# The valuation of american put options. and uniqueness of a weak solution of the valuation of American options. In our approach, Key words and phrases. option pricing, finite element method, free-boundary value problems, variational inequalities .. [6] M. J. Brennan and E. S. Schwartz, The valuation of American put options, Journal of Finances,. 32, , pp.

## The valuation of american put options. This report focuses on solving the problem of pricing American put options in a numerical way. After a rigorous derivation of the Black-Scholes equation through an extensive use of Taylor's expansion, the formula is further modified to be an inequality such that it accounts for the early-exercise property of the American option.

For full functionality of ResearchGate it is necessary to enable JavaScript. Here are the instructions how to enable JavaScript in your web browser. BSc Financial Mathematics, Advisor: This report focuses on solving the problem of pricing American put options in a numerical way. Numerical methods, namely the finite difference method and the moving boundary method are employed to solve for the option price.

In terms of the finite mesh discretisation in stock price and time, three schemes, namely the explicit, implicit and Crank-Nicolson scheme are developed and a comparison of the schemes is performed based on an illustrative example. Apparently , finite difference schemes are capable of solving European option prices with fixed boundaries. The moving boundary technique is, however, appropriate to solve for the American option price as the best exercise boundary is unknown.

It is effectively a combination of the boundary updating process and the finite difference method with fixed boundaries. Starting from an initial boundary, the basic idea is to solve a set of fixed boundary problems in order to get updates for the boundary. A comparison of the traditional approach and the moving boundary approach shows that the latter is superior, especially in terms of the reduced computation time and the output of the boundary price as a by-product.

To assess the effectiveness of the model, empirical analysis on the pricing of American put options is performed using the adapted moving boundary method. Statistical analysis proves the explanatory power of this method. The slight difference between option values estimated by the model and that from actual data is partly due to the ill-posed assumptions.

To further improve the model, stability analysis on the explicit scheme is performed; the non-dividend-payment assumption is relaxed and the value of the relaxation factor in the successive over-relaxation algorithm is optimized to minimize the convergence time.

Replacement of the distribution of stock price and time-varying volatilities are two possible areas remaining open for future discussion. The Valuation of American. More about the model and algorithm Limitations and future improvements This report focuses on solving the problem of pricing American put options in a. After a rigorous derivation of the Black-Scholes equation through an.

In terms of the finite mesh discretisation in. Apparently , finite difference schemes are capable of solving European option prices. It is effectively a combination of the boundary. A comparis on of the traditional approach and the. T o assess t he effective nes s of the model, empirical analysis on the pricing of American.

The slight difference between. To further improve the model, stability analysis on the e xplicit. Replacement of the distribution of stock price and. An option contract is a financial derivative and trading these contracts is a.

Currently, there are various option contracts available on the market,. With an option, an. The holder of a European option can only exercise it on the expiration date while the. American option holder can exercise the contract any time no later than the expiry. A path-dependent option is different as it is priced dependin g on the previous. To be specific, an option buyer bears no obligation but the right to exercise the.

H owever , his counterparty has no right but the responsibility to complete the. Thus, a proper option price is essential to satisfy the needs of both sides. In terms of the option v aluation, European options are easy to price since the exercise. In particular , for a non-dividend-paying.

European op tion, analytical solutions can be de rived from Black and Scholes 19 For the American option, owing t o the. That is why it is currently the major option contract traded on most large.

Hence in an efficient market, it is crucial for the seller to. Moreover , a n investor should make careful calculations to exercise the American. The premium, also known as. However , because of its early-exercise characteristic, more research efforts are. When the price is higher than the exercise price, instead of exercising.

Hence, there is no economic benefit. As a result, this narrows the focus of this research. For these options, early-excising may lead to a higher.

Specifically, the famous Black-Scholes. In addition, there is no fixed exercise boundary in. F or this reason,. Recently , advances in. For example, the finite difference me thod, finite. After a critical review of the past literature , Section 3 will cover a detailed explanation.

Following that, algorithms used in this. Briefly , option prices derived from the adapt ed Black -Scholes formula will be compared. Finally , a detailed discussion of the. Limitations and suggestions for the direction of future studies. Section 8 summarises the entire research project. Li terature R eview. There are various factors which influence the price of an option: Investors would regard it as the benchmark to.

To find an optimal option price associated with these risk factors, extensive. At the beginning of the last century , the sto ck price was assumed by researchers to. However , this model suffers from two major drawbacks: The first one is, by assuming.

The other one is that the stock price might become negative based. During the 19 60s, a number of academics have modified the. This a ss ertion was also supported later. Furthermore, they incorpora ted risk.

However , parameters use d in their model were not specifically defined, which made it. Nevertheless, this work laid the foundation for the. In that paper ,. Black and Scholes derived an explicit formula to price the non-dividend-paying.

European op tion by constructing a riskless portfolio. Though their model was based on. T o solv e the Black-Scholes. In this way , an exact analytical s olution to the approximation to the. The formula suggests that option price wi ll not be affected. Another marked feature of the formula is that it can be easily. In spite of the. For instance, the capital market is not frictionless as there exists.

In addition, trading is not continuous,. It therefore undermines the accuracy of. T o account for the dividend-paying. However , empirical evidence shows that the. Furthermore, the Black-Scholes model is based. Considering this, risk neutrality is an alternative to derive the Black-Scholes formula. In a risk ne utral world, stock price was discounted at. This derivation is systematic but the martingale pricing. A discu ssion of various relevant methods introduced in the past. T o further account for.

One major drawback of this ex tended model is that riskless. In terms of the valuation of American options, the original Blac k-Scholes formula. The rationale is s traightforward since in a riskless portfolio, the. American option is valued higher than its European counterpart due to its.

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