Valuing an American Option-Derivative Pricing in Excel

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:

• 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 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:

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"] American option valuation in Excel[/caption]

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.

Article Source Here: Valuing an American Option-Derivative Pricing in Excel

VALUING A EUROPEAN OPTION

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"] Barrick Gold mining financial data as at Aug 23 2018[/caption]

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"] 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.

 

References

[1] Hull, John C. (2005), Options, Futures and Other Derivatives (6th Ed.), Prentice-Hall

 

Article Source Here: VALUING A EUROPEAN OPTION

A Simple Hedging System with Time Exit

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"] 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.

 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

Historical Default Rates Do Not Predict Future Defaults

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

Are Collateralized Loan Obligations the New Debt Bombs?

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.

...Investors haven’t been able to get enough of the repackaged corporate loans known as collateralized loan obligations. That intense demand, is allowing the managers that put these securities together to sell off pieces of CLOs that by law they previously had to hang on to. These sales are the crest of what could be a $7 billion wave of such deals. Read more

As reported by the Washington Post, money raised from these CLOs is used to finance corporate stock buybacks and dividend payouts.

The most significant and troubling aspect of this buyback boom, however, is that despite record corporate profits and cash flow, at least a third of the shares are being repurchased with borrowed money, bringing the corporate debt to an all-time high, not only in an absolute sense but also in relation to profits, assets and the overall size of the economy.

In recent years, moreover, a greater part of corporate borrowing has come in the form of bank loans that are quickly packaged into securities known as CLOs, or collateralized loan obligations, which are sliced and diced and sold off to sophisticated investors just as home loans were during the mortgage bubble. Bloomberg News recently reported that pension funds and insurance companies, particularly those in Japan, can’t get enough of the CLOs because of the higher yields that they offer. Wells Fargo estimates that a record $150 billion will be issued this year, roughly double last year’s issuance. And as happened with the late-cycle home mortgages in 2007 and 2008, analysts are noticing a marked decline in the quality of loans in the CLO packages, with three-quarters of them now without the standard covenants designed to reduce the chance of default. Read more

But how risky are these collateralized loan obligations?

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.

Post Source Here: Are Collateralized Loan Obligations the New Debt Bombs?

Do Properly Anticipated Prices Fluctuate Randomly? Evidence from VIX Futures Markets

According to Marketwatch, Goldman Sachs strategists just issued a warning regarding the volatility index, VIX,

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

Volatility Index VIX as of May 18 2018

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 more

But 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

Overnight Index Swap Discounting

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:

WATCHING financial markets can be like watching a horror film. A character walks into the darkness alone. A floorboard creaks. The latest spooky sign is the spread between the three-month dollar London interbank offered rate (LIBOR) and the overnight index swap (OIS) rate. It usually hovers at around 0.1%, but has recently climbed to 0.6% (see chart). As it widens, bankers are bracing for a jump scare.

To see why, consider what each rate represents. LIBOR is the rate that banks charge other banks for unsecured loans. The OIS rate measures expectations for the federal funds rate, which is set by the central bank. As LIBOR rises above the OIS rate, that suggests banks fear it is getting riskier to lend to each other. (The gap was 3.65 percentage points in the depths of the crisis, after Lehman Brothers filed for bankruptcy.) Read more

[caption id="attachment_459" align="aligncenter" width="628"] Libor-OIS spread as at May 2, 2018. Source: Bloomberg[/caption]

What exactly is an overnight index swap?

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 N denotes the notional amount of the swap,

R_i-1,i is the forward OIS rate,

Z_i is the discount factor at time t_i

is the daily accrual factor, and

 s_K is the par swap rate of a swap with maturity 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.

Post Source Here: Overnight Index Swap Discounting

Can a Horse Racing System be Applied to the Stock Markets?

Bill Benter is one of the most profitable professional gamblers in the world. According to Wikipedia

William Benter was born and raised in Pittsburgh, Pennsylvania.[2] As he grew up, he wanted to use his mathematical talents to make a profit so immediately after finishing a university physics degree in 1977,[3] he went to the blackjack tables in Las Vegas and used his skills to count cards. He came across the book, Beat the Dealer, by Edward O. Thorp, which helped him improve his methods.[4] Seven years later, he was banned from most of Vegas’ strip’s casinos.[2]

Benter then met with Alan Woods, a like-minded gambler whose expertise in horse racing complemented his own in computers. The two became racing partners and in 1984, moved to Hong Kong.[3] Starting with a mere US$150,000 (equivalent to US$353,331 in 2017), the pair relied on their mathematical skill to create a formula for choosing race winners.[2]

Using his statistical model, Benter identified factors that could lead to successful race predictions. He found that some came out as more important than others.[5] Benter later worked with Robert Moore.

Benter is a visiting professor at the Southampton Management School[6] as part of the Centre for Risk Research and a fellow of the Royal Statistical Society.[7]

Bloomberg recently published an interesting story about his career, 

Benter grew up in a Pittsburgh idyll called Pleasant Hills. He was a diligent student and an Eagle Scout, and he began to study physics in college. His parents had always given him freedom—on vacations, he’d hitchhiked across Europe to Egypt and driven through Russia—and in 1979, at age 22, he put their faith to the test. He left school, boarded a Greyhound bus, and went to play cards in Las Vegas.

Benter had been enraptured by Beat the Dealer, a 1962 book by math professor Edward Thorp that describes how to overcome the house’s advantage in blackjack. Thorp is credited with inventing the system known as card counting: Keep track of the number of high cards dealt, then bet big when it’s likely that high cards are about to fall. It takes concentration, and lots of hands, to turn a tiny advantage into a profit, but it works.

Thorp’s book was a beacon for shy young men with a gift for mathematics and a yearning for a more interesting life. When Benter got to Las Vegas, he worked at a 7-Eleven for $3 an hour and took his wages to budget casinos. The Western—with its dollar cocktails and shabby patrons getting drunk at 10 a.m.—and the faded El Cortez were his turf. He didn’t mind the scruff. It thrilled him to see scientific principles play out in real life, and he liked the hedonistic city’s eccentric characters. It was the era of peak disco, with Donna Summer and Chic’s Le Freak all over the radio. On a good day, Benter might win only about $40, but he’d found his métier—and some new friends. Fellow Thorp acolytes were easy to spot on casino floors, tending to be conspicuously focused and sober. Like them, Benter was a complete nerd. He had a small beard, wore tweedy jackets, and talked a lot about probability theory. Read more

But can a winning horse racing system be applied to the stock markets? Benter himself provided an answer in this video

ByMarketNews