Instruments where the Lattice Model is Used:
Complex securities and embedded derivatives such as convertible debt, profit interests, preferred warrants, purchase options, and securities with unique characteristics, as mentioned below.
Introduction:
A Binomial lattice model is an open-form model with more flexibility around certain features of an asset than the Black-Scholes model. It involves constructing a binomial tree with up and down paths at each node representing steps that consider the different paths the underlying asset could take during the term of the option. At each step, the model can consider different values in various parameters, such as a change in the exercise price over the life of the option, change in volatility or the ability for early exercise. However, a binomial has only two ways it can go at any node, up or down, which poses a limitation to this construct. Similar to Black-Scholes, the model does not handle certain situations well, such as incidences of path dependence.
A lattice-based model is used to value derivatives by employing a binomial tree to compute the various paths for the price of an underlying asset, such as stock in this example. The binomial option pricing model uses an iterative procedure, allowing for the specification of nodes, or points in time, during the time span between the valuation date and the convertible note’s maturity date. A binomial tree graphically plots out the possible values that stock prices can have over different periods.
Characteristics (Indicative list)
- Warrants convertible in the next round of financing where it has interdependent conversion features or with different payoffs at IPO or M&A, etc.
- Preferred Series liquidation rights based on achieving certain milestones of the Company.
- Convertible debt, with an option to get converted to an existing round or in a new financing round with interdependent conversion features.
- Profit interest with performance-based vesting criteria.
- Earn outs with payoffs based on the company’s future revenues/profits.
- Incentive stock options based on the company’s performance and achievement of certain milestones.
- Embedded derivative associated with the convertible notes.
The Company:
- A public company that develops and sells medical aesthetic products in the US and internationally (the “Company”) identified certain embedded derivatives related to the conversion feature of the convertible notes.
- Holders refer to certain holders of the embedded derivative.
- The valuation is for the embedded derivatives associated with its convertible notes on a quarterly basis under ASC 815.
Problem Statement:
- After reviewing the agreement, we noted the following embedded derivative features:
- Embedded Conversion features:
- Conversion at the option of the Holder;
- Successor Major Transaction1 conversion at the option of the Holder;
- The conversion price is $1 per share (1.0% exit fee payable in cash) plus the additional share coefficient times note principal/$2,000 (additional conversion shares).
- Company Share Major Transaction conversion at the option of the Holder;
- The conversion price is $1 per share plus 1.0% of conversion amount/closing price of common stock (base conversion shares) plus the additional share coefficient times note principal/$1,000 (additional conversion shares).
- Embedded Redemption features:
- Redemption upon Major Transaction at the option of the Holder; and
- Redemption upon an event of default at the option of the Holder.
Based on discussions with Management, we noted that events such as Major Transaction and Optional Redemption have de minimis probability; hence embedded derivatives associated with them may not be valued separately. As such, we valued only the optional conversion feature embedded with the Convertible Notes as discussed below.
Preferred Approach:
Without Scenario:
We determined the fair value of the Convertible Notes without the conversion option by discounting the interest payments at each node (time period) and principal repayment at maturity by the applicable yield mentioned below. This was estimated based on the Discounted Cash Flow Method. We have considered the following steps:
1. Interest Calculation:
- Interest for the Convertible Notes has been considered based on the agreement and would be paid off in cash at the end of each quarter. Thus, we have calculated the interest for each month and cumulated it for three months; and
2. Discount Rate (Bond yield):
- As the Company does not have a published debt rating, we estimated a synthetic credit rating for the Company by comparing certain financial ratios and metrics of the Company to those of other issuers which are comparable to the Company with publicly available credit ratings from Standard & Poor’s (S&P) or Moody’s. We compared the Company’s financial ratios, including leverage ratios (debt ratio, debt to equity ratio, and interest coverage ratio), the outcome of DuPont Analysis, etc., to those with the comparable companies’ metrics. Based on the above ratios, we have concluded that BBB is the applicable synthetic credit rating for the Company and utilized the applicable yield for the same rating.
Our Solution:
Key Assumptions – Binomial Lattice Model:
- We will have nodes for each period starting from the Valuation Date and ending at the maturity date.
- Stock Price: The starting stock price will be based on the Company’s closing stock price as of the Valuation Date. Then, we estimated the stock price at each node with an up and down factor, as noted below.
- Up and Down Factor: It is derived based on the volatility considered based on the comparable guideline public companies as noted below.
- Risk-Neutral Up and Down Probability: Based on the risk-free rate and the Up and the Down Factors as noted above.
- Volatility: We analyzed the historical volatility of the Company and guideline comparable companies, noting the relatively small trading volume of the Company’s common stock. We computed the volatility using daily price observations.
- The payoff at each node: At each node, we estimated the maximum payoff for the Holder by comparing the share price (derived above) multiplied by the number of conversion shares with the principal plus the accrued interest.
- Discount Rate: Considering the payoff, the discount rate would be either of the following:
- Risk-free rate: Based on the US Treasury securities, with a term similar to the remaining term of the Convertible Notes. This rate would be considered when the Convertible Notes would convert to the Common Shares; or
- Bond Yield: As stated above in the Discount Rate of the Without section above. This rate would be considered when the payoff would be principal plus accrued interest.
- Node value: Maximum of the payoff of the node or the probability-weighted present value of immediately adjacent nodes in the binomial lattice model.
- We performed backward induction to determine the value of the Convertible Notes as of the Valuation Date under the With scenario.
Challenge:
Besides the complexities associated with modeling the terms associated with the note that cannot be modeled using a standard Black Scholes construct, the auditors required a higher level of accuracy by considering a higher frequency, and hence, we considered 200 nodes/time intervals on a weekly basis.
Conclusion:
We built the lattice model based on 200 nodes as specified above, giving a more accurate result than a model with weekly frequency.
Snapshot of Results:
- See the snippets below for calculation of calculation pertaining to 200 nodes. For ease, we have shown the first few nodes and then the last 2 nodes for the underlying asset lattice (share price), convertible note lattice, and the lattice for intermediate calculations.
