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A convertible bond (CB) with a right of m out of n day provisional call or soft-call becomes callable given that the underlying stock closes above a pre-set trigger price for any m or more days over the n consecutive trading days up to the current day. It is computationally challenge to value the contribution of this embedded option to the price of CB. This paper proposes an approximation based on the idea of an auxiliary reversed binomial (ARB) tree, and shows that the approach can be efficiently implemented under the Cox-Ross-Rubinstein parameterization. Two important insights emerge from ARB. First, the convertible bond is unconditionally callable at higher stock prices, but uniformly not callable at lower prices. Second, the effect of the soft-call is rather localized around the trigger price. Surprisingly, the simple One-Touch, or one out of one, approximation is found to yield CB prices that are very close to those from ARB, even though ARB is found to a better approximation in almost every aspect. Further numerical results suggest that in order to generate well-behaved CB prices, cautions need to be taken while designing the terms of soft-call. Being independent of the finite difference pricing grid, the proposed ARB tree can also be used in association with the tree method or Monte Carlo simulation, and could in principle be applicable to exotic derivatives with similar embedded options.

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... Except an example, how to generate the daily indicator is not explicitly described in that paper. Liu (2008) proposes another approximation method (ARB). The idea is that at any state of the stock, if the stock goes backward m steps on a tree, all historical m-step stock paths can be generated, based on counting the number of paths which satisfy the soft-call trigger condition, the probability of trigger can then be calculated. ...

... The idea is using an auxiliary state variable to keep track of the historical path, but instead of tracking the exactly path, here we will only track the number of days stock is above the barrier (It will be called level in this article). The idea is similar to the ARB tree proposed by Liu (2008), instead of counting the trigger probably of stock at every stock grid point, here we will generate the conditional trigger probability based on the new auxiliary state variable. If walk backward on the tree, this probability can be calculated out. ...

In convertible bond market, it is very common to protect the conversion privilege from being called away too soon by using soft-call constraint, or to protect the bond being converted too soon by using provision convert constraint. The first option will protect the bond holder; the second will be benefit to bond issuer. Both constrains have the common feature that the option can be exercise only when the underlying stock closes above a pre-set barrier for any n or more days over m consecutive trading days up to the exercise day. This feature brings challenge for pricing. This paper will propose an approximation solution by Looking Backward (LB) method. In order to illustrate the idea more clearly, I will focus on the Black model stock dynamic using binomial tree based on Cox-Ross-Rubinstein scheme. The results are compared with the exactly solution given by the author in [1]. The extension to other numerical method such as PDE with more general stock dynamic will also be discussed, and the numerical scheme will be laid out. The idea of the method can be applied to the pricing of other path dependent instruments in general.

... In other words, the m-out-of-n soft call provision is replaced with the 1-out-of-1 soft call provision. Liu (2008) proposes an auxiliary reverse binomial tree method to handle the m-out-of-n soft call provision embedded in a CB with a constant interest rate. Due to the path dependency in the provision, the number of examined paths on the binomial tree grows exponentially, which makes it computationally infeasible. ...

The Chinese convertible bond market has been developing rapidly in the last 10 years. However, some special characteristics of the Chinese convertible bond, such as the soft call/put provision, cause huge difficulty in the valuation. In this paper, we establish a new valuation model for the Chinese convertible bond, based on the available Chinese market data, through a hybrid willow tree approach with consideration of the underlying stock price, stochastic interest rate, and credit risk of the issuer. We employ the Brownian bridge to handle the special characteristics. Finally, we examine our model prices for the daily market closing prices for 20 Chinese convertible bonds traded from 2007 to 2017. The empirical results show the effectiveness of our valuation model under the historical and implied volatilities of the underlying stock price.

... In other words, it replaces the m out of n with 1 out of 1, but it is different from the Chinese CB contract, which the current trading price is excluded from determining the activation of the provision. Liu (Liu, [2008]) proposed an auxiliary reverse binomial tree method to handle the m out of n soft-call provision embedded in a CB with a constant interest rate. Due to the path-dependency, the number of examined paths on the binomial tree grows exponentially, which makes its computation infeasible for long-term CBs. ...

Chinese convertible bond market is developing very fast in recent 10 years. However, studies on the pricing model of this emerging market are limited due to some special features embedded in the CB contract different from the major markets. Among these features, the most difficult one is the Parisian-like m out of n soft-call/put provision because of the moving observation window and high path dependency. In this paper, we consider the stock price, stochastic interest rate and credit risk of the issuer on a two-factor willow tree structure. Then, we employ the Brownian bridge at each day to handle the Parisian-like m out of n soft-call/put provision. Finally, we compare the prices computed by our hybrid method with the daily closing prices of 20 Chinese CBs from 2007 to 2017. It illustrates that our pricing model can be a good benchmark for Chinese CBs pricing. Moreover, as a by-product, our method can also calibrate the implied volatility of stock from the market price of its CB, even though no stock option is traded on Chinese market.

The m out of n day provision (MooN) of convertible bonds is difficult to handle. To approximating the MooN better, this paper proposes an approach named the conditional range probability (CRP). CRP is the simulated probability of the MooN being reached within a price range at a future time, conditional on today’s price of the underlying, and can be incorporated into any conventional derivatives pricing method. For a purposely designed exotic call option with a 20 out of 30 day provision, CRP under finite difference is found to outperform significantly several existing approaches and produce a mean pricing error of 1% over a wide range of initial underlying prices for the exotic call. The result implies that finite difference utilizing CRP will yield excellent approximating prices for convertible bonds.

Pricing algorithms for options with exotic path- dependence using the forward shooting grid approach are characterized by the augmentation of an auxil- iary state vector at each grid node on a lattice tree that simulates the discrete asset price process. The state vector is used to capture the specific path-dependent feature of the option contract. We demonstrate the versatility of the forward shoot- ing grid algorithms by generalizing the approach to price various types of Parisian options, options with reset features, and alpha quantile options. The convergence behaviors of the numerical results obtained by the for- ward shooting grid algorithms are also examined. The advantage of the forward shooting grid approach over the finite-difference approach becomes more appar- ent when the governing differential equation for the option value cannot be expressed in a simple form.

This article presents a simple yet powerful new approach for approximating the value of American options by simulation. The key to this approach is the use of least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniques cannot be used. We illustrate this technique with several realistic examples including valuing an option when the underlying asset follows a jump-diffusion process and valuing an American swaption in a 20-factor string model of the term structure. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

This new and updated edition deals with all aspects of Monte Carlo simulation of complex physical systems encountered in condensed-matter physics, statistical mechanics, and related fields. After briefly recalling essential background in statistical mechanics and probability theory, it gives a succinct overview of simple sampling methods. The concepts behind the simulation algorithms are explained comprehensively, as are the techniques for efficient evaluation of system configurations generated by simulation. It contains many applications, examples, and exercises to help the reader and provides many new references to more specialized literature. This edition includes a brief overview of other methods of computer simulation and an outlook for the use of Monte Carlo simulations in disciplines beyond physics. This is an excellent guide for graduate students and researchers who use computer simulations in their research. It can be used as a textbook for graduate courses on computer simulations in physics and related disciplines. A broad and self-contained overview of Monte Carlo simulations Contains extensive cross-referencing between simulation and relevant theory and between applications of similar algorithms in different contexts Provides many applications, examples, `recipes', and specific case studies

Wall Street is the stuff of legend and a source of nightmares, a force so powerful in American society--and, indeed, in world economics and culture--that it has become an almost universal symbol of both the highest aspirations of commercial success and the basest impulses of greed and deception. How did such a small, concentrated pocket of lower Manhattan came to have such enormous influence in national and world affairs. In this wide-ranging volume, economic historian Charles Geisst answers this question as he provides the first history of Wall Street, ranging from the loose association of traders meeting on New York sidewalks and coffee houses in the late 18th century, to the modern billion-dollar computer-driven colossus of today. Here is a fascinating chronicle of America's securities industry and of its role in our nation's economic development. Geisst's narrative ranges over two centuries, from just after the Revolutionary War, to the California Gold Rush and the economic boom (for the North) of the Civil War, to the great stock market crash of 1929, right up to the recent junk bond frenzy and the merger mania of the 1980s that culminated in the fall of Drexel Burnham. The book traces many themes--the move of industry and business westward in the early 19th century, the rise of the great Robber Barons, the influence of the securities market on incredible growth of industry, particularly in the innovative financing of the railroads and major steel companies and crucial investments in Bell's and Edison's technical innovations. Geisst also looks at the gradual increase in government involvement in Wall Street, revealing how regulation had been minimal at first and many investors had suffered from the abuses of corrupt firms. But with the beginning of the New Deal, the government stepped in to pass a series of laws--centered on the Securities Exchange Commission--that severely restricted the ways that Wall Street firms could operate. Here began a heated debate that still rages today between those who want unfettered license to operate as they please and those who want the government to regulate the market to curb corruption. Of course, "The Street" has always been a breeding ground for characters with brazen nerve, and no history of the stock market would be complete without a look at the most ruthless wheeler dealers. Geisst for instance details the manipulations by which Jay Gould and associates cornered the gold market, leading to the terrifying market crash on "Black Friday" in September 1869. Here too are battles of will between powerful personalities and the determined rise to power of such "self made men" as John Jacob Astor, John D. Rockefeller, and Cornelius "Commodore" Vanderbilt--as well as the connivings of lesser known deal makers like William Crapo "Billy" Durant, reputed to have made $50 million in three months shortly before the stock market crash in 1929. Wall Street is at once a chronicle of the street itself, from the days when the wall was merely a defensive barricade built by Peter Stuyvesant, and in a broader sense it is an engaging economic history of the United States, a tale of profits and losses, endlessly enterprising spirits, and the role Wall Street played in helping America become the most powerful economy in the world.

Preface; 1. Introduction; 2. Some necessary background; 3. Simple
sampling Monte Carlo methods; 4. Importance sampling Monte Carlo
methods; 5. More on importance sampling Monte Carlo methods of lattice
systems; 6. Off-lattice models; 7. Reweighting methods; 8. Quantum Monte
Carlo methods; 9. Monte Carlo renormalization group methods; 10.
Non-equilibrium and irreversible processes; 11. Lattice gauge models: a
brief introduction; 12. A brief review of other methods of computer
simulation; 13. Monte Carlo simulations at the periphery of physics and
beyond; 14. Monte Carlo studies of biological molecules; 15. Outlook;
Appendix; Index.

We introduce adaptive learning behavior into a general-equilibrium life-cycle economy with capital accumulation. Agents form forecasts of the rate of return to capital assets using least-squares autoregressions on past data. We show that, in contrast to the perfect-foresight dynamics, the dynamical system under learning possesses equilibria that are characterized by persistent excess volatility in returns to capital. We explore a quantitative case for theselearning equilibria. We use an evolutionary search algorithm to calibrate a version of the system under learning and show that this system can generate data that matches some features of the time-series data for U.S. stock returns and per-capita consumption. We argue that this finding provides support for the hypothesis that the observed excess volatility of asset returns can be explained by changes in investor expectations against a background of relatively small changes in fundamental factors.

“Soft-call” in convertible bonds (CBs) usually means
that the bond can be recalled by the issuer only if the stock price has
previously closed above a specified trigger price for any 20 out of any
30 consecutive trading days. It is not an easy optionality to value and
no method has been implemented besides Monte Carlo. The problem is not
very well suited to Monte Carlo due to a large number of possible
permutations of stock price closes above or below the trigger over a
year period (i.e., 2<sup>260</sup>) with the result that a Monte Carlo
valuation requires a trade off between being slow and not smooth. The
soft-call feature is typically modeled in the CB industry by presuming
that the bond is called as soon as stock touches the trigger price.
After discussion of the exact solution of this problem (requiring
valuation of component derivatives on, of order, 2<sup>260</sup> grids),
a simple algorithm is presented to approximately value this feature for
the general n out of m case of soft-call. The algorithm requires merely
a subtle change to the call feature of the one-touch model and only one
running of a grid or tree and hence it is very fast. The method boils
down to making the bond “1-touch” callable on some days and
not on others, the precise sequence being a function of the 29 day stock
price close history. It gives smooth functional output (the theoretical
price jumps from day to day) and very compelling qualitative results.
The results are accurate to a dime on the dollar of benefits due to
provisional call, and this is determined by comparison to the exact
solution for easily calculated cases

Convertible bonds can be difficult to value, given that they incorporate elements of both debt and equity. Further complications arise with the presence of additional options such as callability and puttability, and contractual complexities such as trigger prices and soft call provisions, when the ability of the issuing firm to exercise its option to call depends on the history of its stock price. This article explores the valuation of convertible bonds subject to credit risk using an approach based on the numerical solution of linear complementarity problems. Models that do not explicitly specify what happens in the event of a default by the issuer can lead to internal inconsistencies, such as a call by the issuer just before expiration rendering the convertible value independent of the credit risk of the issuer, or implied hedging strategies that are not self-financing. A general and consistent framework for valuing convertible bonds assuming a Poisson default process allows various models for stock price behavior, recovery, and action by holders of the bonds in the event of a default. The numerical algorithm uses a partially implicit method to decouple the system of linear complementarity problems at each time step. Numerical examples illustrating the convergence properties of the algorithm are provided.

China's convertible bond market, Chapter 6 in China's Financial Markets: An Insider's Guide to How the Markets Work

- Q Liu

Liu, Q., 2007. China's convertible bond market, Chapter 6 in China's Financial Markets: An
Insider's Guide to How the Markets Work (Neftci, Salih. N. and Michelle Y. Menager-Xu eds).

Convertible Securities: The Latest Instruments, Portfolio Strategies, and Valuation Analysis

- J P Calamos

Calamos, J. P., 1988. Convertible Securities: The Latest Instruments, Portfolio Strategies, and
Valuation Analysis. McGraw-Hill, NY.