The Black-Scholes Model
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Up to EUR 3,500,000.00 7% Fixed Rate Bonds Due 6 April 2026 ISIN
Up to EUR 3,500,000.00 7% Fixed Rate Bonds due 6 April 2026 ISIN IT0005440976 Terms and Conditions Executed by EPizza S.p.A. 4126-6190-7500.7 This Terms and Conditions are dated 6 April 2021. EPizza S.p.A., a company limited by shares incorporated in Italy as a società per azioni, whose registered office is at Piazza Castello n. 19, 20123 Milan, Italy, enrolled with the companies’ register of Milan-Monza-Brianza- Lodi under No. and fiscal code No. 08950850969, VAT No. 08950850969 (the “Issuer”). *** The issue of up to EUR 3,500,000.00 (three million and five hundred thousand /00) 7% (seven per cent.) fixed rate bonds due 6 April 2026 (the “Bonds”) was authorised by the Board of Directors of the Issuer, by exercising the powers conferred to it by the Articles (as defined below), through a resolution passed on 26 March 2021. The Bonds shall be issued and held subject to and with the benefit of the provisions of this Terms and Conditions. All such provisions shall be binding on the Issuer, the Bondholders (and their successors in title) and all Persons claiming through or under them and shall endure for the benefit of the Bondholders (and their successors in title). The Bondholders (and their successors in title) are deemed to have notice of all the provisions of this Terms and Conditions and the Articles. Copies of each of the Articles and this Terms and Conditions are available for inspection during normal business hours at the registered office for the time being of the Issuer being, as at the date of this Terms and Conditions, at Piazza Castello n. -
University of Illinois at Urbana-Champaign Department of Mathematics
University of Illinois at Urbana-Champaign Department of Mathematics ASRM 410 Investments and Financial Markets Fall 2018 Chapter 5 Option Greeks and Risk Management by Alfred Chong Learning Objectives: 5.1 Option Greeks: Black-Scholes valuation formula at time t, observed values, model inputs, option characteristics, option Greeks, partial derivatives, sensitivity, change infinitesimally, Delta, Gamma, theta, Vega, rho, Psi, option Greek of portfolio, absolute change, option elasticity, percentage change, option elasticity of portfolio. 5.2 Applications of Option Greeks: risk management via hedging, immunization, adverse market changes, delta-neutrality, delta-hedging, local, re-balance, from- time-to-time, holding profit, delta-hedging, first order, prudent, delta-gamma- hedging, gamma-neutrality, option value approximation, observed values change inevitably, Taylor series expansion, second order approximation, delta-gamma- theta approximation, delta approximation, delta-gamma approximation. Further Exercises: SOA Advanced Derivatives Samples Question 9. 1 5.1 Option Greeks 5.1.1 Consider a continuous-time setting t ≥ 0. Assume that the Black-Scholes framework holds as in 4.2.2{4.2.4. 5.1.2 The time-0 Black-Scholes European call and put option valuation formula in 4.2.9 and 4.2.10 can be generalized to those at time t: −δ(T −t) −r(T −t) C (t; S; δ; r; σ; K; T ) = Se N (d1) − Ke N (d2) P = Ft;T N (d1) − PVt;T (K) N (d2); −r(T −t) −δ(T −t) P (t; S; δ; r; σ; K; T ) = Ke N (−d2) − Se N (−d1) P = PVt;T (K) N (−d2) − Ft;T N (−d1) ; where d1 and d2 are given by S 1 2 ln K + r − δ + 2 σ (T − t) d1 = p ; σ T − t S 1 2 p ln K + r − δ − 2 σ (T − t) d2 = p = d1 − σ T − t: σ T − t 5.1.3 The option value V is a function of observed values, t and S, model inputs, δ, r, and σ, as well as option characteristics, K and T . -
More on Market-Making and Delta-Hedging
More on Market-Making and Delta-Hedging What do market makers do to delta-hedge? • Recall that the delta-hedging strategy consists of selling one option, and buying a certain number ∆ shares • An example of Delta hedging for 2 days (daily rebalancing and mark-to-market): Day 0: Share price = $40, call price is $2.7804, and ∆ = 0.5824 Sell call written on 100 shares for $278.04, and buy 58.24 shares. Net investment: (58.24 × $40)$278.04 = $2051.56 At 8%, overnight financing charge is $0.45 = $2051.56 × (e−0.08/365 − 1) Day 1: If share price = $40.5, call price is $3.0621, and ∆ = 0.6142 Overnight profit/loss: $29.12 $28.17 $0.45 = $0.50(mark-to-market) Buy 3.18 additional shares for $128.79 to rebalance Day 2: If share price = $39.25, call price is $2.3282 • Overnight profit/loss: $76.78 + $73.39 $0.48 = $3.87(mark-to-market) What do market makers do to delta-hedge? • Recall that the delta-hedging strategy consists of selling one option, and buying a certain number ∆ shares • An example of Delta hedging for 2 days (daily rebalancing and mark-to-market): Day 0: Share price = $40, call price is $2.7804, and ∆ = 0.5824 Sell call written on 100 shares for $278.04, and buy 58.24 shares. Net investment: (58.24 × $40)$278.04 = $2051.56 At 8%, overnight financing charge is $0.45 = $2051.56 × (e−0.08/365 − 1) Day 1: If share price = $40.5, call price is $3.0621, and ∆ = 0.6142 Overnight profit/loss: $29.12 $28.17 $0.45 = $0.50(mark-to-market) Buy 3.18 additional shares for $128.79 to rebalance Day 2: If share price = $39.25, call price is $2.3282 • Overnight profit/loss: $76.78 + $73.39 $0.48 = $3.87(mark-to-market) What do market makers do to delta-hedge? • Recall that the delta-hedging strategy consists of selling one option, and buying a certain number ∆ shares • An example of Delta hedging for 2 days (daily rebalancing and mark-to-market): Day 0: Share price = $40, call price is $2.7804, and ∆ = 0.5824 Sell call written on 100 shares for $278.04, and buy 58.24 shares. -
Finance: a Quantitative Introduction Chapter 9 Real Options Analysis
Investment opportunities as options The option to defer More real options Some extensions Finance: A Quantitative Introduction Chapter 9 Real Options Analysis Nico van der Wijst 1 Finance: A Quantitative Introduction c Cambridge University Press Investment opportunities as options The option to defer More real options Some extensions 1 Investment opportunities as options 2 The option to defer 3 More real options 4 Some extensions 2 Finance: A Quantitative Introduction c Cambridge University Press Investment opportunities as options Option analogy The option to defer Sources of option value More real options Limitations of option analogy Some extensions The essential economic characteristic of options is: the flexibility to exercise or not possibility to choose best alternative walk away from bad outcomes Stocks and bonds are passively held, no flexibility Investments in real assets also have flexibility, projects can be: delayed or speeded up made bigger or smaller abandoned early or extended beyond original life-time, etc. 3 Finance: A Quantitative Introduction c Cambridge University Press Investment opportunities as options Option analogy The option to defer Sources of option value More real options Limitations of option analogy Some extensions Real Options Analysis Studies and values this flexibility Real options are options where underlying value is a real asset not a financial asset as stock, bond, currency Flexibility in real investments means: changing cash flows along the way: profiting from opportunities, cutting off losses Discounted -
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]. -
Ice Crude Oil
ICE CRUDE OIL Intercontinental Exchange® (ICE®) became a center for global petroleum risk management and trading with its acquisition of the International Petroleum Exchange® (IPE®) in June 2001, which is today known as ICE Futures Europe®. IPE was established in 1980 in response to the immense volatility that resulted from the oil price shocks of the 1970s. As IPE’s short-term physical markets evolved and the need to hedge emerged, the exchange offered its first contract, Gas Oil futures. In June 1988, the exchange successfully launched the Brent Crude futures contract. Today, ICE’s FSA-regulated energy futures exchange conducts nearly half the world’s trade in crude oil futures. Along with the benchmark Brent crude oil, West Texas Intermediate (WTI) crude oil and gasoil futures contracts, ICE Futures Europe also offers a full range of futures and options contracts on emissions, U.K. natural gas, U.K power and coal. THE BRENT CRUDE MARKET Brent has served as a leading global benchmark for Atlantic Oseberg-Ekofisk family of North Sea crude oils, each of which Basin crude oils in general, and low-sulfur (“sweet”) crude has a separate delivery point. Many of the crude oils traded oils in particular, since the commercialization of the U.K. and as a basis to Brent actually are traded as a basis to Dated Norwegian sectors of the North Sea in the 1970s. These crude Brent, a cargo loading within the next 10-21 days (23 days on oils include most grades produced from Nigeria and Angola, a Friday). In a circular turn, the active cash swap market for as well as U.S. -
Yuichi Katsura
P&TcrvvCJ Efficient ValtmVmm-and Hedging of Structured Credit Products Yuichi Katsura A thesis submitted for the degree of Doctor of Philosophy of the University of London May 2005 Centre for Quantitative Finance Imperial College London nit~ýl. ABSTRACT Structured credit derivatives such as synthetic collateralized debt obligations (CDO) have been the principal growth engine in the fixed income market over the last few years. Val- uation and risk management of structured credit products by meads of Monte Carlo sim- ulations tend to suffer from numerical instability and heavy computational time. After a brief description of single name credit derivatives that underlie structured credit deriva- tives, the factor model is introduced as the portfolio credit derivatives pricing tool. This thesis then describes the rapid pricing and hedging of portfolio credit derivatives using the analytical approximation model and semi-analytical models under several specifications of factor models. When a full correlation structure is assumed or the pricing of exotic structure credit products with non-linear pay-offs is involved, the semi-analytical tractability is lost and we have to resort to the time-consuming Monte Carlo method. As a Monte Carlo variance reduction technique we extend the importance sampling method introduced to credit risk modelling by Joshi [2004] to the general n-factor problem. The large homogeneouspool model is also applied to the method of control variates as a new variance reduction tech- niques for pricing structured credit products. The combination of both methods is examined and it turns out to be the optimal choice. We also show how sensitivities of a CDO tranche can be computed efficiently by semi-analytical and Monte Carlo framework. -
RT Options Scanner Guide
RT Options Scanner Introduction Our RT Options Scanner allows you to search in real-time through all publicly traded US equities and indexes options (more than 170,000 options contracts total) for trading opportunities involving strategies like: - Naked or Covered Write Protective Purchase, etc. - Any complex multi-leg option strategy – to find the “missing leg” - Calendar Spreads Search is performed against our Options Database that is updated in real-time and driven by HyperFeed’s Ticker-Plant and our Implied Volatility Calculation Engine. We offer both 20-minute delayed and real-time modes. Unlike other search engines using real-time quotes data only our system takes advantage of individual contracts Implied Volatilities calculated in real-time using our high-performance Volatility Engine. The Scanner is universal, that is, it is not confined to any special type of option strategy. Rather, it allows determining the option you need in the following terms: – Cost – Moneyness – Expiry – Liquidity – Risk The Scanner offers Basic and Advanced Search Interfaces. Basic interface can be used to find options contracts satisfying, for example, such requirements: “I need expensive Calls for Covered Call Write Strategy. They should expire soon and be slightly out of the money. Do not care much as far as liquidity, but do not like to bear high risk.” This can be done in five seconds - set the search filters to: Cost -> Dear Moneyness -> OTM Expiry -> Short term Liquidity -> Any Risk -> Moderate and press the “Search” button. It is that easy. In fact, if your request is exactly as above, you can just press the “Search” button - these values are the default filtering criteria. -
Show Me the Money: Option Moneyness Concentration and Future Stock Returns Kelley Bergsma Assistant Professor of Finance Ohio Un
Show Me the Money: Option Moneyness Concentration and Future Stock Returns Kelley Bergsma Assistant Professor of Finance Ohio University Vivien Csapi Assistant Professor of Finance University of Pecs Dean Diavatopoulos* Assistant Professor of Finance Seattle University Andy Fodor Professor of Finance Ohio University Keywords: option moneyness, implied volatility, open interest, stock returns JEL Classifications: G11, G12, G13 *Communications Author Address: Albers School of Business and Economics Department of Finance 901 12th Avenue Seattle, WA 98122 Phone: 206-265-1929 Email: [email protected] Show Me the Money: Option Moneyness Concentration and Future Stock Returns Abstract Informed traders often use options that are not in-the-money because these options offer higher potential gains for a smaller upfront cost. Since leverage is monotonically related to option moneyness (K/S), it follows that a higher concentration of trading in options of certain moneyness levels indicates more informed trading. Using a measure of stock-level dollar volume weighted average moneyness (AveMoney), we find that stock returns increase with AveMoney, suggesting more trading activity in options with higher leverage is a signal for future stock returns. The economic impact of AveMoney is strongest among stocks with high implied volatility, which reflects greater investor uncertainty and thus higher potential rewards for informed option traders. AveMoney also has greater predictive power as open interest increases. Our results hold at the portfolio level as well as cross-sectionally after controlling for liquidity and risk. When AveMoney is calculated with calls, a portfolio long high AveMoney stocks and short low AveMoney stocks yields a Fama-French five-factor alpha of 12% per year for all stocks and 33% per year using stocks with high implied volatility. -
Managed Futures and Long Volatility by Anders Kulp, Daniel Djupsjöbacka, Martin Estlander, Er Capital Management Research
Disclaimer: This article appeared in the AIMA Journal (Feb 2005), which is published by The Alternative Investment Management Association Limited (AIMA). No quotation or reproduction is permitted without the express written permission of The Alternative Investment Management Association Limited (AIMA) and the author. The content of this article does not necessarily reflect the opinions of the AIMA Membership and AIMA does not accept responsibility for any statements herein. Managed Futures and Long Volatility By Anders Kulp, Daniel Djupsjöbacka, Martin Estlander, er Capital Management Research Introduction and background The buyer of an option straddle pays the implied volatility to get exposure to realized volatility during the lifetime of the option straddle. Fung and Hsieh (1997b) found that the return of trend following strategies show option like characteristics, because the returns tend to be large and positive during the best and worst performing months of the world equity markets. Fung and Hsieh used lookback straddles to replicate a trend following strategy. However, the standard way to obtain exposure to volatility is to buy an at-the-money straddle. Here, we will use standard option straddles with a dynamic updating methodology and flexible maturity rollover rules to match the characteristics of a trend follower. The lookback straddle captures only one trend during its maturity, but with an effective systematic updating method the simple option straddle captures more than one trend. It is generally accepted that when implementing an options strategy, it is crucial to minimize trading, because the liquidity in the options markets are far from perfect. Compared to the lookback straddle model, the trading frequency for the simple straddle model is lower. -
Arbitrage Opportunities with a Delta-Gamma Neutral Strategy in the Brazilian Options Market
FUNDAÇÃO GETÚLIO VARGAS ESCOLA DE PÓS-GRADUAÇÃO EM ECONOMIA Lucas Duarte Processi Arbitrage Opportunities with a Delta-Gamma Neutral Strategy in the Brazilian Options Market Rio de Janeiro 2017 Lucas Duarte Processi Arbitrage Opportunities with a Delta-Gamma Neutral Strategy in the Brazilian Options Market Dissertação para obtenção do grau de mestre apresentada à Escola de Pós- Graduação em Economia Área de concentração: Finanças Orientador: André de Castro Silva Co-orientador: João Marco Braga da Cu- nha Rio de Janeiro 2017 Ficha catalográfica elaborada pela Biblioteca Mario Henrique Simonsen/FGV Processi, Lucas Duarte Arbitrage opportunities with a delta-gamma neutral strategy in the Brazilian options market / Lucas Duarte Processi. – 2017. 45 f. Dissertação (mestrado) - Fundação Getulio Vargas, Escola de Pós- Graduação em Economia. Orientador: André de Castro Silva. Coorientador: João Marco Braga da Cunha. Inclui bibliografia. 1.Finanças. 2. Mercado de opções. 3. Arbitragem. I. Silva, André de Castro. II. Cunha, João Marco Braga da. III. Fundação Getulio Vargas. Escola de Pós-Graduação em Economia. IV. Título. CDD – 332 Agradecimentos Ao meu orientador e ao meu co-orientador pelas valiosas contribuições para este trabalho. À minha família pela compreensão e suporte. À minha amada esposa pelo imenso apoio e incentivo nas incontáveis horas de estudo e pesquisa. Abstract We investigate arbitrage opportunities in the Brazilian options market. Our research consists in backtesting several delta-gamma neutral portfolios of options traded in B3 exchange to assess the possibility of obtaining systematic excess returns. Returns sum up to 400% of the daily interbank rate in Brazil (CDI), a rate viewed as risk-free. -
Option Greeks Demystified
Webinar Presentation Option Greeks Demystified Presented by Trading Strategy Desk 1 Fidelity Brokerage Services, Member NYSE, SIPC, 900 Salem Street, Smithfield, RI 02917. © 2016 FMR LLC. All rights reserved. 764207.1.0 Disclosures Options trading entails significant risk and is not appropriate for all investors. Certain complex options strategies carry additional risk. Before trading options, please read Characteristics and Risks of Standardized Options, and call 800-544- 5115 to be approved for options trading. Supporting documentation for any claims, if applicable, will be furnished upon request. Examples in this presentation do not include transaction costs (commissions, margin interest, fees) or tax implications, but they should be considered prior to entering into any transactions. The information in this presentation, including examples using actual securities and price data, is strictly for illustrative and educational purposes only and is not to be construed as an endorsement, recommendation. Active Trader Pro PlatformsSM is available to customers trading 36 times or more in a rolling 12-month period; customers who trade 120 times or more have access to Recognia anticipated events and Elliott Wave analysis. Greeks are mathematical calculations used to determine the effect of various factors on options. Goal of this webinar Demystify what Options Greeks are and explain how they are used in plain English. What we will cover: What the Greeks are What the Greeks tell us How the Greeks can help with planning option trades How the Greeks can help with managing option trades The Greeks What the Greeks are: • Delta • Gamma • Vega • Theta • Rho But what do they mean? Greeks What do they tell me? In simplest terms, Greeks give traders a theoretical way to judge their exposure to various options pricing inputs.