An interest rate swap (IRS) is a financial derivative instrument that involves an exchange of a fixed interest rate for a floating interest rate. More specifically,
The above description refers to a plain vanilla IRS. However, interest rate swaps can come in many different flavors. These include, (but are not limited to) - Amortizing notional IRS
- Cross-currency swap
- Float-for-float (basis) swap
- Overnight index swap
- Inflation swap etc.
Interest rate swaps are often used to hedge the fluctuation in the interest rate. To value an IRS, fixed and floating legs are priced separately using the discounted cash flow approach. Below is an example of a hypothetical plain vanilla IRS
The values of the fixed, floating legs and the IRS are calculated using an Excel spreadsheet. Table below presents their values Click on the link below to download the Excel spreadsheet. Article Source Here: Interest Rate Swap-Derivative Pricing in Excel
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In 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,
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. Post Source Here: Valuing an American Option Using Binomial Tree-Derivative Pricing in Excel R. 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
is the asset volatility,
We note that both asset ( where denotes the volatility of equity. These 2 equations can be solved simultaneously in order to obtain Having 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
[1] Merton, R. C. 1974, Article Source Here: Credit Risk Management Using Merton Model In 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.
*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 more*
An 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 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].
[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:
In 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.
The stock price is also obtained from Yahoo finance. It is 13.5 as of the valuation date (Aug 22 2018).
The dividend yield is obtained from Yahoo finance. It is 1.2%. Note that for illustration purposes we use continuous instead of discrete dividend.
The 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.
Barone-Adesi, G. and Whaley, R.E. (1987 Originally Published Here: Valuing an American Option-Derivative Pricing in Excel An option is a financial contract that gives you a right, but not an obligation to buy or sell an underlying at a future time and at a pre-determined price. Specifically,
Excellent textbooks and papers have been written on options pricing theory; see for example Reference [1]. In this post we are going to deal with practical aspects of pricing a European option. We do so through a concrete example. We’re going to price a put option on Barrick Gold, a Canadian mining company publicly traded on the Toronto Stock Exchange under the symbol ABX.TO. For this exercise, we assume that the option is of European style with a strike price of $13. (American style option will be dealt with in the next installment). The option expires in 3 years, and the valuation date is August 22, 2018. [caption id="attachment_484" align="aligncenter" width="540"] Barrick Gold mining financial data as at Aug 23 2018[/caption] The important input parameters are:
In 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.
The stock price is also obtained from Yahoo finance. It is 13.5 as at the valuation date.
The dividend yield is obtained from Yahoo finance. It is 1.2%. Note that for illustration purposes we use continuous instead of discrete dividend.
The 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. After obtaining all the required input data, we use QuantlibXL to calculate the price of the option. The calculator returns a price of $3.21. The picture below presents a summary of the valuation inputs and results. [caption id="attachment_482" align="aligncenter" width="306"] European option valuation in Excel[/caption] In the next installment, we’re going to present an example for American option. Follow the link below to download the Excel Workbook.
[1] Hull, John C. (2005),
Originally Published Here: VALUING A EUROPEAN OPTION This post is a follow-up to the previous one on a simple system for hedging long exposure during a market downturn. It was inspired by H. Krishnan’s book
Basically, the paper says that the equity indices exhibit fatter tails in shorter time frames, from 1 to 4 days. We apply this idea to our breakout system. We’d like to see whether the 4-day rule manifests itself in this simple strategy. To do so, we use the same entry rule as before, but with a different exit rule. The entry and exit rules are as follows,
The system was backtested on SPY from 1993 to the present. Graph below shows the average trade PnL as a function of number of days in the trade, [caption id="attachment_350" align="aligncenter" width="485"] Average trade PnL vs. days in trade[/caption] We observe that if we exit this trade within 4 days of entry, the average loss (i.e. the cost of hedging) is in the range of -0.2% to -0.4%, i.e. an average of -0.29% per trade. From day 5, the loss becomes much larger (more than double), in the range of -0.6% to -0.85%. The smaller average loss incurred during the first 4 days might be a result of the fat-tail behaviour. This test shows that there is some evidence that the scaling behaviour demonstrated in Ref [1] still holds true today, and it manifested itself in this system. More rigorous research should be conducted to confirm this.
[1] Gopikrishnan P, Plerou V, Nunes Amaral LA, Meyer M, Stanley HE, Read Full Article Here: A Simple Hedging System with Time Exit Insulated by cheap money from the QE era and bolstered by cash on their balance sheets, it remains rare for companies in Europe and the U.S. to miss debt payments. Among higher-risk speculative-grade firms the default rate fell to 2.9 percent last quarter, and may drop further to 2.1 percent by year-end, according to Moody’s Investors Service. And only one investment-grade firm has defaulted since 2012, data from Standard & Poor’s Global Ratings show.“Default rates are on the floor,” said Fraser Lundie, co-head of credit at Hermes Investment Management. “Fundamentals still broadly stack up.” Read moreHowever, note that the default rate they talked about is historical default rate. It does not predict future defaults. In fact, historical default rate to future probability of default is what historical volatility to implied volatility. Just because the recent historical volatility is low it does not mean that the volatility risk is low. This applies to the credit market too. But default rates aren’t the only thing credit investors care about. Spreads have widened to levels not seen for more than a year as concerns grow of overheating in the U.S. market, trade disputes, rising rates, inflation and the end of the European Central Bank’s bond-buying program.… The credit market may also be downplaying the potential impact of tariffs, analysts at UBS Group AG wrote in a July 24 report. They say investors should be cautious about sectors including tech, industrials, metals and mining. Higher corporate leverage may also lead to an increase in stress among non-cyclical industries such as consumer staples and healthcare, the analysts including Bhanu Baweja wrote.…The end of loose monetary policies may also boost defaults in emerging markets next year, according to Abdul Kadir Hussain, the head of fixed income at Arqaam Capital, a Dubai-based investment bank.ByMarketNews Published via http://harbourfronttechnologies.blogspot.com/ Last year, in a post entitled Credit Derivatives-Is This Time Different we wrote about credit derivatives and their potential impact on the markets. Since then, they have started attracting more and more attention. For example, Bloomberg recently reported that collateralized loan obligations (CLO), a type of complex credit derivatives, are becoming a favorite financing vehicle for corporate America.
As reported by the Washington Post, money raised from these CLOs is used to finance corporate stock buybacks and dividend payouts.
In the current market environment, it’s difficult to evaluate the riskiness of these CLOs. First of all, Value at Risk (VaR), a popular risk measure used by many financial institutions to quantify the risks and manage economic capital, has been developed and tested in a low-interest rate and low-volatility environment. This makes the VaR vulnerable to future change in the market environment. Second, in the calculation of VaR for a credit derivative portfolio, we would have to determine the probabilities of default (PD) and loss given default (LGD) of the borrowers. Both of these quantities are difficult to estimate. Furthermore, the correlation between PD and LGD is not constant and will likely increase during a market stress. All of these factors make the VaR less accurate. Consequently, managing the risks of a CLO portfolio is a non-trivial task. A slight change in the market environment can lead to damaging consequences. Originally Published Here: Are Collateralized Loan Obligations the New Debt Bombs? Goldman analysts Rocky Fishman and John Marshall said that the VIX, which uses options bets on the S&P 500 to reflect expected volatility over the coming 30 days, has been hovering at or below 13, marking its lowest level since around January (though it is tipping up in Monday trade). Its current level takes the gauge of implied volatility, which tends to rise when stocks fall and vice versa, well below its historic average at about 19.5 since the fear index ripped higher in February.Goldman argues that the 5-day intraday swings of the S&P 500 have been out of whack with the price of the cost of a one-month straddle on the index. A straddle is an options bet that allows an investor to profit from a sharp move in an asset, but without wagering on the specific direction of that expected move. In other words, it is an inherent bet on volatility. A straddle can be structured by buying a put option, which confers the owner the right but not the obligation to sell an asset at a given time and price, and a call option, which offers the comparable right to buy an underlying asset, at the same expiration date and strike price. Read more
But is the volatility index predictable? How about VIX futures and ETFs?A recent research article raised some interesting questions, The VIX index is not traded on the spot market. Hence, in contrast to other futures markets, the VIX futures contract and spot index are not linked by a no-arbitrage condition. We examine (a) whether predictability in the VIX index carries over to the futures market, and (b) whether there is independent time series predictability in VIX futures prices.The answer is no. The answer to both questions is no. Samuelson (1965) was right: VIX futures prices properly anticipate predictability in volatility, and are themselves unpredictable. Read moreBut then why do we trade VIX futures and ETFs?We think that the reasons might be: - Trading the spot VIX is difficult,
- When trading VIX futures and ETFs, we exchange the predictability of the spot index for a little extra return stemming from the volatility risk premium.
ByMarketNews Published via http://harbourfronttechnologies.blogspot.com/ The overnight index swap (OIS) has come into the spotlight recently, due to the widening of the Libor-OIS spread. For example, the Economist recently reported:
[caption id="attachment_459" align="aligncenter" width="628"] Libor-OIS spread as at May 2, 2018. Source: Bloomberg[/caption]
An overnight index swap is a fixed/floating interest rate swap that involves the exchange of the overnight rate compounded over a specified term and a fixed rate. The floating leg of the swap is related to an index of an overnight reference rate, for example Canadian Overnight Repo Rate Average (CORRA) in Canada or Fed Funds rate in the US. Usually, for swaps with maturities of 1 year or less there is only one payment. Beyond the tenor of 1 year, there are multiple payments at regular intervals. At the inception of the swap, the par swap rate makes the value of swap zero. That is, the net present value (NPV) of the fixed leg equals the NPV of the floating leg, where
t_{i}is the daily accrual factor, and
t._{K}The OIS discount factors (DF) are often used to value interest rate derivatives that require a posting of collateral. The OIS discount factor curve is built by bootstrapping from the short maturity and long maturity overnight index swap rates in order of increasing maturity. The processes for backing out the discount factors from the short and long maturity swap rates are, however, different. In the short end of the curve, given that there is only 1 payment, the discount factor is calculated based on the spot rates. At the long end of the curve, the DF curve is determined as follows, - Payment dates are generated at each 6 months (or a year, depending on the currency) from the time zero up to 30 years,
- Par swap rates are determined at each payment date. To obtain the par swap rates for the payment dates where there are no swap quotes, one linearly interpolates the par swap rates in order to complete the long end of the swap curve,
- Using the par swap rates at each payment date, discount factors are obtained by solving a recursive equation.
This is just an introduction to OIS discounting. The process for building an OIS discount curve involves many technical details. We are happy to answer your questions. Article Source Here: Overnight Index Swap Discounting |