scholarly journals A Quasi-Closed-Form Solution for the Valuation of American Put Options

2020 ◽  
Vol 8 (4) ◽  
pp. 62
Author(s):  
Cristina Viegas ◽  
José Azevedo-Pereira
Keyword(s):  
Closed Form ◽  
Method Of Lines ◽  
Closed Form Solution ◽  
Form Solution ◽  
Option Value ◽  
Put Options ◽  
American Put Option ◽  
Put Option ◽  
American Put Options ◽  
American Put

This study develops a quasi-closed-form solution for the valuation of an American put option and the critical price of the underlying asset. This is an important area of research both because of a large number of transactions for American put options on different underlying assets (stocks, currencies, commodities, etc.) and because this type of evaluation plays a role in determining the value of other financial assets such as mortgages, convertible bonds or life insurance policies. The procedure used is commonly known as the method of lines, which is considered to be a formulation in which time is discrete rather than continuous. To improve the quality of the results obtained, the Richardson extrapolation is applied, which allows the convergence of the outputs to be accelerated to values close to reality. The model developed in this paper derives an explicit formula of the finite-maturity American put option. The results obtained, besides allowing us to quickly determine the option value and the critical price, enable the graphical representation—in two and three dimensions—of the option value as a function of the other components of the model.

Pricing of perpetual American put option with sub-mixed fractional Brownian motion

2019 ◽  
Vol 22 (4) ◽  
pp. 1145-1154
Author(s):  
Feng Xu ◽  
Shengwu Zhou
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Asset Price ◽  
Put Options ◽  
No Arbitrage ◽  
American Put Option ◽  
Put Option ◽  
American Put Options ◽  
Mixed Fractional Brownian Motion ◽  
American Put

Abstract The pricing problem of perpetual American put options is investigated when the underlying asset price follows a sub-mixed fractional Brownian motion process. First of all, the sub-mixed fractional Black-Scholes partial differential equation is established by using the delta hedging method and the principle of no arbitrage. Then, by solving the free boundary problem, we get the pricing formula of the perpetual American put option.

scholarly journals Joshi’s Split Tree for Option Pricing

Risks ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 81
Author(s):  
Guillaume Leduc ◽  
Merima Nurkanovic Hot
Keyword(s):  
Closed Form ◽  
Numerical Study ◽  
Convergence Theory ◽  
Two Phase ◽  
Binomial Tree ◽  
Put Options ◽  
American Put Options ◽  
Binomial Trees ◽  
American Case ◽  
American Put

In a thorough study of binomial trees, Joshi introduced the split tree as a two-phase binomial tree designed to minimize oscillations, and demonstrated empirically its outstanding performance when applied to pricing American put options. Here we introduce a “flexible” version of Joshi’s tree, and develop the corresponding convergence theory in the European case: we find a closed form formula for the coefficients of 1/n and 1/n3/2 in the expansion of the error. Then we define several optimized versions of the tree, and find closed form formulae for the parameters of these optimal variants. In a numerical study, we found that in the American case, an optimized variant of the tree significantly improved the performance of Joshi’s original split tree.

scholarly journals Examining the Efficiency of American Put Option Pricing by Monte Carlo Methods with Variance Reduction

2018 ◽  
Vol 10 (2) ◽  
pp. 10
Author(s):  
George Chang
Keyword(s):  
Monte Carlo ◽  
Option Pricing ◽  
Variance Reduction ◽  
American Options ◽  
Balance Sheet ◽  
Put Options ◽  
Control Variate ◽  
American Put Option ◽  
Put Option ◽  
American Put

We apply the Monte Carlo simulation algorithm developed by Broadie and Glasserman (1997) and the control variate technique first introduced to asset pricing via simulation by Boyle (1977) to examine the efficiency of American put option pricing via this combined method. The importance and effectiveness of variance reduction is clearly demonstrated in our simulation results. We also found that the control variates technique does not work as well for deep-in-the-money American put options. This is because deep-in-the-money American options are more likely to be exercised early, thus the value of the American options are less in line (or less correlated) with those of their European counterparts. the same FPESS can also be observed when investigators partition large datasets into smaller datasets to address a variety of auditing questions. In this study, we fill the empirical gap in the literature by investigating the sensitivity of the FPESS to partitioned datasets. We randomly selected 16 balance-sheet datasets from: China Stock Market Financial Statements Database™, that tested to be Benford Conforming noted as RBCD. We then explore how partitioning these datasets affects the FPESS by repeated randomly sampling: first 10% of the RBCD and then selecting 250 observations from the RBCD. This created two partitioned groups of 160 datasets each. The Statistical profile observed was: For the RBCD there were no indications of Non-Conformity; for the 10%-Sample there were no overall indications that Extended Procedures would be warranted; and for the 250-Sample there were a number of indications that the dataset was Non-Conforming. This demonstrated clearly that small datasets are indeed likely to create the FPESS. We offer a discussion of these results with implications for audits in the Big-Data context where the audit In-charge would find it necessary to partition the datasets of the client. 

scholarly journals Optimal exercise of American put options near maturity: A new economic perspective

2021 ◽  
Author(s):  
Anna Battauz ◽  
Marzia De Donno ◽  
Janusz Gajda ◽  
Alessandro Sbuelz
Keyword(s):  
Stock Price ◽  
Price Level ◽  
Intrinsic Value ◽  
European Option ◽  
Put Options ◽  
Put Option ◽  
American Put Options ◽  
New Perspective ◽  
American Put

AbstractThe critical price $$S^{*}\left( t\right) $$ S ∗ t of an American put option is the underlying stock price level that triggers its immediate optimal exercise. We provide a new perspective on the determination of the critical price near the option maturity T when the jump-adjusted dividend yield of the underlying stock is either greater than or weakly smaller than the riskfree rate. Firstly, we prove that $$S^{*}\left( t\right) $$ S ∗ t coincides with the critical price of the covered American put (a portfolio that is long in the put as well as in the stock). Secondly, we show that the stock price that represents the indifference point between exercising the covered put and waiting until T is the European-put critical price, at which the European put is worth its intrinsic value. Finally, we prove that the indifference point’s behavior at T equals $$S^{*}\left( t\right) $$ S ∗ t ’s behavior at T when the stock price is either a geometric Brownian motion or a jump-diffusion. Our results provide a thorough economic analysis of $$S^{*}\left( t\right) $$ S ∗ t and rigorously show the correspondence of an American option problem to an easier European option problem at maturity .

scholarly journals CLOSED FORM OPTIMAL EXERCISE BOUNDARY OF THE AMERICAN PUT OPTION

2021 ◽  
Vol 24 (01) ◽  
pp. 2150004
Author(s):  
YERKIN KITAPBAYEV
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Stock Price ◽  
Time Dependent ◽  
Strike Price ◽  
American Put Option ◽  
Optimal Exercise Boundary ◽  
Put Option ◽  
Exercise Boundary ◽  
American Put

We present three models of stock price with time-dependent interest rate, dividend yield, and volatility, respectively, that allow for explicit forms of the optimal exercise boundary of the finite maturity American put option. The optimal exercise boundary satisfies the nonlinear integral equation of Volterra type. We choose time-dependent parameters of the model so that the integral equation for the exercise boundary can be solved in the closed form. We also define the contracts of put type with time-dependent strike price that support the explicit optimal exercise boundary.

scholarly journals Neural Network Pricing of American Put Options

Risks ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 73
Author(s):  
Raquel M. Gaspar ◽  
Sara D. Lopes ◽  
Bernardo Sequeira
Keyword(s):  
Mean Squared Error ◽  
General Motors ◽  
Gamble Company ◽  
Market Prices ◽  
Put Options ◽  
Put Option ◽  
Squared Error ◽  
American Put Options ◽  
American Put ◽  
Better Than

In this study, we use Neural Networks (NNs) to price American put options. We propose two NN models—a simple one and a more complex one—and we discuss the performance of two NN models with the Least-Squares Monte Carlo (LSM) method. This study relies on American put option market prices, for four large U.S. companies—Procter and Gamble Company (PG), Coca-Cola Company (KO), General Motors (GM), and Bank of America Corp (BAC). Our dataset is composed of all options traded within the period December 2018 until March 2019. Although on average, both NN models perform better than LSM, the simpler model (NN Model 1) performs quite close to LSM. Moreover, the second NN model substantially outperforms the other models, having an RMSE ca. 40% lower than the presented by LSM. The lower RMSE is consistent across all companies, strike levels, and maturities. In summary, all methods present a good accuracy; however, after calibration, NNs produce better results in terms of both execution time and Root Mean Squared Error (RMSE).

SHOULD AN AMERICAN OPTION BE EXERCISED EARLIER OR LATER IF VOLATILITY IS NOT ASSUMED TO BE A CONSTANT?

2011 ◽  
Vol 14 (08) ◽  
pp. 1279-1297 ◽  
Author(s):  
SONG-PING ZHU ◽  
WEN-TING CHEN
Keyword(s):  
Stochastic Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Put Options ◽  
American Put Option ◽  
Put Option ◽  
Exercise Price ◽  
Volatility Model ◽  
American Put

In this paper, we present a correction to Merton (1973)'s well-known classical case of pricing perpetual American put options by considering the same pricing problem under a stochastic volatility model with the assumption that the volatility is slowly varying. Two analytic formulae for the option price and the optimal exercise price of a perpetual American put option are derived, respectively. Upon comparing the results obtained from our analytic approximations with those calculated by a spectral collocation method, it is shown that our current approximation formulae provide fast and reasonably accurate numerical values of both option price and the optimal exercise price of a perpetual American put option, within the validity of the assumption we have made for the asymptotic expansion. We shall also show that the range of applicability of our formulae is remarkably wider than it was initially aimed for, after the original assumption on the order of the "volatility of volatility" being somewhat relaxed. Based on the newly-derived formulae, the quantitative effect of the stochastic volatility on the optimal exercise strategy of a perpetual American put option has also been discussed. A most noticeable and interesting result is that there is a special cut-off value for the spot variance, below which a perpetual American put option priced under the Heston model should be held longer than the case of the same option priced under the traditional Black-Scholes model, when the price of the underlying is falling.

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