Numerical Methods for Option Pricing
DOI:
https://doi.org/10.61173/7nsbn224Keywords:
Trinomial tree model, completing market, pricing optionAbstract
Nowadays, financial markets have become more complex and have given rise to more research opportunities, one of which focuses on research related to the pricing of various financial instruments used in options trading. In this essay, a gradual process of reflection is used to deepen the option pricing theory. Firstly, an effective mathematical method for pricing option contracts is using the binomial tree model, a relatively straightforward way of calculating the value of an option. However, it has only two upward and downward trends and has limitations that are the most different from the actual market conditions. That is why the study will have to continue in-depth to get as close as possible to the real situation in the market. The next step is to make the model more in line with market reality by adding a possible rate of change, resulting in the Trinomial Tree Model. The model incorporates the possibility of future price increases, decreases, or stabilization (the third change). The Trinomial Tree Model follows the no-arbitrage principle and removes the assumption of a risk-free market opportunity. This paper derives a method for constraining market prices under risk-neutral conditions. This is crucial for investors seeking profitable outcomes in options trading. The main objective of this research-based paper is to develop a complete theory of option pricing in a one-step trinomial tree model.
References
change, the second equation when there is a m amount of [1] Korn, R. and Muller, S. (2010a). Recent Developments in change in the value of the stock (uncertainty of the price Applied Probability. and Statistics: Dedicated to the Memory goes up, down, or no change), and the third equation of Jürgen Lehn. [online] scholar.google.com. Available at: when the value of the stock falls by a d amount. And the https://scholar.google.com/scholar?hl=zh-CN&as_sdt=0 simulation of this portfolio holds when the starting price %2C5&q=history+of+binomial+tree+model&btnG=#d= is equal to the price of the option’s payoff at this stage. By gs_qabs&t=1696673413811&u=%23p%3Da-Ss_Y5zuIgJ using the replication strategy (NB, NS), we obtain three [Accessed 7 Oct. 2023]. equations: [2] Paul Clifford, Yan Wang, Oleg Zaboronski. Pricing options using trinomial trees. Working paper, available from https:// N B (1 + r ) + N S S0u = F ( S0u ), warwick.ac.uk/fac/sci/maths/people/staff/oleg_zaboronski/fm/ N B (1 + r ) + N S S0 m = F ( S0 m), (6) trinomial_tree_2010_kevin.pdf (retrieved on August 1st, 2023). B S N (1 + r ) + N S0 d = F ( S0 d ). 2010. [3] Fei Lung Yuen, Hailiang Yang. Option pricing with regime Unfortunately, equations with two unknowns and three switching by. trinomial tree method. Journal of Computational equations do not always have a. solution. Indeed, there and Applied Mathematics. Volume 233, Issue 8, Pages 1821- are two variables NB, NS with three equations; therefore, 1833. 2010. it is not always possible to solve these equations and [4] John Hull. Options, Futures, and Other Derivatives (Tenth determine the option price this way. edition). Harlow, UK Pearson. 2018. It is calculated that there can be a solution only if the [5] Platen, E. (2002). Arbitrage in continuous complete markets. option price is actually within this range: Advances in Applied Probability, 34(3), pp.540–558. doi:https:// doi.org/10.1239/aap/1033662165.
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