- Valuing an American Option Using Binomial Tree-Derivative Pricing in ExcelIn a previous post, we provided an example of pricing American options using an analytical approximation. Such a pricing model is fast and accurate enough for risk management purposes. However, sometimes more accurate results are required. For this purpose, the binomial (lattice) model can be used. Wikipedia describes the binomial tree model as follows,In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. The binomial model was first proposed by Cox, Ross and Rubinstein in 1979. Essentially, the model uses a "discrete-time" (lattice based) model of the varying price over time of the underlying financial instrument...The binomial pricing model traces the evolution of the option's key underlying variables in discrete-time. This is done by means of a binomial lattice (tree), for a number of time steps between the valuation and expiration dates. Each node in the lattice represents a possible price of the underlying at a given point in time.Valuation is performed iteratively, starting at each of the final nodes (those that may be reached at the time of expiration), and then working backwards through the tree towards the first node (valuation date). The value computed at each stage is the value of the option at that point in time.We utilized the lattice model previously to price convertible bonds. In this post, we’re going to use it to value an American equity option. We use the same input parameters as in the previous example. Using our Excel workbook, we obtain a price of $3.30, which is smaller than the price determined by the analytical approximation (Barone-Andesi-Whaley) approach.[caption id="attachment_561" align="aligncenter" width="335"] American option valuation in Excel using Binomial Tree[/caption]Click on the link below to download the Excel Workbook.Originally Published Here: Valuing an American Option Using Binomial Tree-Derivative Pricing in Excel
- Credit Risk Management Using Merton ModelR. Merton published a seminal paper [1] that laid the foundation for the development of structural credit risk models. In this post, we’re going to provide an example of how it can be used for managing credit risks.Within the Merton model, equity of a firm is considered a call option on its asset, and it is expressed as follows,where E denotes the equity of the firm, V is the firm’s asset, is the asset volatility, B is the notional amount of the debt, r is the risk-free interest rate, andWe note that both asset (V) and its volatility are not observable. However, the asset volatility can be related to equity and its volatility through the following equation,where denotes the volatility of equity.These 2 equations can be solved simultaneously in order to obtain V and its volatility which are then used to determine the credit spreadHaving the credit spread, we will be able to calculate the probability of default (PD). Loss given default (LGD) can also be derived under Merton framework.Graph below shows the term structures of credit spread under various scenarios for the leverage ratio (B/V).[caption id="attachment_541" align="aligncenter" width="564"] Term structure of credit spread[/caption]It’s worth mentioning that the Merton model usually underestimates credit spreads. This is due to several factors such as the volatility risk premium, firm’s idiosyncratic risks and the assumptions embedded in the Merton model. This phenomenon is called the credit spread puzzle. Research is being conducted actively in order to improve the model.References[1] Merton, R. C. 1974, On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of Finance, Vol. 29, pp. 449–470. Originally Published Here: Credit Risk Management Using Merton Model
- Valuing an American Option-Derivative Pricing in ExcelIn the previous installment, we presented a concrete example of pricing a European option. In this follow-up post we are going to provide an example of valuing American options.The key difference between American and European options relates to when the options can be exercised:A European option may be exercised only at the expiration date of the option, i.e. at a single pre-defined point in time.An American option on the other hand may be exercised at any time before the expiration date. Read moreAn exact analytical solution exists for European options. For American options, however, we have to use numerical methods such as Binomial Tree (i.e. Lattice) or approximations. The post entitled How to Price a Convertible Bond provides an example of the Binomial Tree approach.The Binomial Tree model is an accurate one. However, its main drawback is that it’s slow. Consequently, several researchers have developed approximate solutions that are faster. In this example we’re going to use the Barone-Andesi-Whaley approximation [1].The Barone-Adesi and Whaley Model has the advantages of being fast, accurate and inexpensive to use. It is most accurate for options that will expire in less than one year.The Black-Scholes Model is appropriate for European options, that is, options that may be exercised only on the expiration date. The Barone-Adesi and Whaley Model is designed for American options, which are options that may be exercised at any time before they expire. The Barone-Adesi and Whaley Model takes the value computed by the Black-Scholes Model and adds the value of the early exercise option that is available on American option.. Read more[caption id="attachment_518" align="aligncenter" width="621"] Government of Canada Benchmark Bond Yield. Source: Bank of Canada[/caption]Recall that the important inputs are:VolatilityIn this example we are going to use historical volatility. We retrieve the historical stock data from Yahoo finance. We then proceed to calculate the daily returns and use them to determine the annual volatility. The resulting volatility is 43%. Detailed calculation is provided in the accompanying Excel workbook.Stock priceThe stock price is also obtained from Yahoo finance. It is 13.5 as of the valuation date (Aug 22 2018).DividendThe dividend yield is obtained from Yahoo finance. It is 1.2%. Note that for illustration purposes we use continuous instead of discrete dividend.Interest rateThe risk-free interest rate is retrieved from Bank of Canada website. Since the tenor of the option is 3 years, we’re going to use the 3-year benchmark yield. It is 2.13% as at the valuation date.We use the Excel calculator again and obtain a price of $3.32 for the American put option.[caption id="attachment_519" align="aligncenter" width="336"] American option valuation in Excel[/caption]Click on the link below to download the Excel Workbook. ReferencesBarone-Adesi, G. and Whaley, R.E. (1987) Efficient Analytic Approximation of American Option Values The Journal of Finance, 42, 301-320.Article Source Here: Valuing an American Option-Derivative Pricing in Excel
- A Credit Crunch Looms For The U.S. MarketsFrom Palisade Research: Another week and another signal flashing red to deal with. The credit market – in my opinion – is indicating an inevitable ‘crunch’ coming up. And even worse – we’re seeing the global dollar shortage deepening. Many readers know… Read more ›
- Banks Are Creating A 2007-Like Topping PatternTechnical analyst Chris Kimble looks at a long-term chart of an important bank stock index and sees potential trouble ahead. The left chart above looks at the Bank Index (BKX) over the past 13-years. In 2007, the index diverged with the broad market… Read more ›
- What We Learned From The Latest IMF MeetingFrom Invesco: The International Monetary Fund (IMF) took a decidedly bearish tone during its annual meeting in Bali earlier this month — in fact, I would say it was the grimmest gathering of the IMF that I’ve ever seen. I… Read more ›
- Taiwan Stocks Are Looking Good AgainFrom Dana Lyons: Taiwan’s stock market is hitting interesting long-term levels. The global equity correction has unleashed some not insignificant damage on a lot of international markets. It also has (potentially) produced opportunities. Last year was a good year for… Read more ›
- Big Dividends For Uncertain MarketsFrom Contrarian Outlook: Dividends or growth? Why choose? There’s a widespread belief that stocks and funds can deliver red-hot capital gains or substantial income, but not both. Fortunately for us that’s not true. It is possible to collect big dividends and capital appreciation. I’m… Read more ›
- Expect Noisy Action For The S&P 500 This FallFrom Larry McMillan: After two horrendous days on October 10th and 11th, the market experienced a solid oversold bounce. Some buy signals were even generated from that bounce, but now it seems to be failing again without having fulfilled even… Read more ›
- What BDCs To Buy And AvoidFrom Contrarian Outlook: Do you want to generate income that increases along with interest rates, with the potential upside from private equity investments? A Business Development Company (BDC), a type of closed-end investment company, could be the answer you’re looking for. BDCs… Read more ›
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