Is Asset Dynamics Priced In Correctly by Black-Scholes-Merton Model?

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,

volatility trading strategies mean reverting asset

where β represents the impact of past shocks and h is a small constant. We note that and in case β=0 the price dynamics becomes GBM.

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,

volatility trading strategies trending asset

From the above equation, it can be seen that

  • When the underlying asset is mean reverting, i.e. β<0, which is often the case for equity indices, the MA(1) volatility becomes smaller. Therefore if we use BSM with σ as input for volatility, it will overestimate the option price.
  • Conversely, when the asset is trending, i.e. β>0, BSM underestimates the option price.
  • Time to maturity, τ, also affects the degree of over- underpricing. Longer-dated options will be affected more by the autocorrelation factor.


[1] Liao, S.L. and Chen, C.C. (2006), Journal of Futures Markets, 26, 85-102.

Originally Published Here: Is Asset Dynamics Priced In Correctly by Black-Scholes-Merton Model?

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Valuing a Fixed Rate Bond-Derivative Pricing in Python

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.

US swap curve as at Oct 31 2018

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).

derivative pricing in python

Click on the link below to download the python code.



[1]  Quantlib Python Cookbook, Balaraman and Ballabio, Leanpub, 2017

Post Source Here: Valuing a Fixed Rate Bond-Derivative Pricing in Python

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Merton Credit Risk Model, a Case Study

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.

Merton Credit Risk Model

Junior Gold Miners ETF as at Nov 28 2018. Source:

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.

Article Source Here: Merton Credit Risk Model, a Case Study

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Interest Rate Swap-Derivative Pricing in Excel

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)

  • 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

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

Interest rate swap derivative in Excel

Click on the link below to download the Excel spreadsheet.

Originally Published Here: Interest Rate Swap-Derivative Pricing in Excel

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Valuing an American Option Using Binomial Tree-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.

Derivative Pricing in Excel Binomial Tree Lattice

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.

Derivative Pricing in Excel Binomial Tree Workbook

American option valuation in Excel using Binomial Tree

Click on the link below to download the Excel Workbook.

Originally Published Here: Valuing an American Option Using Binomial Tree-Derivative Pricing in Excel

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Credit Risk Management Using Merton Model

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,

credit risk management structural model

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, and

We 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,

credit risk management structural modelwhere 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 spread

credit risk management structural model

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).

credit risk management structural model

Term structure of credit spread

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.


[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

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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

Financial derivative pricing interest rate derivative

Government of Canada Benchmark Bond Yield. Source: Bank of Canada

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.

Stock price

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.

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.

Derivative Pricing in Excel - Valuing an American Option

American option valuation in Excel

Click on the link below to download the Excel Workbook.



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

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