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Register a new userAn Alternative Approach to Evaluate American Options Price Using HJM Approach
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Vajira Manathunga
Publication date
2021
fields
Financial
and research's language is
English
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Developments in finance industry and academic research has led to innovative financial products. This paper presents an alternative approach to price American options. Our approach utilizes famous cite{heath1992bond} (HJM) technique to calculate American option written on an asset. Originally, HJM forward modeling approach was introduced as an alternative approach to bond pricing in fixed income market. Since then, cite{schweizer2008term} and cite{carmona2008infinite} extended HJM forward modeling approach to equity market by capturing dynamic nature of volatility. They modeled the term structure of volatility, which is commonly observed in the market place as opposed to constant volatility assumption under Black - Scholes framework. Using this approach, we propose an alternative value function, a stopping criteria and a stopping time. We give an example of how to price American put option using proposed methodology.
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We call a given American option representable if there exists a European claim which dominates the American payoff at any time and such that the values of the two options coincide in the continuation region of the American option. This concept has interesting implications from a probabilistic, analytic, financial, and numeric point of view. Relying on methods from Jourdain and Martini (2001, 2002), Chrsitensen (2014) and convex duality, we make a first step towards verifying representability of American options.
The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as begin{equation*} V^{omega}_{text{A}^{text{Put}}}(s) = sup_{tauinmathcal{T}} mathbb{E}_{s}[e^{-int_0^tau omega(S_w) dw} (K-S_tau)^{+}], end{equation*} where $mathcal{T}$ is a family of stopping times, $omega$ is a discount function and $mathbb{E}$ is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process $S_t$ is a geometric Levy process with negative exponential jumps, i.e. $S_t = s e^{zeta t + sigma B_t - sum_{i=1}^{N_t} Y_i}$. The asset-dependent discounting is reflected in the $omega$ function, so this approach is a generalisation of the classic case when $omega$ is constant. It turns out that under certain conditions on the $omega$ function, the value function $V^{omega}_{text{A}^{text{Put}}}(s)$ is convex and can be represented in a closed form; see Al-Hadad and Palmowski (2021). We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of $omega$ such that $V^{omega}_{text{A}^{text{Put}}}(s)$ takes a simplified form.
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Continuous-time random walks are a well suited tool for the description of market behaviour at the smallest scale: the tick-to-tick evolution. We will apply this kind of market model to the valuation of perpetual American options: derivatives with no maturity that can be exercised at any time. Our approach leads to option prices that fulfil financial formulas when canonical assumptions on the dynamics governing the process are made, but it is still suitable for more exotic market conditions.
We study the explosion of the solutions of the SDE in the quasi-Gaussian HJM model with a CEV-type volatility. The quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low dimensional Markovian representation which simplifies their numerical implementation and simulation. We show rigorously that the short rate in these models explodes in finite time with positive probability, under certain assumptions for the model parameters, and that the explosion occurs in finite time with probability one under some stronger assumptions. We discuss the implications of these results for the pricing of the zero coupon bonds and Eurodollar futures under this model.
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