Derivative Valuation, Risk Management, Volatility Trading
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A lot of research has been devoted to answering the question: do options price in the volatility risks correctly? The most noteworthy phenomenon (or bias) is called the volatility risk premium, i.e. options implied volatilities tend to overestimate future realized volatilities. Much less attention is paid, however, to the underlying asset dynamics, i.e. to answering the question: do options price in the asset dynamics correctly? Note that within the usual BSM framework, the underlying asset is assumed to follow a GBM process. So to answer the above question, it’d be useful to use a different process to model the asset price. We found an interesting article on this subject [1]. Instead of using GBM, the authors used a process where the asset returns are auto-correlated and then developed a closed-form formula to price the options. Specifically, they assumed that the underlying asset follows an MA(1) process,
where β represents the impact of past shocks and h is a small constant. We note that After applying some standard pricing techniques, a closed-form option pricing formula is derived which is similar to BSM except that the variance (and volatility) contains the autocorrelation coefficient,
From the above equation, it can be seen that
References [1] Liao, S.L. and Chen, C.C. (2006), Journal of Futures Markets, 26, 85-102. Original Post Here: Is Asset Dynamics Priced In Correctly by Black-Scholes-Merton Model?
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Debt instruments are an important part of the capital market. In this post, we are going to provide an example of pricing a fixed-rate bond. A fixed rate bond is a long term debt paper that carries a predetermined interest rate. The interest rate is known as coupon rate and interest is payable at specified dates before bond maturity. Due to the fixed coupon, the market value of a fixed-rate bond is susceptible to fluctuations in interest rates, and therefore has a significant amount of interest rate risk. That being said, the fixed-rate bond, although a conservative investment, is highly susceptible to a loss in value due to inflation. The fixed-rate bond’s long maturity schedule and predetermined coupon rate offers an investor a solidified return, while leaving the individual exposed to a rise in the consumer price index and overall decrease in their purchasing power. The coupon rate attached to the fixed-rate bond is payable at specified dates before the bond reaches maturity; the coupon rate and the fixed-payments are delivered periodically to the investor at a percentage rate of the bond’s face value. Due to a fixed-rate bond’s lengthy maturity date, these payments are typically small and as stated before are not tied into interest rates. Read more We are going to price a hypothetical bond as at October 31, 2018. We first build a zero coupon curve. Picture below shows the market rates as at the valuation date. [caption id="attachment_595" align="aligncenter" width="628"] We utilize the deposit rates (leftmost column) to construct the zero curve up to 12-month maturity. We then use this zero curve to price the following hypothetical fixed rate bond: Currency: USD Maturity: 1 year Payment frequency: semi-annual Coupon rate: 3% We use Python [1] to build a bond pricer. Picture below shows the result returned by the Python program. The price is $99.94 (per $100 notional).
Click on the link below to download the python code.
References [1] Quantlib Python Cookbook, Balaraman and Ballabio, Leanpub, 2017 Post Source Here: Valuing a Fixed Rate Bond-Derivative Pricing in Python In a previous post entitled Credit Risk Management Using Merton Model we provided a brief theoretical description of the Merton structural credit risk model. Note that, The Merton model is an analysis model – named after economist Robert C. Merton – used to assess the credit risk of a company's debt. Analysts and investors utilize the Merton model to understand how capable a company is at meeting financial obligations, servicing its debt, and weighing the general possibility that it will go into credit default…Loan officers and stocks analysts utilize the Merton model to analyze a corporation's risk of credit default. This model allows for easier valuation of the company and also helps analysts determine if the company will be able to retain solvency by analyzing maturity dates and debt totals. Read more In this installment, we are going to present a case study based upon the Merton credit risk model. [caption id="attachment_588" align="aligncenter" width="628"] Our Client is a junior gold miner. They are looking to raise additional capital in order to finance the production of gold. The client currently has an outstanding liability in the form of future discount for the payment of deliverable gold. According to a contingency clause, the Buyer can exercise a certain amount of cash into a secured obligation. This conversion will increase the leverage of the Company, thus leading to higher credit risks. We determined the increase in the credit risks by using the Merton structural credit model. We estimated that if the Buyer exercises the cash conversion clause, the credit spread of the Client will increase by approximately 20%. We thus helped the Client better negotiate the deal with their counterparty. Originally Published Here: Merton Credit Risk Model, a Case Study 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, An interest rate swap's (IRS's) effective description is a derivative contract, agreed between two counterparties, which specifies the nature of an exchange of payments benchmarked against an interest rate index. The most common IRS is a fixed for floating swap, whereby one party will make payments to the other based on an initially agreed fixed rate of interest, to receive back payments based on a floating interest rate index. Each of these series of payments is termed a 'leg', so a typical IRS has both a fixed and a floating leg. The floating index is commonly an interbank offered rate (IBOR) of specific tenor in the appropriate currency of the IRS, for example LIBOR in USD, GBP, EURIBOR in EUR or STIBOR in SEK. To completely determine any IRS a number of parameters must be specified for each leg; the notional principal amount (or varying notional schedule), the start and end dates and date scheduling, the fixed rate, the chosen floating interest rate index tenor, and day count conventions for interest calculations. Read more 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)
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 Maturity: 5 years Notional: 10 Million EUR Fixed rate: 3.5% Floating rate: Euribor 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 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, 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"] 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 E denotes the equity of the firm, V is the firm’s asset, B is the notional amount of the debt, r is the risk-free interest rate, and
We note that both asset (V) and its volatility
These 2 equations can be solved simultaneously in order to obtain V and its volatility
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"] 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. 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. The key difference between American and European options relates to when the options can be exercised:
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 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"] Recall that the important inputs are: Volatility 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. Stock price The stock price is also obtained from Yahoo finance. It is 13.5 as of the valuation date (Aug 22 2018). Dividend The dividend yield is obtained from Yahoo finance. It is 1.2%. Note that for illustration purposes we use continuous instead of discrete dividend. Interest rate 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"] Click on the link below to download the Excel Workbook.
References Barone-Adesi, G. and Whaley, R.E. (1987) Efficient Analytic Approximation of American Option Values The Journal of Finance, 42, 301-320. 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, ... an option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price (market price) of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium. The seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer (owner) "exercises" the option. An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put. Read more 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"] The important input parameters are: Volatility 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. Stock price The stock price is also obtained from Yahoo finance. It is 13.5 as at the valuation date. Dividend The dividend yield is obtained from Yahoo finance. It is 1.2%. Note that for illustration purposes we use continuous instead of discrete dividend. Interest rate 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"] In the next installment, we’re going to present an example for American option. Follow the link below to download the Excel Workbook.
References [1] Hull, John C. (2005), Options, Futures and Other Derivatives (6th Ed.), Prentice-Hall
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 The Second Leg Down, in which he referred to an interesting research paper [1] on the power-law behaviour of the equity indices. The paper states, We find that the distributions for ∆t ≤4 days (1560 mins) are consistent with a power-law asymptotic behavior, characterized by an exponent α≈ 3, well outside the stable Levy regime 0 < α <2. .. For time scales longer than ∆t ≈4 days, our results are consistent with slow convergence to Gaussian behavior. 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, Short at the close when Close of today < lowest Close of the last 10 days Cover at the close T days after entry (T=1,2,... 10) 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"] 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. References [1] Gopikrishnan P, Plerou V, Nunes Amaral LA, Meyer M, Stanley HE, Scaling of the distribution of fluctuations of financial market indices, Phys Rev E, 60, 5305 (1999). Read Full Article Here: A Simple Hedging System with Time Exit
Yesterday, Bloomberg published an article arguing that the current credit risk is low because the default rate is low,
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 more However, 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/ |
and in case β=0 the price dynamics becomes GBM.
US swap curve as at Oct 31 2018[/caption]
Junior Gold Miners ETF as at Nov 28 2018. Source: stockcharts.com[/caption]
American option valuation in Excel using Binomial Tree[/caption]
is the asset 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.
which are then used to determine the credit spread
Term structure of credit spread[/caption]
Government of Canada Benchmark Bond Yield. Source: Bank of Canada[/caption]
American option valuation in Excel[/caption]
Barrick Gold mining financial data as at Aug 23 2018[/caption]
European option valuation in Excel[/caption]
Average trade PnL vs. days in trade[/caption]
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