, instead of pricing the DKI_Binar_Put directly, we can price DKO_Binary_Put and then subtract it from a Binary_Put. In regards to our task, the standard binary put is a European option, which can be priced by a continuous BS model and the barriers have discrete monitored frequencies
3.Grid Construction Way
In order to capture the critical events, one way is to put the critical values on the grid nodes. Although there is a way to shift the grid to put the strike on the grid at the cost of the initial stock price not being on the grid any more, the grid shift method is not suitable to barriers option becau
we have to put both barriers on the grid nodes.
Still, bad on an equally spaced grid, we offer an innovative and general method to meet the requirements. We t the barriers as boundaries firstly and then divide the distance equally. So, both the barriers are definitely on the grid nodes, at the cost that the initial stock price may not be on the grid nodes, nor at the center of the grid.
At the same time, some papers suggest we should simply put the critical values just between the nodes, so we also test this.
4.Improved Crank-Nicolson Method
The Crank-Nicolson scheme has faster convergence (quadratic in time and space, compared to linear in time, quadratic in space for EFD and IFD) while maintaining the stability of the implicit method. But, CN time-stepping can have problems if the time-step is larger than the explicit time-step size since C-N is not a positive coefficient method. Normally, this problem is not vere. However, the following situations can cau difficulties: digital payoffs, barriers. In the cas, we can obrve slow convergence (not at the cond order rate) and obvious oscillations. Regarding our option, we have to improve CN.
a.Rannacher time-stepping
Rannacher (1984) suggests a payoff smoothed (if required) and after each rough initial state, we take fully implicit finite time-steps (two in implementation), and u C-N thereafter. Note that the methods are not guaranteed to preclude oscillations, but we are guaranteed to get cond order convergence. Second order convergence does not imply no oscillations. In practice, the methods work remarkably well.
The rationale is that high frequency error components will be dampened by the implicit steps, leading to smooth convergence. The expected rate of convergence remains quadratic since only a finite number of implicit steps are taken. Furthermore, this type of time-stepping can help eliminate oscillations in the solution derivative values. Effective hedging of the underlying contract is then made easier.
b.Discontinuous Payoff
Discretely monitored barriers introduce discontinuities at obrvation dates, while the payoff itlf is discontinuous for digital options.
D. M. Pooley (2002) suggests that discontinuities in the payoff function (or its derivatives) can cau inaccuracies for numerical schemes when pricing financial contracts. D. M. Pooley (2002) discusd three techniques: averaging the initial data, shifting the grid, and a projection method and thinks the techniques are not sufficient to restore expected behaviour. D. M. Pooley (2002) concludes that when combined with a special time-stepping method, high accuracy is achieved.鲌
As a discrete monitored binary option, the option should suffer all the possible problems. But as the strike price is just the same as the barrier (the boundary), so according to the constructed grid, there is no evidence of any discontinuous payoff. Therefore the discontinuous payoff is not applicable here.
However, generally, discrete payoffs remain a significant problem for option pricing.
Pricing Results
When discretizing in time, many methods are available. Explicit schemes are typically simple to implement, but suffer from stability issues. Implicit methods are unconditionally stable, but only exhibit linear convergence. If possible, it is advantageous to u Crank-Nicolson timestepping to achieve quadratic convergence. However, Crank-Nicolson timestepping is prone to spurious oscillati
ons if twice the maximum stable
explicit timestep is exceeded. Further, if discontinuous initial conditions are prent, the expected quadratic convergence may not be realized.
1.Indirect method gives us an amazing good result, which convergent quickly.
Direct and indirect pricing gives us different values, which converge to each other generally. Regarding the option, indirect pricing quickly gives us a near-accurate stable answer, due to the very high knock-in probability. Due to the high probability, the double knock-in barriers option is nearly a standard binary option.
In this way, if we u the indirect method, it means we u a standard binary option as a benchmark to calculate. At the same time, as we u direct method, this means we u zero as the benchmark. It is obvious that the option value is near to that of a standard binary option, so the convergence is very fast.
2.Generally, OnNodes method gives us a better result
The reason is obvious that all the critical events are very nsitive to the timing. When we put the crit
ical value on the nodes, it is easy to capture the event in time.
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A special reason is that all the critical values are on the boundaries, which means we do not need any trade-offs to put this critical value on node or that value on node.
3.Improved CN method does give us a smooth, but wor result in discrete
monitoring
In continuous monitoring it ems Improved CN method is the same as the CN - there are no oscillations with either method. In discrete monitoring, Improved CN method does give us a better result. There is no oscillation except for oscillation in the between-nodes method of Improved CN.
4 method gives a better convergence speed and improved CN method has a
monotonic convergence.
Compared to other methods, CN and Improved CN have faster convergence speed. At the same time, IFD is always stable and has no oscillation. Improved CN also has no oscillation.
5.Extrapolation improves convergence for IFD and improved CN, but which does
not make any real n.
It is obvious that IFD and improved CN have montonic convergence, so it is possible to u extrapolation to speed the convergence. At the same time, it is obvious also that even with traditional extrapolation, the results couldn’t be as good as CN method.
Figure 1: Continuous Monitored Option Pricing
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Figure 2: Discrete Monitored Option Pricing
Multinominal Solution
Alford, J. and N.Webber (2001) investigate multinonimal method and that the heptanomial lattice is the fastest and most accurate of the lattices of higher order, and recommend its u as standard in many one factor lattice implementations. In order to ensure that convergence achieves its theoretical rates, Alford, J. and N.Webber (2001) suggests smoothing to ensure that the payoff function is sufficiently differentiable, and
