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Authors
Miquel Montero
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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.
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Perpetual American options are financial instruments that can be readily exercised and do not mature. In this paper we study in detail the problem of pricing this kind of derivatives, for the most popular flavour, within a framework in which some of the properties |volatility and dividend policy| of the underlying stock can change at a random instant of time, but in such a way that we can forecast their final values. Under this assumption we can model actual market conditions because most relevant facts usually entail sharp predictable consequences. The effect of this potential risk on perpetual American vanilla options is remarkable: the very equation that will determine the fair price depends on the solution to be found. Sound results are found under the optics both of finance and physics. In particular, a parallelism among the overall outcome of this problem and a phase transition is established.
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.
This paper examines the valuation of American capped call options with two-level caps. The structure of the immediate exercise region is significantly more complex than in the classical case with constant cap. When the cap grows over time, making extensive use of probabilistic arguments and local time, we show that the exercise region can be the union of two disconnected set. Alternatively, it can consist of two sets connected by a line. The problem then reduces to the characterization of the upper boundary of the first set, which is shown to satisfy a recursive integral equation. When the cap decreases over time, the boundary of the exercise region has piecewise constant segments alternating with non-increasing segments. General representation formulas for the option price, involving the exercise boundaries and the local time of the underlying price process, are derived. An efficient algorithm is developed and numerical results are provided.
A make-your-mind-up option is an American derivative with delivery lags. We show that its put option can be decomposed as a European put and a new type of American-style derivative. The latter is an option for which the investor receives the Greek Theta of the corresponding European option as the running payoff, and decides an optimal stopping time to terminate the contract. Based on this decomposition and using free boundary techniques, we show that the associated optimal exercise boundary exists and is a strictly increasing and smooth curve, and analyze the asymptotic behavior of the value function and the optimal exercise boundary for both large maturity and small time lag.
In this paper, we will discuss an approximation of the characteristic function of the first passage time for a Levy process using the martingale approach. The characteristic function of the first passage time of the tempered stable process is provided explicitly or by an indirect numerical method. This will be applied to the perpetual American option pricing and the barrier option pricing. Numerical illustrations are provided for the calibrated parameters using the market call and put prices.
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