A Derivation of the Black–Scholes Pricing Equations for Vanilla Options

Total Page:16

File Type:pdf, Size:1020Kb

A Derivation of the Black–Scholes Pricing Equations for Vanilla Options A Derivation of the Black–Scholes Pricing Equations for Vanilla Options Junior quant: ‘Should I be surprised that μ drops out?’ Senior quant: ‘Not if you want to keep your job.’ This appendix describes the procedure for deriving closed-form expressions for the prices of vanilla call and put options, by analytically performing integrals derived in Chapter 2. In that chapter, we derive two integral expressions, either of which may be used to calculate the Black–Scholes value of a European option. One of the integral expressions yields the value in terms of the transformed variable X (Equation 2.64): ∞ ( − )2 1 − X X ¯ v(X,τ)= √ e 2σ2τ P(X )dX −∞ 2πσ2τ The other integral expression yields the value in terms of the original financial variable S (spot price) (Equation 2.65): ∞ 2 − ( − ) 1 1 (lnF − lnS ) V(S,t) = e rd Ts t exp − P(S )dS 2 σ 2( − ) 0 2πσ (Te − t) S 2 Te t We will here perform the integral in Equation 2.64 in order to obtain the pricing formulae. The transformed payoff function P¯ (X) takes the following form for a vanilla option: ¯ X P(X) = max 0,φ F0e − K (A.1) where φ is the option trait (+1foracall;−1foraput)andF0 is the arbitrary quantity that was introduced in the transformation process for the purpose of dimensional etiquette. The zero floor in the payoff function has the result that the integrand vanishes for a semi-infinite range of X values. For φ =+1, the integrand vanishes for X < ln(K/F0), whereas for φ =−1, the integrand vanishes for X > ln(K/F0). In general, we may write the 215 216 FX Barrier Options integral in terms of lower limit a and upper limit b: ( − )2 b − X X 1 2 X v(X,τ) = √ e 2σ τ max 0,φ F0e − K dX (A.2) a 2πσ2τ where the limits a and b depend on the option trait φ in the following way: ( ln K φ =+1 a = F0 (A.3) −∞ φ =−1 ( ∞ φ =+1 b = (A.4) ln K φ =−1 F0 Let us write the two terms in Equation A.2 explicitly as such: v(X,τ) = I1 − I2 (A.5) where ( − )2 b − X X . 1 2 X I1 = φF0 √ e 2σ τ e dX (A.6) a 2πσ2τ and ( − )2 b − X X . 1 2 I2 = φK √ e 2σ τ dX (A.7) a 2πσ2τ The integrand of Expression I2 is the probability density function (PDF) of a normal distribution with mean X and variance σ 2τ. This can be written in terms of the special function N(·), which is the cumulative distribution function (CDF) of a standard normal distribution: − − = φ b √X − a √X I2 K N σ τ N σ τ (A.8) In Expression I1, completion of the square in the exponent gives: 2 X − X+σ2τ 1 2 b − X+ σ τ 1 2 I1 = φF0e 2 √ e 2σ τ dX (A.9) a 2πσ2τ As with I2, the integrand is again the probability density function (PDF) of a normal distribution, and the variance is again σ 2τ, but this time the mean is (X + σ 2τ). Again, this can be written in terms of N(·): 2 2 + 1 σ 2τ b − X − σ τ a − X − σ τ = φ X 2 √ − √ I1 F0e N σ τ N σ τ (A.10) Derivation of the Black–Scholes Pricing Equations for Vanilla Options 217 The closed-form expressions for I1 and I2 given by Equations A.10 and A.8 respectively can now be inserted into Equation A.5 to give a closed-form expression for v. Since the quantities a and b depend on the option trait φ (see Equations A.3 and A.4), we will separate the call and put cases. For calls (φ =+1), I1, I2 and v are given as follows: ⎛ ⎛ ⎞⎞ ln K − X − σ 2τ + 1 σ 2τ F0 = X 2 ⎝ (∞) − ⎝ √ ⎠⎠ I1 (call) F0e N N σ τ ⎛ ⎛ ⎞⎞ ln K − X − σ 2τ + 1 σ 2τ F0 = X 2 ⎝ − ⎝ √ ⎠⎠ F0e 1 N σ τ ⎛ ⎞ −ln K + X + σ 2τ + 1 σ 2τ F0 = X 2 ⎝ √ ⎠ F0e N σ τ (A.11) ⎛ ⎛ ⎞⎞ ln K − X = ⎝ (∞) − ⎝ F0√ ⎠⎠ I2 (call) K N N σ τ ⎛ ⎛ ⎞⎞ ln K − X = ⎝ − ⎝ F0√ ⎠⎠ K 1 N σ τ ⎛ ⎞ −ln K + X = ⎝ F√0 ⎠ KN σ τ (A.12) ⎛ ⎞ ⎛ ⎞ −ln K + X + σ 2τ −ln K + X + 1 σ 2τ F0 F0 ⇒ = X 2 ⎝ √ ⎠ − ⎝ √ ⎠ vcall F0e N σ τ KN σ τ (A.13) Forputs(φ =−1), I1, I2 and v are given as follows: ⎛ ⎛ ⎞ ⎞ ln K − X − σ 2τ + 1 σ 2τ F0 =− X 2 ⎝ ⎝ √ ⎠ − (−∞)⎠ I1 (put) F0e N σ τ N ⎛ ⎞ ln K − X − σ 2τ + 1 σ 2τ F0 =− X 2 ⎝ √ ⎠ F0e N σ τ (A.14) 218 FX Barrier Options ⎛ ⎛ ⎞ ⎞ ln K − X =− ⎝ ⎝ F0√ ⎠ − (−∞)⎠ I2 (put) K N σ τ N ⎛ ⎞ ln K − X =− ⎝ F0√ ⎠ KN σ τ (A.15) ⎛ ⎞ ⎛ ⎞ ln K − X − σ 2τ ln K − X + 1 σ 2τ F0 F0 ⇒ =− X 2 ⎝ √ ⎠ + ⎝ √ ⎠ vput F0e N σ τ KN σ τ (A.16) The similarities between Equations A.13 and A.16 allow us to recombine the call and put results into a single vanilla result, like so: ⎡ ⎛ ⎞ ⎛ ⎞⎤ X − ln K + σ 2τ X − ln K + 1 σ 2τ F0 F0 = φ⎣ X 2 ⎝φ √ ⎠ − ⎝φ √ ⎠⎦ vvanilla F0e N σ τ KN σ τ (A.17) We now have a closed-form expression for the transformed value variable v(X,τ).To obtain an expression for the original value variable V(S,t), it only remains for us to undo the four transformations of Section 2.7.1 one by one. Undoing Transformation 4 gives us an expression for undiscounted vanilla prices in terms of the forward: ⎡ ⎛ ⎞ ⎛ ⎞⎤ ln F + 1 σ 2τ ln F − 1 σ 2τ ˜ ( τ)= φ⎣ ⎝φ K √ 2 ⎠ − ⎝φ K √ 2 ⎠⎦ U F, FN σ τ KN σ τ (A.18) Undoing Transformation 3 gives us an expression for undiscounted prices in terms of spot: ⎡ ⎛ ⎞ S + ( − + 1 σ 2)τ ( − )τ ln K rd rf 2 ( τ)= φ⎣ rd rf ⎝φ √ ⎠ U S, Se N σ τ ⎛ ⎞⎤ S 1 2 ln + (rd − rf − σ )τ − ⎝φ K √ 2 ⎠⎦ KN σ τ (A.19) Derivation of the Black–Scholes Pricing Equations for Vanilla Options 219 Undoing Transformation 2 gives us an expression for discounted prices in terms of spot: ⎡ ⎛ ⎞ S + ( − + 1 σ 2)τ − τ ln K rd rf 2 ˜ ( τ)= φ⎣ rf ⎝φ √ ⎠ V S, Se N σ τ ⎛ ⎞⎤ S + ( − − 1 σ 2)τ − τ ln K rd rf 2 − rd ⎝φ √ ⎠⎦ Ke N σ τ (A.20) Lastly, undoing Transformation 1 gives us an expression for discounted prices in terms of spot, with explicit reference to the time variable t: ⎡ ⎛ ⎞ S + ( − + 1 σ 2)( − ) − ( − ) ln K rd rf 2 T t V(S,t) = φ⎣Se rf T t N⎝φ √ ⎠ σ T − t ⎛ ⎞⎤ S + ( − − 1 σ 2)( − ) − ( − ) ln K rd rf 2 T t −Ke rd T t N⎝φ √ ⎠⎦ (A.21) σ T − t If it seems that we have laboured the working, it is for a reason: each of the forms of expression we have presented can be useful in its own right. All of the forms of expression shown above may be found in the literature. The long expressions that form the arguments of the normal CDF are commonly given their own symbols. For example, following the conventions in Hull [2], we define: S + ( − + 1 σ 2)( − ) ln K rd rf 2 T t d1 = √ (A.22) σ T − t S + ( − − 1 σ 2)( − ) ln K rd rf 2 T t d2 = √ (A.23) σ T − t whereupon our formula for the discounted prices in terms of spot becomes: −r (T−t) −r (T−t) V(S,t) = φ Se f N(φd1) − Ke d N(φd2) (A.24) The value V here is for an option with unit Foreign principal (Af = 1); to get the value for non-unit-principal options, we simply need to multiply V by Af. B Normal and Lognormal Probability Distributions B.1 Normal distribution N In the case where Z follows a normal distribution, its density function fZ has the form: ( − μ )2 N ( ) = 1 − z Z fZ z exp (B.1) πσ2 2σ 2 2 Z Z where z may take any real value. μ σ 2 The mean of the distribution equals Z , and its variance equals Z .Thestandard normal distribution is a normal distribution which has mean equal to zero and variance equal to one. We denote the PDF and CDF of the standard normal distribution by special functions n(·) and N(·) respectively: 1 z2 n(z) = √ exp − (B.2) 2π 2 z 1 x2 N(z) = √ exp − dx (B.3) −∞ 2π 2 B.2 Lognormal distribution LN In the case where Z follows a lognormal distribution, its density function fZ has the form: ( − μ )2 LN ( ) = 1 1 − lnz ln Z fZ z exp (B.4) πσ2 z 2σ 2 2 Z Z where z > 0. 220 C Derivation of the Local Volatility Function C.1 Derivation in terms of call prices Our aim here is to derive an expression for the local volatility (lv) function σ(S,t) that appears in the local volatility model of Equation 4.21: dS = (rd − rf)S dt + σ(S,t)S dWt Central to the derivation is the probability density function (PDF) of spot. This quantity provides the crucial link between the dynamics of spot and the values of options. We introduced the PDF in the special case of the Black–Scholes model, in Section 2.7.2.
Recommended publications
  • Schedule Rc-L – Derivatives and Off-Balance Sheet Items
    FFIEC 031 and 041 RC-L – DERIVATIVES AND OFF-BALANCE SHEET SCHEDULE RC-L – DERIVATIVES AND OFF-BALANCE SHEET ITEMS General Instructions Schedule RC-L should be completed on a fully consolidated basis. In addition to information about derivatives, Schedule RC-L includes the following selected commitments, contingencies, and other off-balance sheet items that are not reportable as part of the balance sheet of the Report of Condition (Schedule RC). Among the items not to be reported in Schedule RC-L are contingencies arising in connection with litigation. For those asset-backed commercial paper program conduits that the reporting bank consolidates onto its balance sheet (Schedule RC) in accordance with ASC Subtopic 810-10, Consolidation – Overall (formerly FASB Interpretation No. 46 (Revised), “Consolidation of Variable Interest Entities,” as amended by FASB Statement No. 167, “Amendments to FASB Interpretation No. 46(R)”), any credit enhancements and liquidity facilities the bank provides to the programs should not be reported in Schedule RC-L. In contrast, for conduits that the reporting bank does not consolidate, the bank should report the credit enhancements and liquidity facilities it provides to the programs in the appropriate items of Schedule RC-L. Item Instructions Item No. Caption and Instructions 1 Unused commitments. Report in the appropriate subitem the unused portions of commitments. Unused commitments are to be reported gross, i.e., include in the appropriate subitem the unused amount of commitments acquired from and conveyed or participated to others. However, exclude commitments conveyed or participated to others that the bank is not legally obligated to fund even if the party to whom the commitment has been conveyed or participated fails to perform in accordance with the terms of the commitment.
    [Show full text]
  • Lecture 7 Futures Markets and Pricing
    Lecture 7 Futures Markets and Pricing Prof. Paczkowski Lecture 7 Futures Markets and Pricing Prof. Paczkowski Rutgers University Spring Semester, 2009 Prof. Paczkowski (Rutgers University) Lecture 7 Futures Markets and Pricing Spring Semester, 2009 1 / 65 Lecture 7 Futures Markets and Pricing Prof. Paczkowski Part I Assignment Prof. Paczkowski (Rutgers University) Lecture 7 Futures Markets and Pricing Spring Semester, 2009 2 / 65 Assignment Lecture 7 Futures Markets and Pricing Prof. Paczkowski Prof. Paczkowski (Rutgers University) Lecture 7 Futures Markets and Pricing Spring Semester, 2009 3 / 65 Lecture 7 Futures Markets and Pricing Prof. Paczkowski Introduction Part II Background Financial Markets Forward Markets Introduction Futures Markets Roles of Futures Markets Existence Roles of Futures Markets Futures Contracts Terminology Prof. Paczkowski (Rutgers University) Lecture 7 Futures Markets and Pricing Spring Semester, 2009 4 / 65 Pricing Incorporating Risk Profiting and Offsetting Futures Concept Lecture 7 Futures Markets and Buyer Seller Pricing Wants to buy Prof. Paczkowski Situation - but not today Introduction Price will rise Background Expectation before ready Financial Markets to buy Forward Markets 1 Futures Markets Buy today Roles of before ready Futures Markets Strategy 2 Buy futures Existence contract to Roles of Futures hedge losses Markets Locks in low Result Futures price Contracts Terminology Prof. Paczkowski (Rutgers University) Lecture 7 Futures Markets and Pricing Spring Semester, 2009 5 / 65 Pricing Incorporating
    [Show full text]
  • FINANCIAL DERIVATIVES SAMPLE QUESTIONS Q1. a Strangle Is an Investment Strategy That Combines A. a Call and a Put for the Same
    FINANCIAL DERIVATIVES SAMPLE QUESTIONS Q1. A strangle is an investment strategy that combines a. A call and a put for the same expiry date but at different strike prices b. Two puts and one call with the same expiry date c. Two calls and one put with the same expiry dates d. A call and a put at the same strike price and expiry date Answer: a. Q2. A trader buys 2 June expiry call options each at a strike price of Rs. 200 and Rs. 220 and sells two call options with a strike price of Rs. 210, this strategy is a a. Bull Spread b. Bear call spread c. Butterfly spread d. Calendar spread Answer c. Q3. The option price will ceteris paribus be negatively related to the volatility of the cash price of the underlying. a. The statement is true b. The statement is false c. The statement is partially true d. The statement is partially false Answer: b. Q 4. A put option with a strike price of Rs. 1176 is selling at a premium of Rs. 36. What will be the price at which it will break even for the buyer of the option a. Rs. 1870 b. Rs. 1194 c. Rs. 1140 d. Rs. 1940 Answer b. Q5 A put option should always be exercised _______ if it is deep in the money a. early b. never c. at the beginning of the trading period d. at the end of the trading period Answer a. Q6. Bermudan options can only be exercised at maturity a.
    [Show full text]
  • Analysis of Securitized Asset Liquidity June 2017 an He and Bruce Mizrach1
    Analysis of Securitized Asset Liquidity June 2017 An He and Bruce Mizrach1 1. Introduction This research note extends our prior analysis2 of corporate bond liquidity to the structured products markets. We analyze data from the TRACE3 system, which began collecting secondary market trading activity on structured products in 2011. We explore two general categories of structured products: (1) real estate securities, including mortgage-backed securities in residential housing (MBS) and commercial building (CMBS), collateralized mortgage products (CMO) and to-be-announced forward mortgages (TBA); and (2) asset-backed securities (ABS) in credit cards, autos, student loans and other miscellaneous categories. Consistent with others,4 we find that the new issue market for securitized assets decreased sharply after the financial crisis and has not yet rebounded to pre-crisis levels. Issuance is below 2007 levels in CMBS, CMOs and ABS. MBS issuance had recovered by 2012 but has declined over the last four years. By contrast, 2016 issuance in the corporate bond market was at a record high for the fifth consecutive year, exceeding $1.5 trillion. Consistent with the new issue volume decline, the median age of securities being traded in non-agency CMO are more than ten years old. In student loans, the average security is over seven years old. Over the last four years, secondary market trading volumes in CMOs and TBA are down from 14 to 27%. Overall ABS volumes are down 16%. Student loan and other miscellaneous ABS declines balance increases in automobiles and credit cards. By contrast, daily trading volume in the most active corporate bonds is up nearly 28%.
    [Show full text]
  • Section 1256 and Foreign Currency Derivatives
    Section 1256 and Foreign Currency Derivatives Viva Hammer1 Mark-to-market taxation was considered “a fundamental departure from the concept of income realization in the U.S. tax law”2 when it was introduced in 1981. Congress was only game to propose the concept because of rampant “straddle” shelters that were undermining the U.S. tax system and commodities derivatives markets. Early in tax history, the Supreme Court articulated the realization principle as a Constitutional limitation on Congress’ taxing power. But in 1981, lawmakers makers felt confident imposing mark-to-market on exchange traded futures contracts because of the exchanges’ system of variation margin. However, when in 1982 non-exchange foreign currency traders asked to come within the ambit of mark-to-market taxation, Congress acceded to their demands even though this market had no equivalent to variation margin. This opportunistic rather than policy-driven history has spawned a great debate amongst tax practitioners as to the scope of the mark-to-market rule governing foreign currency contracts. Several recent cases have added fuel to the debate. The Straddle Shelters of the 1970s Straddle shelters were developed to exploit several structural flaws in the U.S. tax system: (1) the vast gulf between ordinary income tax rate (maximum 70%) and long term capital gain rate (28%), (2) the arbitrary distinction between capital gain and ordinary income, making it relatively easy to convert one to the other, and (3) the non- economic tax treatment of derivative contracts. Straddle shelters were so pervasive that in 1978 it was estimated that more than 75% of the open interest in silver futures were entered into to accommodate tax straddles and demand for U.S.
    [Show full text]
  • How Much Do Banks Use Credit Derivatives to Reduce Risk?
    How much do banks use credit derivatives to reduce risk? Bernadette A. Minton, René Stulz, and Rohan Williamson* June 2006 This paper examines the use of credit derivatives by US bank holding companies from 1999 to 2003 with assets in excess of one billion dollars. Using the Federal Reserve Bank of Chicago Bank Holding Company Database, we find that in 2003 only 19 large banks out of 345 use credit derivatives. Though few banks use credit derivatives, the assets of these banks represent on average two thirds of the assets of bank holding companies with assets in excess of $1 billion. To the extent that banks have positions in credit derivatives, they tend to be used more for dealer activities than for hedging activities. Nevertheless, a majority of the banks that use credit derivative are net buyers of credit protection. Banks are more likely to be net protection buyers if they engage in asset securitization, originate foreign loans, and have lower capital ratios. The likelihood of a bank being a net protection buyer is positively related to the percentage of commercial and industrial loans in a bank’s loan portfolio and negatively or not related to other types of bank loans. The use of credit derivatives by banks is limited because adverse selection and moral hazard problems make the market for credit derivatives illiquid for the typical credit exposures of banks. *Respectively, Associate Professor, The Ohio State University; Everett D. Reese Chair of Banking and Monetary Economics, The Ohio State University and NBER; and Associate Professor, Georgetown University. We are grateful to Jim O’Brien and Mark Carey for discussions.
    [Show full text]
  • Local Volatility Modelling
    LOCAL VOLATILITY MODELLING Roel van der Kamp July 13, 2009 A DISSERTATION SUBMITTED FOR THE DEGREE OF Master of Science in Applied Mathematics (Financial Engineering) I have to understand the world, you see. - Richard Philips Feynman Foreword This report serves as a dissertation for the completion of the Master programme in Applied Math- ematics (Financial Engineering) from the University of Twente. The project was devised from the collaboration of the University of Twente with Saen Options BV (during the course of the project Saen Options BV was integrated into AllOptions BV) at whose facilities the project was performed over a period of six months. This research project could not have been performed without the help of others. Most notably I would like to extend my gratitude towards my supervisors: Michel Vellekoop of the University of Twente, Julien Gosme of AllOptions BV and Fran¸coisMyburg of AllOptions BV. They provided me with the theoretical and practical knowledge necessary to perform this research. Their constant guidance, involvement and availability were an essential part of this project. My thanks goes out to Irakli Khomasuridze, who worked beside me for six months on his own project for the same degree. The many discussions I had with him greatly facilitated my progress and made the whole experience much more enjoyable. Finally I would like to thank AllOptions and their staff for making use of their facilities, getting access to their data and assisting me with all practical issues. RvdK Abstract Many different models exist that describe the behaviour of stock prices and are used to value op- tions on such an underlying asset.
    [Show full text]
  • Master Thesis the Use of Interest Rate Derivatives and Firm Market Value an Empirical Study on European and Russian Non-Financial Firms
    Master Thesis The use of Interest Rate Derivatives and Firm Market Value An empirical study on European and Russian non-financial firms Tilburg, October 5, 2014 Mark van Dijck, 937367 Tilburg University, Finance department Supervisor: Drs. J.H. Gieskens AC CCM QT Master Thesis The use of Interest Rate Derivatives and Firm Market Value An empirical study on European and Russian non-financial firms Tilburg, October 5, 2014 Mark van Dijck, 937367 Supervisor: Drs. J.H. Gieskens AC CCM QT 2 Preface In the winter of 2010 I found myself in the heart of a company where the credit crisis took place at that moment. During a treasury internship for Heijmans NV in Rosmalen, I experienced why it is sometimes unescapable to use interest rate derivatives. Due to difficult financial times, banks strengthen their requirements and the treasury department had to use different mechanism including derivatives to restructure their loans to the appropriate level. It was a fascinating time. One year later I wrote a bachelor thesis about risk management within energy trading for consultancy firm Tensor. Interested in treasury and risk management I have always wanted to finish my finance study period in this field. During the master thesis period I started to work as junior commodity trader at Kühne & Heitz. I want to thank Kühne & Heitz for the opportunity to work in the trading environment and to learn what the use of derivatives is all about. A word of gratitude to my supervisor Drs. J.H. Gieskens for his quick reply, well experienced feedback that kept me sharp to different levels of the subject, and his availability even in the late hours after I finished work.
    [Show full text]
  • Seeking Income: Cash Flow Distribution Analysis of S&P 500
    RESEARCH Income CONTRIBUTORS Berlinda Liu Seeking Income: Cash Flow Director Global Research & Design Distribution Analysis of S&P [email protected] ® Ryan Poirier, FRM 500 Buy-Write Strategies Senior Analyst Global Research & Design EXECUTIVE SUMMARY [email protected] In recent years, income-seeking market participants have shown increased interest in buy-write strategies that exchange upside potential for upfront option premium. Our empirical study investigated popular buy-write benchmarks, as well as other alternative strategies with varied strike selection, option maturity, and underlying equity instruments, and made the following observations in terms of distribution capabilities. Although the CBOE S&P 500 BuyWrite Index (BXM), the leading buy-write benchmark, writes at-the-money (ATM) monthly options, a market participant may be better off selling out-of-the-money (OTM) options and allowing the equity portfolio to grow. Equity growth serves as another source of distribution if the option premium does not meet the distribution target, and it prevents the equity portfolio from being liquidated too quickly due to cash settlement of the expiring options. Given a predetermined distribution goal, a market participant may consider an option based on its premium rather than its moneyness. This alternative approach tends to generate a more steady income stream, thus reducing trading cost. However, just as with the traditional approach that chooses options by moneyness, a high target premium may suffocate equity growth and result in either less income or quick equity depletion. Compared with monthly standard options, selling quarterly options may reduce the loss from the cash settlement of expiring calls, while selling weekly options could incur more loss.
    [Show full text]
  • Implied Volatility Modeling
    Implied Volatility Modeling Sarves Verma, Gunhan Mehmet Ertosun, Wei Wang, Benjamin Ambruster, Kay Giesecke I Introduction Although Black-Scholes formula is very popular among market practitioners, when applied to call and put options, it often reduces to a means of quoting options in terms of another parameter, the implied volatility. Further, the function σ BS TK ),(: ⎯⎯→ σ BS TK ),( t t ………………………………(1) is called the implied volatility surface. Two significant features of the surface is worth mentioning”: a) the non-flat profile of the surface which is often called the ‘smile’or the ‘skew’ suggests that the Black-Scholes formula is inefficient to price options b) the level of implied volatilities changes with time thus deforming it continuously. Since, the black- scholes model fails to model volatility, modeling implied volatility has become an active area of research. At present, volatility is modeled in primarily four different ways which are : a) The stochastic volatility model which assumes a stochastic nature of volatility [1]. The problem with this approach often lies in finding the market price of volatility risk which can’t be observed in the market. b) The deterministic volatility function (DVF) which assumes that volatility is a function of time alone and is completely deterministic [2,3]. This fails because as mentioned before the implied volatility surface changes with time continuously and is unpredictable at a given point of time. Ergo, the lattice model [2] & the Dupire approach [3] often fail[4] c) a factor based approach which assumes that implied volatility can be constructed by forming basis vectors. Further, one can use implied volatility as a mean reverting Ornstein-Ulhenbeck process for estimating implied volatility[5].
    [Show full text]
  • 307439 Ferdig Master Thesis
    Master's Thesis Using Derivatives And Structured Products To Enhance Investment Performance In A Low-Yielding Environment - COPENHAGEN BUSINESS SCHOOL - MSc Finance And Investments Maria Gjelsvik Berg P˚al-AndreasIversen Supervisor: Søren Plesner Date Of Submission: 28.04.2017 Characters (Ink. Space): 189.349 Pages: 114 ABSTRACT This paper provides an investigation of retail investors' possibility to enhance their investment performance in a low-yielding environment by using derivatives. The current low-yielding financial market makes safe investments in traditional vehicles, such as money market funds and safe bonds, close to zero- or even negative-yielding. Some retail investors are therefore in need of alternative investment vehicles that can enhance their performance. By conducting Monte Carlo simulations and difference in mean testing, we test for enhancement in performance for investors using option strategies, relative to investors investing in the S&P 500 index. This paper contributes to previous papers by emphasizing the downside risk and asymmetry in return distributions to a larger extent. We find several option strategies to outperform the benchmark, implying that performance enhancement is achievable by trading derivatives. The result is however strongly dependent on the investors' ability to choose the right option strategy, both in terms of correctly anticipated market movements and the net premium received or paid to enter the strategy. 1 Contents Chapter 1 - Introduction4 Problem Statement................................6 Methodology...................................7 Limitations....................................7 Literature Review.................................8 Structure..................................... 12 Chapter 2 - Theory 14 Low-Yielding Environment............................ 14 How Are People Affected By A Low-Yield Environment?........ 16 Low-Yield Environment's Impact On The Stock Market........
    [Show full text]
  • Tax Treatment of Derivatives
    United States Viva Hammer* Tax Treatment of Derivatives 1. Introduction instruments, as well as principles of general applicability. Often, the nature of the derivative instrument will dictate The US federal income taxation of derivative instruments whether it is taxed as a capital asset or an ordinary asset is determined under numerous tax rules set forth in the US (see discussion of section 1256 contracts, below). In other tax code, the regulations thereunder (and supplemented instances, the nature of the taxpayer will dictate whether it by various forms of published and unpublished guidance is taxed as a capital asset or an ordinary asset (see discus- from the US tax authorities and by the case law).1 These tax sion of dealers versus traders, below). rules dictate the US federal income taxation of derivative instruments without regard to applicable accounting rules. Generally, the starting point will be to determine whether the instrument is a “capital asset” or an “ordinary asset” The tax rules applicable to derivative instruments have in the hands of the taxpayer. Section 1221 defines “capital developed over time in piecemeal fashion. There are no assets” by exclusion – unless an asset falls within one of general principles governing the taxation of derivatives eight enumerated exceptions, it is viewed as a capital asset. in the United States. Every transaction must be examined Exceptions to capital asset treatment relevant to taxpayers in light of these piecemeal rules. Key considerations for transacting in derivative instruments include the excep- issuers and holders of derivative instruments under US tions for (1) hedging transactions3 and (2) “commodities tax principles will include the character of income, gain, derivative financial instruments” held by a “commodities loss and deduction related to the instrument (ordinary derivatives dealer”.4 vs.
    [Show full text]