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Financial Numerical Recipes in C

Discussion of algorithms and C++ code for advanced financial calculations by Bernt Arne Ødegaard.

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/


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Financial Numerical Recipes (in C++)

Financial Numerical Recipes (in C++) In finance, there are areas where formulas tend to get involved. Sometimes it may be easier to follow an exact computer routine.

http://finance.bi.no/~bernt/gcc_prog/recipes/


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Valuation inputs for Norway - finance.bi.no

Valuation inputs for Norway Bernt Arne Ødegaard 31 March 2020 1 Consensus estimates To choose parameters relevant for valuations, there are different methods available.

http://finance.bi.no/~bernt/teach/corp_fin_2020/lectures/survey_risikopremien/lectures.pdf


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C++ in Finance

C++ programs for Finance I have made some programs for common uses in finance. They may be of interest to others, so I have made them public under the GNU Public Licence . C++ Financial Algoritms (Financial Numerical Recipes)

http://finance.bi.no/~bernt/gcc_prog/


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Setup. - BI finance

For any particular contingent claim, the terms of the claim will give a number of boundary conditions that determines the form of the pricing formula.. We will start by discussing the original example solved by Black, Scholes, Merton: European call and put options.

http://finance.bi.no/~bernt/gcc_prog/algoritms_v1/algoritms/node7.html


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European call and put options, The Black Scholes analysis.

European call and put options, The Black Scholes analysis. A call (put) option gives the holder the right, but not the obligation, to buy (sell) some underlying asset at a given price , called the exercise price, on or before some given date .. If the option is European, it can only be used (exercised) at the maturity date.

http://finance.bi.no/~bernt/gcc_prog/algoritms_v1/algoritms/node8.html


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Extending the Black Scholes formula - BI finance

where is the regular Black Scholes formula.. If the inequality is not fulfilled, one performs the calculation shown in formula 8.2 and implemented in code 8.3. Options on futures Black's model For an European option written on a futures contract, we use an adjustment of the Black Scholes solution, which was developed in Black (1976).Essentially we replace with in the Black Scholes formula, and ...

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node9.html


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Empirics of the Oslo Stock Exchange. Basic, descriptive ...

Empirics of the Oslo Stock Exchange. Basic, descriptive, results 1980-2019 Bernt Arne Ødegaard University of Staangerv (UiS) Mar 2020 We give some basic empirical characteristics of the Oslo Stock Exchange in the period after 1980.

http://finance.bi.no/~bernt/wps/empirics_ose_basics/empirics_ose_basics_2020_03.pdf


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Bernt Arne Ødegaard - Handelshøyskolen BI

Bernt Arne Ødegaard This is the homepage for Professor Bernt Arne Ødegaard, at the University of Stavanger, in Stavanger, Norway.

http://finance.bi.no/~bernt/


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44921 omsl - finance.bi.no

Title: 44921_omsl.indd Author: JarleS Created Date: 5/5/2008 8:37:12 AM

http://finance.bi.no/~bernt/wps/liquidity_ose/Norges_Bank_Working_Paper_2008_9.pdf


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Binomial approximation, dividends.

Binomial approximation, dividends. If the underlying asset is a stock paying dividends during the maturity of the option, the terms of the option is not adjusted to reflect this cash payment, which means that the option value will reflect the dividend payments.

http://finance.bi.no/~bernt/gcc_prog/algoritms_v1/algoritms/node16.html


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Asset pricing data at OSE - Handelshøyskolen BI

Asset pricing data at OSE This page contain some calculated data useful for academic asset pricing investigations at the Oslo Stock Exchange. Source of data

http://finance.bi.no/~bernt/financial_data/ose_asset_pricing_data/index.html


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Binomial option pricing - BI finance

Binomial option pricing Option and other derivative pricing is one of the prime ``success stories'' of modern finance. An option is a derivative security, the cash flows from the security is a function of the price of some other security, typically called the underlying security. A call option is a right, but not obligation, to buy a given quantity of the underlying security at a given price ...

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node6.html


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FinancialNumericalRecipesinC

1.2 ThestructureofaC++program ThefirstthingtorealizeaboutC ++ isthatitisastronglytypedlanguage. Everythingmustbedeclared beforeitisused,bothvariablesandfunctions. C ...

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes.pdf


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Basic Option Pricing, the Black Scholes formula.

This can be used in solving the Black Scholes pde above, since they define a boundary condition for the pde. Analytical option prices, Black Scholes case. The pde with the boundary condition was shown by Black and Scholes to have an analytical solution of functional form shown in formula 6.1.. The calculation is shown in code 6.1. Proving the Black Scholes formulas correctness

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node7.html


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Alternatives to the Black Scholes type option formula

In implementing this formula, we need to terminate the infinite sum at some point. But since the factorial function is growing at a much higher rate than any other, that is no problem, terminating at about should be on the conservative side. To avoid numerical difficulties, use the following method for calculation of

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node15.html


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Presentations

Presentations Bernt Arne Ødegaard 31 March 2020 Some hints on doing presentations (case presentations) Be prepared > Bring slides ourY internal case discussion will give you an overview of the case.

http://finance.bi.no/~bernt/teach/corp_fin_2020/lectures/writing/presentations.pdf


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The term structure of interest rates and an object lesson

The term structure of interest rates and an object lesson In this chapter we look at various algorithms that has been used to estimate a ``term structure,'' i e a relation between length of period for investment and interest rate.

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node4.html


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Option pricing by simulation - BI finance

where is a transformation of the original expectation. One way to estimate the value of the call is to simulate a large number of sample values of according to the assumed price process, and find the estimated call price as the average of the simulated values. By appealing to a law of large numbers, this average will converge to the actual call value, where the rate of convergence will depend ...

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node12.html


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Finite Differences - finance.bi.no

Finite Differences The method of choice for any engineer given a differential equation to solve is to numerically approximate it using a finite difference scheme, which is to approximate the continous differential equation with a discrete difference equation, and solve this difference equation.. In the following we implement the two schemes described in chapter 14.7 of Hull (1993), the ...

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node11.html


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Empirics of the Oslo Stock Exchange. Descriptive Basics

Empirics of the Oslo Stock Exchange. Basic, descriptive, results. 1980-2019 We give some basic empirical characteristics of the Oslo Stock Exchange in the period after 1980.

http://finance.bi.no/~bernt/wps/empirics_ose_basics/


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PROBLEM SET: Interest Exercise 1.

B) $725. C) $736. D) $1086. Exercise 6. [1] If an investment providing a nominal return of 12.25% only o ers a real rate of return of 5.70%, then the in ation

http://finance.bi.no/~bernt/teach/corp_fin_2020/lectures/interest_rates/pset_interest_rates_2_sol.pdf


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Option pricing with binomial approximations

Option pricing with binomial approximations Introduction We have shown binomial calculations given an up and down movement in chapter 5.However, binomial option pricing can also be viewed as an approximation to a continuous time distribution by judicious choice of the constants and .To do so one has to ask: Is it possible to find a parametrization (choice of and ) of a binomial process which ...

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node10.html


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The Capital Asset Pricing Model (CAPM)

The setup of CAPM In equilibrium all assets must be owned by somebody. m must be the market portfolio consisting of all risky assets in proportion to their relative market values. Consequence: The only relevant risk for a given asset i is how it contributes to the risk of the market portfolio m, which is the covariability between the return on i and the market portfolio m,

http://finance.bi.no/~bernt/teach/corp_fin_2020/lectures/capm/lecture_slides_capm_corporate_finance.pdf


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Stock Market Liquidity and the Business Cycle

Stock Market Liquidity and the Business Cycle RANDI N˛S, JOHANNES A. SKJELTORP, and BERNT ARNE ˜DEGAARD ABSTRACT In the recent nancial crisis we saw liquidity in the stock market drying up as a precursor

http://finance.bi.no/~bernt/publications/2010_jf_liquidity_business_cycle/liquidity_business_cycle_oct_2010.pdf


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A quadratic approximation to American prices due to Barone ...

The case we look at here is an option written on a commodity paying as a continuous payout. Here is the (unknown) formula that determines the price of the contingent claim. For an European option this can be explicitly solved, with the adjusted Black Scholes formula as the solution, but for American options, which may be exercised early, there is no known analytical solution.

http://finance.bi.no/~bernt/gcc_prog/algoritms_v1/algoritms/node24.html


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BI finance

The Corporate Governance Program was a research program funded by the Norwegian Research Council in the period 1997-2005.. The program director was Bernt Arne Ødegaard . Objectives ; Projects and Publications ; People ; Presentations ; Press Coverage ; Related Publications

http://finance.bi.no/~bernt/governance/index.html


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The Capital Asset Pricing Model (CAPM)

where β∗ is the firms asset beta. Exercise 2. A firm is considering investing in two investment projects, A and B. Project A has a low beta, and project B has a high beta.

http://finance.bi.no/~bernt/teach/corp_fin_2020/lectures/capm/lecture_capm_corporate_finance_as_problems.pdf


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Black Scholes bond pricing. - BI finance

Black Scholes bond pricing. The Black Scholes model can be used under restrictive assumptions, but the constant volatility assumption of the bond price is unrealistic.

http://finance.bi.no/~bernt/gcc_prog/algoritms_v1/algoritms/node56.html


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PROBLEM SET: Bond Pricing Exercise 1.

PROBLEM SET: Bond Pricing Exercise 1. Explain why the yield of a bond that trades at a discount exceeds the bond's coupon rate. Exercise 2. 6-11. Suppose that Ally Financial Inc. issued a bond with 10 years until maturit,y a face aluev of $1000, and

http://finance.bi.no/~bernt/teach/corp_fin_2020/lectures/bond_pricing/pset_bond_pricing_2.pdf


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Mathematical operators. - Handelshøyskolen BI

Mathematical operators. Exponentiation. The usual operators are what you expect, but exponentiation is not a part of the language, these and other mathematical operators are implemented as functions in the math library pow(x,n) calculates . exp(r) calculates . Increment and decrement. When we want to increase the value of one item by one, in most languages this is written:

http://finance.bi.no/~bernt/gcc_prog/algoritms_v1/algoritms/node65.html


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Mean variance portfolios. - BI finance

Short sales constraints Up: Mean Variance Analysis. Previous: Introduction. Contents Index Mean variance portfolios. In the case where there are no short sales constraints, the minimum variance portfolio for any given expected return has an analytical solution and is therefore easy to generate.

http://finance.bi.no/~bernt/gcc_prog/algoritms_v1/algoritms/node48.html


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Empirical data in C++

C++ Classes for empirical financial data I am using C++ for most of my econometric work, and have made a number of general utilities for that purpose, mainly for dealing with time series observations.

http://finance.bi.no/~bernt/gcc_prog/empirical/


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Investor Short-Termism and Firm Value

1 Introduction It is often claimed that investor short-termism causes firm managers to neglect profitable long-term investment. Fuller and Jensen (2002) argue that Wall Street is partly responsible for this

http://finance.bi.no/~bernt/wps/ownership_duration/durationpaper_aug_2008.pdf


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American Options. - BI finance

American Options. We now compare the American versions of the same algoritms, the only difference being the check for exercise at each point. Computer algoritm, implicit finite differences.

http://finance.bi.no/~bernt/gcc_prog/algoritms_v1/algoritms/node19.html


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University of Stavanger (UiS) Exam MØA 370 Cases in Finance

University of Stavanger (UiS) Exam MØA 370 Cases in Finance Exam Fall 2019 Exercise 1. Conceptual questions [10] 1. We will occasionally calculate a concept of net debt by subtracting a firms’ cash and financial assets from the firms debt (of various kinds).

http://finance.bi.no/~bernt/teach/moa370_2020/exams/exam_fall_2019.pdf


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The Impact of Collateral on Swap Rates - BI finance

cash is certainly not the only form of collateral, it is the most popular form of accepted collateral (ISDA (2000), p. 2).4 In a discrete-time setting, we show that collateraliza- tion and marking-to-market result in intermediate cash flows in the swap contract that

http://finance.bi.no/seminar/Sundaresan.pdf


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Term Structure Models - BI finance

Term Structure Models We now expand on the analysis of the term structure in chapter 3.As shown there, the term structure is best viewed as an abstract function of term to maturity, equally well described by the prices of zero coupon bonds (discount factors), yield on zero coupon bonds (spot rates) or forward rates.

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node20.html


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Using a library for matrix algebra - BI finance

Using a library for matrix algebra What really distinguishes C ++ from standard C is the ability to extend the language by creating classes and collecting these classes into libraries. A library is a collection of classes and routines for one particular purpose. We have already seen this idea when creating the date and term_structure classes. However, one should not necessarily always go ahead ...

http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node16.html


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Empirics on the OSE - Handelshøyskolen BI

Empirics on the OSE. Bernt Arne Ødegaard, University of Stavanger . Much of my research concerns empirical investigations of various aspects of the Oslo Stock Exchange.

http://finance.bi.no/~bernt/empirics_ose/index.html