In recent years, numerical methods for valuing options such as binomial tree models, Monte Carlo simulation and finite difference methods are use for a wide range of financial purposes. This paper illustrates and compares the three numerical methods. On one hand, it provided general description of the three methods separately involved their definitions, merits and drawbacks and determinants of each method. On the other hand, this paper makes a concrete comparison in valuing options between the three numerical methods. Overall, the three numerical methods have proven to be valuable and efficient methods to value options.
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Introduction
In recent years, option valuation methods are very important in the theory of finance and increased wildly in the practice field. The various approaches on option prices valuation included binomial tree models, Monte Carlo simulation and finite difference methods. Binomial models are suggested by Cox, Ross and Rubinstein (1979). Boyle (1977) firstly discussed Monte Carlo simulation and then it has been used by both Johnson and Shanno(1985) and Hull and White(1987) to value options when it is a stochastic process. Finite difference methods are discussed by Schwartz (1977), Brennan and Schwartz (1979), and Courtadon (1982) (Hull and White, 1988). This essay aims to provide a comparison and contrast among the three numerical methods mentioned above. All these numerical methods focus on the objectives of both calculation accuracy and speed. The only way for any given method to achieve better accuracy and speed is to calculate with many times (Hull and White, 1988). For one thing, this essay provides general description about binomial trees, Monte Carlo simulation and finite difference methods and defines benefits and drawbacks of each method. For another thing, it makes contrast on the valuation option prices involved American and European options.
Binomial tree models
Hull and White (1988) provide a general description about binomial trees. They concluded that” Binomial model is a particular case of a more general set of multivariate multinomial models”. All multivariate multinomial models are characteristics as lattice approaches such as binomial and trinomial lattice models(Hull and White, 1988).And the binomial trees, a valuation option approach, which involved separating option into a large number of small time intervals of length Δt. The assumption of this method is that the asset price changed from its initial value to two new values, both upward and downward movement, Su and Sd separately. The probability of an upward movement was indicated as p, while the probability of a downward movement is 1-p and the parameter u, d, p are used to value option prices. (Hull, 2008)
The binomial model focused on option replication. For the binomial trees, the only way to reproduce the payoff of an option is to trade a portfolio involved the stock and the risk-free asset. Within other lattice approaches, involved the trinomial tree model, do not admit option replication(Figlewski&Gao, 1999).However, the fair value of option can be valued under the basic assumptions of option pricing which is the world is risk-neutral. (Hull, 2008)
In this case, the fair value can be valued simply by computing the expected values within the risk neutral distribution and discounting at the risk-free interest rate (Hull, 2008).When the world is risk-neutral, any approximation procedure which is based on a probability distribution and rough risk neutral distribution and make convergence to its limit, can be used to value options prices properly. Therefore, it is necessary to use trinomial tree model even a more complex structure without lack of the ability to calculating unique option payoffs (Figlewski&Gao, 1999).
What is also worth mentioning about the application of binomial tree is that there exists known payouts involved dividends (Hull and White, 1988). Dividend policy was based on the principle that the stock maintains a constant yield on each ex-dividend date which was denoted by δ (Cox et.al, 1979)
Essentially, binomial and trinomial models are powerful, intuitive methods to value both American and European option. Moreover, it also provides asymptotically exact approximation based on Black-Scholes assumptions (Figlewski&Gao, 1999).
Consider the efficiency and accuracy of this method, the binomial method is more efficient and accurate when there are a small number of options values without dividends. However it lacks of efficient in a situation where effects of cash dividends should be analysed. Actually, the fixed dividend yield generated an improper hedge ratio despite that the assumption of fixed dividend yield is an efficient and accurate approximation. Furthermore, the binomial tree models are inefficient in valuing American options compared with European option. And it is less efficient and accurate than finite difference methods for multiple options valuation. This is because it has a conditional starting point (Geske&Shastri, 1985).
Monte Carlo simulation
Monte Carlo simulation is a useful numerical method to implement for various kinds of purposes of finance such as securities valuation. For the valuation of option, Monte Carlo simulation use risk-neutral measure (Hull, 2008). For example, a call option is a security whose expected payoffs depend on not only one basic security. The value of a derivative security can be obtained by discounted the expected payoff in the risk-neutral world at the riskless rate (Boyle, et.al, 1997).
Boyle et.al (1997) stated that “this approach comprises several steps in the following. Firstly, Simulate sample paths of the underlying state variables (e.g., underlying asset prices and interest rates) over the relevant time horizon. Stimulate these according to the risk-neutral measure. Secondly, evaluate the discounted cash flows of a security on each sample path, as determined by the structure of the security in question. Thirdly, average the discounted cash flows over sample paths”
There is a tendency that high-dimensional integral is becoming more and more necessary to evaluate in the derivative security. Monte Carlo simulation is widely used in the option valuation due to the increases of high dimension (Ibanez &Zapatero, 2004). Regarding the integral of the function f(x) over the d-dimensional unit hypercube, the simple Monte Carlo estimate of the integral is equivalent to the average value of the function f over n random points from the unit hypercube. When n tends to be infinite, this estimate converges to the true value of the integral. Furthermore, the distinct advantage of this method compared with other numerical approaches is that the error convergence rate is independent dimension. In addition, the function f should be square integrable and this is the only restriction which is relative and slight ((Boyle, et.al, 1997).
Monte Carlo simulation is simple, flexible. It can be easily modified to adapt different processes which involved governing stock returns. Moreover, compared other methods, it has distinct merit in some specific circumstances. Essentially Monte Carlo simulation can be used when the process of generating future stock value movement determined the final stock value. This process mentioned above is created on a computer and aims to generate a series of stock price trajectories which is used to obtain the evaluation of option. In addition, the standard deviation also can be used simultaneously in order to make sure the accuracy of the results (Boyle, 1977).
However, there are some disadvantages of this method. In recent years, some new techniques were developed so as to overcome the disadvantages. One key drawback is that it is wasteful to calculate many times and difficult to control situations when there are early exercise opportunities (Hull, 2008). Different variances reduction techniques involved control variate approach and antithetic variate method are used to solve these problems. Furthermore, deterministic sequences also known as low-discrepancy sequences or quasi-random sequences are used to accelerate the valuation of multi-dimensional integrals, (Boyle, et.al, 1997).
Quasi-Monte Carlo methods are suggested as a new approach to supplement Monte Carlo simulation. It uses deterministic sequences rather than random sequences. These sequences are used to obtain convergence with known error bounds¼ˆJoy¼Œet.al. 1996¼‰
Until recently, Monte Carlo simulation has not been used in American options. The key problem is that payoff depends on some sources of uncertainty. The optimal exercise frontier of American options is uncertain (Barraquand &Martineau, 1995).
Finite difference methods
Hull (2008) provides a general description of finite difference methods. He concluded that “finite difference methods value a derivative by solving the differential equation that the derivative satisfies.” Finite difference methods are classified into two ways those are implicit and explicit finite difference method. The former approach is related the value of option at time t+Δt to three alternative values at time t, while the latter one is related the value of option at time t to three alternative values at time t+Δt (Hull& White, 1990).
The explicit finite difference method is equivalent to a trinomial lattice approach. Compared with the two finite difference methods, the distinct advantage of explicit finite difference method is that it has fewer boundary conditions than the implicit way. For instance, to implement implicit method, considering the price of a derivative security S, it is vital to specify boundary conditions for the derivative security whether minimising or maximising price. By contrast, the explicit method, regarded as a trinomial lattice approach, does not need specific boundary conditions (Hull& White, 1990).
There are two alternative problems of partial differential equations. The first, known as boundary value problems where a wide range of boundary conditions must be specified, the second, known as initial value problems where only a fraction of valuation required to be specified. There is a fact that most option valuation problems are initial value problems. The explicit finite difference method is the most appropriate method to solve initial value problems because implicit finite method used extra boundary condition which was produced errors (Hull& White, 1990).
Furthermore, consider the efficiency and accuracy of valuing option, the explicit finite difference method, with logarithmical transformation, is more efficient than the implicit method. This is because it does not need the solution solved a series of simultaneous equations (Geske&Shastri, 1985).
In addition, for the finite difference method and jump process, the simple explicit difference approximation is harmonized with a three-point jump process, while the more complex implicit difference approximation corresponds a generalized jump process which is based on that the value of derivative security will jump to infinite future values, not just three points(Brennan&Schwartz, 1978)
Finite difference approach can be used in the same situation as binomial tree approach. They can control American and European option and cannot easily used when the payoff of an option depends on the past history of the state variable. Furthermore, finite difference methods can be used in the situation where there are some state variable¼ˆHull 2008). However, the binomial tree method is more intuitive and easily implemented than the finite difference methods. Therefore, financial economists tend to use binomial tree methods when there are a small number of option values. In contrast, finite difference methods are frequently used and more efficient in a situation where there are a large number of option values (Geske&Shastri, 1985).
The comparison between the three methods
Overall, compared with the three numerical methods of valuing option, Monte Carlo simulation should be seen as a supplement methods for the binomial tree models and finite difference methods. This is because the increase of a variety of complexity in financial instruments (Boyle, 1977). Furthermore, binomial and finite difference methods are implemented with low dimension of problems and standard dynamics, while Monte Carlo simulation is the proper methods to solve high dimension problems and stochastic parameters (Ibanez &Zapatero, 2004)
The binomial tree models and finite difference methods are classified as backward methods and can easily handle early exercise opportunities. On the contrary, Monte Carlo simulation is a looking forward method and may be opposed with backward induction (Ibanez &Zapatero, 2004)
For the two similar methods, finite difference approach is equivalent to a trinomial lattice method. They are both useful for American and European options and tend not to be used in a situation where the options’ payoff depends on the past history of state variables. However, there also are some differences between them. Binomial tree methods can be used to calculate a small number of values of options, while finite difference methods can be used and more efficient and accurate when there exit a large number of option values. In addition, binomial tree models are more intuitive and readily completed than the finite difference methods
Monte Carlo simulation is a powerful and flexible method to value various options. In principle, Monte Carlo simulation is calculated a multi-dimension integral and this is becoming an attractiveness compared other numerical methods. It can be used to solve the problem of high dimension. The drawbacks should not be neglected. The computation with many times and cannot easily handle the situation where there are early exercise opportunities. Based the traditional Monte Carlo simulation, a new approach was developed, known as Quasi-Monte Carlo methods to improve the efficiency of Monte Carlo method. The basic theorem is to use deterministic number rather than random.
However, it has not been used in valuing American options due to the optimal exercise frontier is uncertain. One way to value American option is to achieve combination of Monte Carlo simulation and dynamic programming (Ibanez &Zapatero, 2004)
Conclusion
To sum up, with the complexity of numerical computation, numerical methods are wildly used to value derivative security. This paper provided general description and specific comparison between the three numerical methods mentioned above. Binomial tree models, known as lattice approach, are a powerful and intuitive tool to value both American and European option with and without dividend. When there are a small number of option values, binomial method is more efficient and accurate. On the contrary, it is inefficient in a situation where effects of cash dividend should be analysed.
Finite difference method can be seen as the trinomial lattice approach. They are used with the problems of low dimension and have been regarded as efficient and accurate methods to value American and European options. Compared with binomial tree models, finite difference methods is more efficient and accurate when practicers computing a large number of values of options.
Monte Carlo simulation can be seen as a supplement tool for the two methods mentioned above to value options. It can be used with high dimensional problems whereas other two methods are used with low dimensional problems. The flows of Monte Carlo simulation are that it consumes time for calculating and cannot readily handle the situation where there are early exercise opportunities. In this case, Quasi-Monte Carlo methods based on traditional Monte Carlo simulation utilise deterministic sequences known as quasi-random sequences. These sequences provide an opportunity to acquire convergence with known error bounds.
Referenc:
Barraquand¼ŒJ.& Martineau, D. (1995)”Numerical Valuation of High Dimensional Multivariate American Securities “The Journal of Financial and Quantitative Analysis, Vol. 30, No. 3 pp. 383 -405
Boyle, P.P., “Option: A Monte Carlo Approach,” Journal of Financial Economics, Volume:4, pp: 323-338
Boyle, P. Broadie, M. and Glasserman,P.(1997) “Monte Carlo methods for security pricing,” Journal of Economic Dynamics and Control, Volume 21, Issues 8-9,29,pp:1267-1321
Brennan, M.J. & Schwartz, E.S., (1978)”Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis,” The Journal of Financial and Quantitative Analysis, Vol. 13, No. 3 pp. 461 -474
Cox, J.C., Ross, S.A. and Rubinstein. M.(1979) “Option pricing: A simplified approach,” Journal of Financial Economics, Volume 7, Issue 3, pp: 229-263
Figlewski,S.&Gao,B.(1999)”The adaptive mesh model: a new approach to efficient option pricing,” Journal of Financial Economics, Volume 53, Issue 3, pp: 313-351
Geske,R. &Shastri, K.(1985) “Valuation by Approximation: A Comparison of Alternative Option Valuation Techniques,” The Journal of Financial and Quantitative Analysis, Vol. 20, No. 1 pp. 45- 71
Hull, J.(2008) Option, Futures, and Other Derivatives,7th edition, Upper Saddle River: Pearson Prentice Hall
Hull, J, &White, A.(1988) “The Use of the Control Variate Technique in Option Pricing,” Journal of Financial and Quantitative Analysis. Vol. 23, Issue. 3; p. 237-251
Hull, J, &White, A. (1990) “Valuing Derivative Securities Using the Explicit Finite Difference Method,”Journal of Financial and Quantitative Analysis. Vol. 25, No. 1; pp: 87-100
Ibanez, A. &Zapatero, F. (2004) “Monte Carlo Valuation of American Options through Computation of the Optimal Exercise Frontier,” Journal of Financial and Quantitative analysis Vol.39, No. 2, pp: 253-275
Joy, C., Boyle, P.P. and Tan, K.S.(1996)” Quasi-Monte Carlo Methods in Numerical Finance,” Management Science.Vol.42, No.6,pp:926-938
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