Fx Opções E Estruturado Produtos Pdf

Opções de FX e Produtos Estruturados 1 Opções de FX e Produtos Estruturados Uwe Wystup 7 de Abril de 2006 3 Conteúdo 0 Prefácio Escopo deste Livro O Leitor Sobre o Autor Agradecimentos Opções de Câmbio Uma Viagem pelo Histórico de Opções Questões Técnicas para Opções de Baunilha Valor Uma Nota sobre o Adiante Gregos Identidades Homogeneidade Baseadas Relações Cotação Strike em termos de Delta Volatilidade em termos de Delta Volatilidade e Delta para uma greve dada Greves em termos de Deltas Volatilidade Histórico Volatilidade Histórico Correlação Volatilidade Sorriso At-The-Money Volatilidade Interpolação Volatilidade Smile Convenções At-The - Dinheiro Definição Interpolação da Volatilidade nos Pilares de Vencimento Interpolação da Volatilidade Espalhamento entre Pilares de Maturidade Volatilidade Fontes Volatilidade Cones Estocástica Volatilidade 4 4 ​​Exercícios de Wystup Estratégias Básicas contendo Opções de Baunilha Call and Put Risco de Reversão Risco Reversão Flip Straddle Strangle Borboleta Seagull Exercícios First G Eneration Exotics Opções de Barreira Opções Digitais, Opções de Toque e Rebates Composto e Parcelamento Opções Asiáticas Opções Lookback Opções de Avanço, Ratchet e Cliquet Opções de Energia Quanto Opções Exercícios Exotics de Segunda Geração Corredores Faders Opções de Barreira Exótica Opções de Pagamento-Mais Opções de câmbio Cestas Opções melhores e piores Opções Opções e adiantamentos sobre a variação média de harmônicos e swaps de volatilidade Exercícios Produtos estruturados Produtos futuros Avançado direto Avançar para frente Avançar para frente Avançar para frente Avançar para frente de tubarão Avançar para o tubarão 5 Opções de FX e produtos estruturados Butterfly Forward Range Forward Range Acréscimo Forward Acumulativo Forward Boomerang Forward Amortização Forward Auto-Renovação Forward Duplo Shark Forward Forward Iniciar Chooser Forward Estilo Livre Forward Boostado SpotForward Tempo Opção Exercícios Série de Estratégias Shark Forward Série Colar Série Extra Exercícios Depósitos e Empréstimos Duplos Moeda DepositLoan Desempenho Depósitos vinculados Túnel DepositLoan Corredor DepositLoan Turbo DepositLoan Tower DepósitoLoan Exercícios Taxa de juros e Cross Currency Swaps Cruz Moeda Swap Hanseatic Swap Turbo Cross Moeda Swap Buffered Cruz Moeda Swap Flip Swap Corridor Swap Duplo-No-Touch ligado Swap Intervalo Redefinir Trocar Troca Exercícios Notas de Participação Notas de Participação de Ouro Notas de Cotações Swap de Emitentes Moving Strike Turbo Spot Unlimited 6 6 Wystup 2.6 Produtos FX Híbridos Questões Práticas Os Traders Regra do Dedo Custo das Observações de Vanna e Volga Verificação de consistência Abreviaturas para a Primeira Geração Exotics Volatilidade para Reversões de Risco , Borboletas e Valor Teórico Opções de Barreira Preços Opções de Double Barrier Preços Opções de Double-No-Touch Preços Opções de Estilo Europeu Não-Touch Probabilidade O custo de negociação e sua Implicação no Preço de Mercado de Onetouch Opções Exemplo Outras Aplicações Exercícios Oferta Ask Spr Eads One Touch Spreads Vanilla Spreads Spreads for First Generation Exotics Lance Mínimo Ask Spread Oferta Pedir Preços Exercícios Liquidação O Modelo Black-Scholes para a Liquidação de Liquidação em Dinheiro Real Entrega Opções de Liquidação com Exercícios de Entrega Diferidos Sobre o Custo dos Anúncios de Fixação Atrasada A Fixação de Moeda do Análise do Modelo de Banco Central Europeu e Análise de Resultados Análise de EUR-USD Conclusão 7 Opções de FX e Produtos Estruturados 7 4 Contabilidade de Cobertura de acordo com IAS Introdução Instrumentos Financeiros Visão Geral Definição Geral Activos Financeiros Passivos Financeiros Compensação de Activos Financeiros e Passivos Financeiros Instrumentos de Capital Composto Instrumentos Financeiros Derivados Derivados Embebidos Classificação de Instrumentos Financeiros Avaliação de Instrumentos Financeiros Reconhecimento Inicial Medição Inicial Medição Subsequente Desreconhecimento Visão Geral de Contabilidade Hedge Tipos de Hedges Requisitos Básicos Parar Métodos de Contabilidade de Cobertura para Testi Ng Hedge Efectividade Justo Valor Hedge Cash Flow Hedge Testing for Efficiency - Um Estudo de Caso do Forward Plus Simulação de Taxas de Câmbio Cálculo do Forward Plus Valor Cálculo das Taxas de Juro Cálculo da Taxa de Transação Previsto Value Offset Ratio - Teste Prospectivo para Eficácia Medida de Redução de Variância - Teste Prospectivo de Efetividade Análise de Regressão - Teste Prospectivo de Efetividade Resultado Teste Retrospectivo para Efetividade Conclusão Fontes Originais Relevantes para Padrões Contábeis Exercícios 8 8 Wystup 5 Mercados de Câmbio Uma Visita pelo Mercado Declaração do Grupo GFI (Fenics), 25 Outubro Entrevista com a ICY Software, 14 de Outubro Entrevista com a Bloomberg, 12 de Outubro Entrevista com a Murex, 8 de Novembro Entrevista com a SuperDerivatives, 17 de Outubro Entrevista com a Lucht Probst Associates, 27 de Fevereiro Software e Requisitos do Sistema Fenics Position Rietary Negociação Negociação por Vendas Inter Banco Vendas Vendas Vendas Institucionais Vendas Corporativas Private Banking Opções de Câmbio Listadas Joke de Negociação 9 Capítulo 0 Prefácio 0.1 Escopo deste Manual A gestão de tesouraria de empresas internacionais envolve lidar com fluxos de caixa em diferentes moedas. Portanto, o serviço natural de um banco de investimento consiste em uma variedade de mercado monetário e produtos cambiais. Este livro explica os produtos mais populares e estratégias com um foco em tudo além de opções de baunilha. Ele explica todas as opções de FX, estruturas comuns e soluções sob medida em exemplos com um foco especial na aplicação com vistas de comerciantes e vendas, bem como de uma perspectiva de cliente corporativo. Ele contém negócios realmente negociados com motivações correspondentes explicando por que as estruturas foram negociadas. Desta forma, o leitor recebe um sentimento de como construir novas estruturas para atender às necessidades dos clientes. Os exercícios são destinados a praticar o material. Vários deles são realmente difíceis de resolver e podem servir como incentivos para pesquisas futuras e testes. Soluções para os exercícios não são parte deste livro, no entanto eles serão publicados na página web do livro, 0.2 O Pré-requisito de Leitores é algum conhecimento básico de mercados de FX, como por exemplo tirado do Book Foreign Exchange Primer por Shami Shamah, Wiley 2003, ver 90. Os leitores-alvo são estudantes de pós-graduação e faculdade de programas de engenharia financeira, que podem usar este livro como um livro para um curso chamado produtos estruturados ou opções de moeda exóticos. 9 10 10 Traders Wystup, Trainee Structurers, Product Developers, Vendas e Quants com interesse na linha de produtos FX. Para eles, pode servir como fonte de idéias e também como um guia de referência. Tesoureiros de empresas interessadas em administrar seus livros. Com este livro à mão eles podem estruturar suas próprias soluções. Os leitores mais interessados ​​nos aspectos quantitativos e de modelagem são recomendados para ler Risco de Câmbio por J. Hakala e U. Wystup, Risk Publications, Londres, 2002, ver 50. Este livro explica várias opções de FX exóticas com um foco especial no subjacente Modelos e matemática, mas não contém quaisquer estruturas ou clientes corporativos ou investidores ver. 0.3 Figura 1: Uwe Wystup, professor de Finanças Quantitativas na HfB Business School de Finanças e Gestão em Frankfurt, Alemanha. Uwe Wystup é também CEO da MathFinance AG, uma rede global de quants especializada em Finanças Quantitativas, Consultoria de Opções Exóticas e Produção de Software Front Office. Anteriormente, ele era Engenheiro Financeiro e Estruturador na Equipe de Negociação de Opções de FX do Commerzbank. Antes trabalhou para o Deutsche Bank, Citibank, UBS e Sal. Oppenheim jr. Amp Cie. Ele é fundador e gerente do site MathFinance. de e da Newsletter MathFinance. Uwe é doutor em finanças matemáticas pela Carnegie Mellon University. Ele também palestras sobre matemática finanças para Goethe University Frankfurt, organiza o Frankfurt MathFinance Colóquio e é diretor fundador do Frankfurt MathFinance Institute. Ele deu vários seminários sobre opções exóticas, finanças computacionais e modelagem de volatilidade. Sua área de especialização são os aspectos quantitativos eo design de produtos estruturados de mercados de câmbio estrangeiros. Ele publicou um livro sobre Risco de Câmbio e artigos em Finanças e Estocástica e o Jornal de Derivativos. Uwe deu muitas apresentações em universidades e bancos ao redor do mundo. Mais informações sobre o seu curriculum vitae e uma lista detalhada de publicações está disponível em 0.4 Agradecimentos Gostaria de agradecer aos meus antigos colegas no pregão, sobretudo Gustave Rieunier, Behnouch Mostachfi, Noel Speake, Roman Stauss, Tamaacutes Korchmaacuteros, Michael Braun, Andreas Weber, Tino Senge, Juumlrgen Hakala e todos os meus colegas e co-autores, especialmente Christoph Becker, Susanne Griebsch, Christoph Kuumlhn, Sebastian Krug, Marion Linck, Wolfgang Schmidt e Robert Tompkins. Chris Swain, Rachael Wilkie e muitas outras publicações Wiley merecem respeito como eles estavam lidando com a minha velocidade um pouco lento na conclusão deste livro. Nicole van de Locht e Choon Peng Toh merecem uma medalha para a leitura séria prova séria. 13 Capítulo 1 Opções de Câmbio Os Produtos Estruturados FX são combinações lineares feitas sob medida de Opções de FX, incluindo opções de baunilha e exóticas. Recomendamos o livro por Shamah 90 como uma fonte para aprender sobre mercados de FX com um foco em convenções de mercado, spot, forward e swap contratos, opções de baunilha. Para preços e modelagem de opções de FX exóticas sugerimos Hakala e Wystup 50 ou Lipton 71 como companheiros úteis para este livro. O mercado dos produtos estruturados é limitado ao mercado dos ingredientes necessários. Assim, geralmente há produtos estruturados principalmente negociados os pares de moedas que podem ser formados entre USD, JPY, EUR, CHF, GBP, CAD e AUD. Neste capítulo, começamos com um breve histórico de opções, seguido por uma seção técnica sobre opções de baunilha e volatilidade, e lidamos com combinações lineares de opções de baunilha comumente usadas. Em seguida, vamos ilustrar os ingredientes mais importantes para os produtos estruturados FX: a primeira e segunda geração exóticos. 1.1 Uma viagem através da história das opções As primeiras opções e futuros foram negociados na Grécia antiga, quando as azeitonas foram vendidas antes de atingirem a maturação. Em seguida, o mercado evoluiu da seguinte maneira. Século XVI Desde as tulipas do século 15, que foram apreciadas por sua aparência exótica, foram cultivadas na Turquia. O chefe dos jardins médicos reais em Viena, Áustria, foi o primeiro a cultivar essas tulipas turcas com êxito na Europa. Quando ele fugiu para a Holanda por causa da perseguição religiosa, ele pegou os bulbos. Como novo chefe dos jardins botânicos de Leiden, na Holanda, cultivou várias variedades novas. Foi a partir desses jardins que os comerciantes avarento roubou as lâmpadas para comercializá-los, porque as tulipas eram um grande símbolo de status. Século 17 Os primeiros futuros em tulipas foram negociados em 1634, as pessoas podiam 13 14 14 Wystup comprar cepas de tulipa especial pelo peso de suas lâmpadas, para as lâmpadas o mesmo valor foi escolhido como para o ouro. Junto com o comércio regular, os especuladores entraram no mercado e os preços dispararam. Um bulbo da linhagem Semper Octavian valia dois vagões de trigo, quatro cargas de centeio, quatro bois gordos, oito porcos gordos, doze ovelhas gordas, duas barricas de vinho, quatro barris de cerveja, dois barris de manteiga, 1.000 quilos de queijo , Uma cama de casal com linho e um vagão considerável. As pessoas deixaram suas famílias, venderam todos os seus pertences e até mesmo emprestaram dinheiro para se tornarem comerciantes de tulipas. Quando em 1637, este mercado supostamente livre de risco caiu, comerciantes, bem como particulares, entraram em falência. O governo proibiu o comércio especulativo do período tornou-se famoso como Tulipmania. Século XVIII Em 1728, a Royal West Indian e a Guiné Company, monopolista do comércio com as ilhas do Caribe e da costa africana, emitiram as primeiras opções de ações. Aqueles eram opções na compra da ilha francesa de Ste. Croix, no qual foram planejadas plantações de açúcar. O projeto foi realizado em 1733 e estoques de papel foram emitidos em conjunto com as ações, as pessoas compraram uma parcela relativa da ilha e os valores, bem como os privilégios e os direitos da empresa. Século XIX Em 1848, 82 empresários fundaram o Chicago Board of Trade (CBOT). Hoje é o maior e mais antigo mercado de futuros do mundo inteiro. A maioria dos documentos escritos foram perdidos no grande incêndio de 1871, no entanto, é comum acreditar que os primeiros futuros padronizados foram negociados como de CBOT agora negocia vários futuros e forwards, não apenas T-bonds e títulos do Tesouro, mas também opções e ouro. Em 1870, foi fundada a Bolsa de Algodão de Nova York. Em 1880, o padrão ouro foi introduzido. Século XX Em 1914, o padrão-ouro foi abandonado por causa da guerra. Em 1919, o Chicago Produce Exchange, encarregado de comercializar produtos agrícolas foi renomeado para Chicago Mercantile Exchange. Hoje é o mercado de futuros mais importante para Eurodólar, câmbio e pecuária. Em 1944, o sistema de Bretton Woods foi implementado numa tentativa de estabilizar o sistema monetário. Em 1970, o sistema de Bretton Woods foi abandonado por várias razões. Em 1971, foi introduzido o Acordo Smithsoniano sobre taxas de câmbio fixas. Em 1972, o Mercado Monetário Internacional (IMM) negociou futuros sobre moedas, moedas e metais preciosos. Em 1973, a CBOE (Chicago Board of Exchange) em primeiro lugar trocou opções de compra quatro anos mais tarde também colocar opções. O Acordo Smithsonian foi abandonado as moedas seguidas flutuantes gerenciados. Em 1975, o CBOT vendeu o primeiro futuro de taxa de juros, o primeiro futuro sem nenhum ativo real subjacente. Em 1978, a bolsa holandesa negociou os primeiros derivados financeiros padronizados. Em 1979, foi implementado o Sistema Monetário Europeu e introduzida a Unidade Monetária Europeia (ECU). Em 1991, foi assinado o Tratado de Maastricht sobre uma moeda comum e uma política económica na Europa. Em 1999, o euro foi introduzido, mas os países ainda usavam o dinheiro de suas antigas moedas, enquanto as taxas de câmbio eram mantidas fixas. Em 2002, o Euro foi introduzido como novo dinheiro na forma de dinheiro. 1.2 Questões Técnicas para Opções de Baunilha Consideramos o modelo de movimento browniano geométrico ds t (rdrf) S t dt sigmas t dw t (1.1) para a taxa de câmbio subjacente cotada em FOR-DOM (foreign-domestic), o que significa que uma unidade de Os custos em moeda estrangeira FOR-DOM unidades da moeda nacional. No caso de EUR-USD com um ponto de. Isso significa que o preço de um EUR é USD. A noção de estrangeiros e domésticos não se refere à localização da entidade negociadora, mas apenas a esta convenção de cotação. Denotamos a taxa de juros externa (contínua) por rf ea taxa de juros interna (contínua) por r d. Em um cenário de equidade, r f representaria uma taxa contínua de dividendos. A volatilidade é denotada por sigma, e Wt é um movimento padrão de Brownian. Os caminhos de amostra são exibidos na Figura 1.1. Consideramos este modelo padrão, não porque ele reflete as propriedades estatísticas da taxa de câmbio (na verdade, não), mas porque é amplamente utilizado na prática e sistemas de front office e serve principalmente como uma ferramenta para comunicar preços em opções de FX . Estes preços são geralmente cotados em termos de volatilidade no sentido deste modelo. Aplicando a regra de Itocirc a ln S t resulta a seguinte solução para o processo S t S t S 0 exp sigma2) t sigmaw t, (1.2) que mostra que S t é log-normalmente distribuída, mais precisamente, ln S t é normal Com média ln S 0 (rdrf 1 2 sigma2) t e variância sigma 2 t. Outros pressupostos do modelo são 16 16 Wystup Figura 1.1: Caminhos simulados de um movimento browniano geométrico. A distribuição da mancha S T no tempo T é log-normal. 1. Não há arbitragem 2. A negociação é sem atrito, sem custos de transação 3. Qualquer posição pode ser tomada a qualquer momento, curta, fração arbitrária, sem restrições de liquidez O payoff para uma opção de baunilha (European put ou call) é dado Por F phi (s TK), (1.3) onde os parâmetros contratuais são a batida K, o tempo de expiração T eo tipo phi, uma variável binária que toma o valor 1 no caso de uma chamada e 1 no caso de uma chamada colocar. O valor x indica a parte positiva de x, ou seja, xmax (0, x) 0 x Valor No modelo de Black-Scholes o valor da recompensa F no instante t se o ponto estiver em x é denotado por v (t, x ) E pode ser calculada como a solução do diferencial parcial de Black-Scholes 17 Equações de FX e Produtos Estruturados 17 equação vtrdv (rdrf) xv x sigma2 x 2 vxx 0, (1,4) v (t, x) F. (1,5 ) Ou equivalentemente (Feynman-Kac-Theorem) como o valor esperado descontado da função de payoff, v (x, K, T, t, sigma, rd, rf, phi) er dtau IEF. (1.6) Esta é a razão pela qual a engenharia financeira básica se preocupa principalmente em resolver equações diferenciais parciais ou expectativas de computação (integração numérica). O resultado é a fórmula de Black-Scholes Nós abreviamos v (x, K, T, t, sigma, rd, rf, phi) phie rdtau fn (phid) KN (phid). (1,7) x: preço atual do tau subjacente T t: tempo até a maturidade f IES T S tx xe (r d r f) tau. Preço a termo da subjacente teta plusmn rdrf sigma plusmn sigma 2 d plusmn ln x K sigmatheta plusmntau sigma tau ln f K plusmn sigma 2 2 tau sigma tau n (t) 1 2pi e 1 2 t2 n (t) N (x) xn (T) dt 1 N (x) A fórmula de Black-Scholes pode ser derivada usando a representação integral da Equação (1.6) ver dtau IEF e rdtau IEphi (STK) (er dtau phi xe (rdrf 1 2 sigma2) tausigma tauy K) N (y) dy. (1.8) O próximo tem que lidar com a parte positiva e então completar o quadrado para obter a fórmula de Black-Scholes. Uma derivação baseada na equação diferencial parcial pode ser feita usando resultados sobre a bem-estudada equação de calor. 18 18 Wystup Uma Nota sobre o Forward O preço forward f é a greve que torna o valor zero do tempo do contrato a termo F S T f (1.9) igual a zero. Segue-se que f IES T xe (r d r f) T, ou seja, o preço a termo é o preço esperado do subjacente no tempo T em uma configuração neutra em termos de risco (drift do movimento browniano geométrico é igual ao custo de carry r d r f). A situação r d gt rf é chamada contango, ea situação r d lt r f é chamada de backwardation. Observe que no modelo de Black-Scholes a classe de curvas de preços a termo é bastante restrita. Por exemplo, não podem ser incluídos efeitos sazonais. Observe que o valor do contrato a termo após o tempo zero é geralmente diferente de zero, e uma vez que uma das contrapartes é sempre curta, pode haver risco de inadimplência da parte curta. Um contrato de futuros previne este caso perigoso: é basicamente um contrato a termo, mas as contrapartes têm de uma conta de margem para garantir que a quantidade de dinheiro ou mercadoria devida não exceda um limite especificado gregos gregos são derivados da função de valor em relação ao modelo E parâmetros do contrato. Eles são uma informação importante para os comerciantes e tornaram-se informações padrão fornecidas pelos sistemas de front-office. Mais detalhes sobre os gregos e as relações entre os gregos são apresentados em Hakala e Wystup 50 ou Reiss e Wystup 84. Para opções de baunilha nós lista alguns deles agora. (Ponto) Delta. V x phie rf tau N (phid) (1.10) Avançar Delta. Delta sem deriva. V f phie r dtau N (phid) (1.11) phin (phid) (1.12) Gamma. (X) x 2 xsigma tau (1.13) 19 Opções de FX e Produtos Estruturados 19 Velocidade. 3 v x 3 e r f tau n (d) x 2 sigma tau () d sigma tau 1 (1,14) Theta. V t e r f tau n (d) xsigma 2 tau phir f xe r f tau N (phid) r d Ke rdtau N (phid) (1.15) Charm. 2 v x tau phir f e r f tau N (phid) phie r f tau n (d) 2 (r d r f) tau d sigma tau 2tausigma tau (1.16) Cor. 3 v x 2 tau e r f tau n (d) 2xtausigma tau 2r f tau (r d r f) tau d sigma tau 2tausigma d tau (1.17) Vega. V sigma xe rf tau taun (d) (1.18) Volga. 2 v sigma 2 xe rf tau taun (d) d d sigma (1.19) Volga também é chamado às vezes vomma ou volgamma. Vanna. 2 v sigma x e r f tau n (d) d sigma (1.20) Rho. V r d phiktaue rdtau N (phid) (1.21) v r phixtaue r f tau N (phid) (1.22) 20 20 Wystup Dual Delta. Dual Gamma. V K phie r dtau N (phid) (1.23) 2 v e r dtau n (d) K 2 Ksigma tau (1.24) Dual Theta. V T (1.25) Identidades A paridade de put-call é a relação d plusmn d (1.26) sigma sigma d plusmn tau (1.27) rd sigma d plusmn tau (1.28) rf sigma xe rf tau n (d) Ke rdtau n (D). (1.29) N (phid) IP phis T phik (1.30) N (phid) IP phis T phi f 2 (1.31) Kv (x, K, T, t, sigma, rd, K, T, t, sigma, rd, rf, 1) xe rf tau Ke r dtau, (1.32) que é apenas uma maneira mais complicada de escrever a equação trivial xx x. A paridade delta put-call é v (x, K, T, t, sigma, rd, rf, 1) xv (x, K, T, t, sigma, rd, rf, 1) x e r f tau. (1.33) Em particular, aprendemos que o valor absoluto de um delta de entrada e de um delta de chamada não é exatamente o somatório de um, mas apenas de um número positivo e r f tau. Eles somam até um aproximadamente se quer o tempo de expiração tau é curto ou se a taxa de juros estrangeira rf é perto de zero. Considerando que a escolha K f produz valores idênticos para call e put, buscamos o strike deltasmétrico que produz deltas absolutamente idênticos (spot, forward ou driftless). Esta condição implica d 0 e, portanto, fe sigma2 2 T, (1.34), em cujo caso o delta absoluto é erf tau 2. Em particular, aprendemos que sempre gt f, ou seja, não pode haver uma put e uma chamada com valores idênticos E deltas. Note-se que a greve é ​​geralmente escolhida como a greve do meio ao negociar um straddle ou uma borboleta. Da mesma forma, a greve dual-delta-simétrica circK fe sigma2 2 T pode ser derivada da condição d Relações com base na homogeneidade Podemos desejar medir o valor do subjacente em uma unidade diferente. Isso obviamente afetará a fórmula de precificação de opções da seguinte forma. (1.35) Diferenciando ambos os lados com respeito (a) (a) Para a e, em seguida, estabelecendo um 1 rendimentos v xv x Kv K. (1.36) A comparação dos coeficientes de x e K nas Equações (1.7) e (1.36) leva a resultados sugestivos para o delta vx e delta duplo v. A homogeneidade é a razão por trás da simplicidade das fórmulas delta, cuja computação tediosa pode ser salva dessa maneira. Podemos realizar uma computação similar para os parâmetros afetados pelo tempo e obter a equação óbvia v (x, K, T, t, sigma, rd, rf, phi) v (x, K, T a, ta, asigma, ar d , Ar f, phi) para todos a gt 0. (1.37) Diferenciar ambos os lados em relação a e, em seguida, definir a 1 produz 0 tau t sigmav sigma rdv rd rfv rf. (1.38) É claro que isso também pode ser verificado por computação direta. O uso geral de tais equações é gerar benchmarks de verificação dupla ao computar gregos. Esses métodos de homogeneidade podem ser facilmente estendidos a outras opções mais complexas. Por meio da simetria put-call, entende-se a relação (ver 6, 7,16 e 19) v (x, K, T, t, sigma, rd, rf, Sigma, rd, rf, 1). (1.39) 22 22 Wystup A greve do put eo strike da chamada resultam em uma média geométrica igual à frente f. O forward pode ser interpretado como um espelho geométrico que reflete uma chamada em um certo número de puts. Note que para as opções de dinheiro (K f) a simetria put-call coincide com o caso especial da paridade put-call onde a chamada e a put têm o mesmo valor. A computação direta mostra que as taxas de simetria v v tauv (1.40) r d r f para as opções de baunilha. Essa relação, de fato, é válida para todas as opções européias e uma ampla classe de opções dependentes de trajetórias, como mostrado em 84. Pode-se verificar diretamente a relação da simetria estrangeira-doméstica 1 xv (x, K, T, t, sigma, rd , Rf, phi) Kv (1 x, 1K, T, t, sigma, rf, rd, phi). (1.41) Essa igualdade pode ser vista como uma das faces da simetria put-call. A razão é que o valor de uma opção pode ser computado tanto em um cenário doméstico, bem como em um cenário estrangeiro. Consideramos o exemplo de modelagem S t da taxa de câmbio da EUR / USD. Em Nova York, a opção de compra (STK) custa v (x, K, T, t, sigma, r usd, r eur, 1) USD e portanto v (x, K, T, t, sigma, r usd, r Eur, 1) x () 1. EUR. Esta opção de compra de EUR também pode ser vista como uma opção de USD-put com pagamento K 1 KST Esta opção custa Kv (1, 1, T, t, sigma, rx K eur, r usd, 1) T e 1 S t têm a mesma volatilidade. Naturalmente, o valor de Nova York eo valor de Frankfurt deve concordar, o que leva a (1.41). Nós também aprenderemos mais tarde, que esta simetria é apenas um possível resultado baseado na mudança de numeraire Cotação Cotação da equação de taxa de câmbio subjacente (1.1) é um modelo para a taxa de câmbio. A citação é uma questão permanentemente confusa, por isso vamos esclarecer isso aqui. A taxa de câmbio significa quanto da moeda nacional são necessários para comprar uma unidade de moeda estrangeira. Por exemplo, se considerarmos o EURUSD como uma taxa de câmbio, então a cotação padrão é EUR-USD, onde USD é a moeda nacional e EUR é a moeda estrangeira. O termo doméstico não está de forma alguma relacionado com a localização do comerciante ou de qualquer país. Significa meramente a moeda numeraire. Os termos doméstico, numeraire ou moeda base são sinônimos como são estrangeiros e subjacentes. Ao longo deste livro denotamos com o slash () o par de moedas e com um traço (-) a cotação. A barra () não significa uma divisão. Por exemplo, o EURUSD também pode ser cotado em EUR-USD, o que significa quantos USD são necessários para comprar um EUR, ou em USD-EUR, o que significa quantos euros são necessários para comprar um USD. Existem certas cotações padrão de mercado listadas na Tabela 1.1. 23 Opções de câmbio e produtos estruturados 23 par de moedas cotação padrão cotação de cotação GBPUSD GBP-CHF EURUSD GBP-CHF EURUSD EUR-JPY GBP-CHF EURJPY EUR-JPY EURCHF EUR-JPY JPY USD-JPY JPY USD-CHF Tabela 1.1: Mercado padrão Cotação dos principais pares de moedas com preços spot amostra Língua de negociação Language Llamamos a um milhão de dólares, um bilhão por quintal. Isto é porque um bilhão é chamado milliarde em francês, alemão e outras línguas. Para a Libra Esterlina, um milhão também é freqüentemente chamado de quid. Alguns pares de moeda têm nomes. Por exemplo, o GBPUSD é chamado de cabo, porque a informação de taxa de câmbio costumava ser enviada através de um cabo no oceano Atlântico entre a América ea Inglaterra. O EURJPY é chamado de cruz, porque é a taxa cruzada do USDJPY negociado mais líquido e da EURUSD. Determinadas moedas também têm nomes, p. O Dólar da Nova Zelândia NZD é chamado de kiwi, o Dólar Australiano AUD é chamado Aussie, as moedas escandinavas DKR, NOK e SEK são chamados Scandies. As taxas de câmbio são geralmente cotadas até cinco valores relevantes, e. Em EUR-USD, podemos observar uma citação de O último dígito 5 é chamado de pip, o dígito médio 3 é chamado de grande figura, como as taxas de câmbio são freqüentemente exibidos nos andares de negociação ea grande figura, que é exibido em maior tamanho, É a informação mais relevante. Os dígitos deixados à figura grande são sabidos de qualquer maneira, os pips à direita da figura grande são frequentemente negligenciáveis. Para deixar claro, uma subida de USD-JPY por 20 pips será e um aumento de 2 grandes números será Cotação de preços de opção Valores e preços de opções de baunilha pode ser citado nas seis maneiras explicadas na Tabela 1.2. 24 24 Nome da Wystup valor do símbolo em unidades de exemplo caixa doméstico d DOM 29.148 USD moeda estrangeira f PARA 24.290 EUR doméstico d DOM por unidade de DOM USD estrangeiro f FOR por unidade de FOR EUR pips nacionais d pips DOM por unidade de FOR USD pips por EUR pips estrangeiros fp FOR por unidade de DOM EUR pips por USD Tabela 1.2: Tipos de cotação de mercado padrão para valores de opção. No exemplo que tomamos FOREUR, DOMUSD, S 0. Rd 3,0, rf 2,5, sigma 10, K. T 1 ano, phi 1 (chamada), notional 1, 000, 000 EUR 1, 250, 000 USD. Para os pips, a cotação USD pips por EUR também é por vezes referida como USD por 1 EUR. Da mesma forma, os pips EUR por USD também podem ser cotados como EUR por 1 USD. A fórmula de Black-Scholes cita d pips. Os outros podem ser calculados usando a seguinte instrução. D pips 1 S 0 S 0 1 f K S d 0 S f pips 0 K d pips (1,42) Delta e Convenção Premium O delta spot de uma opção europeia sem prémio é bem conhecido. Ele será chamado raw delta delta crua agora. Pode ser cotado em qualquer uma das duas moedas envolvidas. O delta é usado para comprar ou vender mancha no montante correspondente, a fim de proteger a opção até a primeira ordem. Por uma questão de consistência, o prémio deve ser incorporado na cobertura delta, uma vez que um prémio em moeda estrangeira já cobrirá parte do risco delta da opção. Para deixar isso claro, consideremos EUR-USD. Na teoria padrão de arbitragem, v (x) denota o valor ou o prêmio em USD de uma opção com 1 EUR nocional, se o spot estiver em x, eo delta v x bruto denota o número de EUR a comprar para a cobertura delta. Portanto, xv x é o número de USD para vender. Se agora o prémio for pago em EUR em vez de em USD, então já temos vx EUR, eo número de euros para comprar tem de ser reduzido por este montante, ou seja, se EUR é a moeda premium, precisamos comprar vxvx EUR para A cobertura delta ou vender equivalentemente xv xv USD. 25 FX Opções e Produtos Estruturados 25 Toda a história de cotações de FX torna-se geralmente uma bagunça, porque precisamos primeiro classificar qual moeda é doméstica, que é estrangeira, qual é a moeda nocional da opção e qual é a moeda premium. Infelizmente isso não é simétrico, uma vez que a contraparte pode ter outra noção de moeda nacional para um determinado par de moedas. Assim, no mercado interbancário profissional há uma noção de delta por par de moedas. Normalmente é o delta do lado esquerdo da tela Fenics se a opção for trocada no prêmio do lado esquerdo, que normalmente é o delta padrão e do lado direito se ele for negociado com o prêmio do lado direito, p. EURUSD lhs, USDJPY lhs, EURJPY lhs, AUDUSD rhs, etc. Desde OTM opções são negociadas a maior parte do tempo a diferença não é enorme e, portanto, não cria um enorme risco local. Além disso, o padrão delta por par de moedas de um lado da mão esquerda delta em Fenics para a maioria dos casos é usado para citar opções de volatilidade. Isso deve ser especificado por moeda. Esta noção interbancária padrão deve ser adaptada ao verdadeiro delta-risk do banco para um sistema automatizado de negociação. Para moedas em que a moeda livre de risco do banco é a moeda base da moeda, é claro que o delta é o delta bruto da opção e para o prémio de risco este prémio deve ser incluído. In the opposite case the risky premium and the market value must be taken into account for the base currency premium, such that these offset each other. And for premium in underlying currency of the contract the market-value needs to be taken into account. In that way the delta hedge is invariant with respect to the risky currency notion of the bank, e. g. the delta is the same for a USD-based bank and a EUR-based bank. Example We consider two examples in Table 1.3 and 1.4 to compare the various versions of deltas that are used in practice. delta ccy prem ccy Fenics formula delta EUR EUR lhs delta raw P EUR USD rhs delta raw USD EUR rhs flip F4 (delta raw P )SK USD USD lhs flip F4 (delta raw )SK Table 1.3: 1y EUR call USD put strike K for a EUR based bank. Market data: spot S . volatility sigma 12, EUR rate r f 3.96, USD rate r d 3.57. The raw delta is 49.15EUR and the value is 4.427EUR. 26 26 Wystup delta ccy prem ccy Fenics formula delta EUR EUR lhs delta raw P EUR USD rhs delta raw USD EUR rhs flip F4 (delta raw P )SK USD USD lhs flip F4 delta raw SK Table 1.4: 1y call EUR call USD put strike K for a EUR based bank. Market data: spot S . volatility sigma 12, EUR rate r f 3.96, USD rate r d 3.57. The raw delta is 94.82EUR and the value is 21.88EUR Strike in Terms of Delta Since v x phie r f tau N (phid ) we can retrieve the strike as K x exp . (1.44) Volatility in Terms of Delta The mapping sigma phie r f tau N (phid ) is not one-to-one. The two solutions are given by sigma plusmn 1 tau(d d ). (1.45) tau Thus using just the delta to retrieve the volatility of an option is not advisable Volatility and Delta for a Given Strike The determination of the volatility and the delta for a given strike is an iterative process involving the determination of the delta for the option using at-the-money volatilities in a first step and then using the determined volatility to re determine the delta and to continuously iterate the delta and volatility until the volatility does not change more than 0.001 between iterations. More precisely, one can perform the following algorithm. Let the given strike be K. 1. Choose sigma 0 at-the-money volatility from the volatility matrix. 2. Calculate n1 (Call(K, sigma n )). 3. Take sigma n1 sigma( n1 ) from the volatility matrix, possibly via a suitable interpolation. 4. If sigma n1 sigma n lt , then quit, otherwise continue with step 2. 27 FX Options and Structured Products 27 In order to prove the convergence of this algorithm we need to establish convergence of the recursion n1 e r f tau N (d ( n )) (1.46) ( e r f ln(sk) tau (rd r f 1 ) 2 N sigma2 ( n ))tau sigma( n ) tau for sufficiently large sigma( n ) and a sufficiently smooth volatility smile surface. We must show that the sequence of these n converges to a fixed point 0, 1 with a fixed volatility sigma sigma( ). This proof has been carried out in 15 and works like this. We consider the derivative The term n1 e r f tau n(d ( n )) d ( n ) n sigma( n ) sigma( n ). (1.47) n e r f tau n(d ( n )) d ( n ) sigma( n ) converges rapidly to zero for very small and very large spots, being an argument of the standard normal density n. For sufficiently large sigma( n ) and a sufficiently smooth volatility surface in the sense that n sigma( n ) is sufficiently small, we obtain sigma( n ) n q lt 1. (1.48) Thus for any two values (1) n1, (2) n1, a continuously differentiable smile surface we obtain (1) n1 (2) n1 lt q (1) n (2) n, (1.49) due to the mean value theorem. Hence the sequence n is a contraction in the sense of the fixed point theorem of Banach. This implies that the sequence converges to a unique fixed point in 0, 1, which is given by sigma sigma( ) Greeks in Terms of Deltas In Foreign Exchange markets the moneyness of vanilla options is always expressed in terms of deltas and prices are quoted in terms of volatility. This makes a ten-delta call a financial object as such independent of spot and strike. This method and the quotation in volatility makes objects and prices transparent in a very intelligent and user-friendly way. At this point we list the Greeks in terms of deltas instead of spot and strike. Let us introduce the quantities phie r f tau N (phid ) spot delta, (1.50) phie r dtau N (phid ) dual delta, (1.51) 28 28 Wystup which we assume to be given. From these we can retrieve Interpretation of Dual Delta d phin 1 (phie r f tau ), (1.52) d phin 1 ( phie r dtau ). (1.53) The dual delta introduced in (1.23) as the sensitivity with respect to strike has another - more practical - interpretation in a foreign exchange setup. We have seen in Section that the domestic value v(x, K, tau, sigma, r d, r f, phi) (1.54) corresponds to a foreign value v( 1 x, 1 K, tau, sigma, r f, r d, phi) (1.55) up to an adjustment of the nominal amount by the factor xk. From a foreign viewpoint the delta is thus given by ( ) phie rdtau N phi ln( K ) (r x f r d sigma2 tau) sigma tau ( phie rdtau N phi ln( x ) (r K d r f 1 ) 2 sigma2 tau) sigma tau , (1.56) which means the dual delta is the delta from the foreign viewpoint. We will see below that foreign rho, vega and gamma do not require to know the dual delta. We will now state the Greeks in terms of x, . r d, r f, tau, phi. Value. (Spot) Delta. v(x, . r d, r f, tau, phi) x x e r f tau n(d ) e r dtau n(d ) (1.57) Forward Delta. v f v x (1.58) e (r f r d )tau (1.59) 29 FX Options and Structured Products 29 Gamma. 2 v e r f tau n(d ) x 2 x(d d ) (1.60) Taking a trader s gamma (change of delta if spot moves by 1) additionally removes the spot dependence, because Gamma trader x 2 v e r f tau n(d ) 100 x 2 100(d d ) (1.61) Speed. 3 v e r f tau n(d ) x 3 x 2 (d d ) (2d 2 d ) (1.62) Theta. 1 v x t e r f tau n(d )(d d ) 2tau e r f tau n(d ) r f r d e r dtau n(d ) (1.63) Charm. Color. Vega. Volga. 2 v x tau 3 v x 2 tau phir f e r f tau N (phid ) phie r f tau n(d ) 2(r d r f )tau d (d d ) 2tau(d d ) (1.64) e r f tau n(d ) 2r f tau (r d r f )tau d (d d ) d 2xtau(d d ) 2tau(d d ) (1.65) v sigma xe r f tau taun(d ) (1.66) 2 v sigma 2 xe r f tau taun(d ) d d d d (1.67) 30 30 Wystup Vanna. 2 v sigma x e r f tau taud n(d ) (1.68) d d Rho. Dual Delta. v e rf tau n(d ) xtau (1.69) r d e r dtau n(d ) v xtau (1.70) r f v K (1.71) Dual Gamma. K 2 2 v K 2 x 2 2 v x 2 (1.72) Dual Theta. v T v t (1.73) As an important example we consider vega. Vega in Terms of Delta The mapping v sigma xe r f tau taun(n 1 (e r f tau )) is important for trading vanilla options. Observe that this function does not depend on r d or sigma, just on r f. Quoting vega in foreign will additionally remove the spot dependence. This means that for a moderately stable foreign term structure curve, traders will be able to use a moderately stable vega matrix. For r f 3 the vega matrix is presented in Table Volatility Volatility is the annualized standard deviation of the log-returns. It is the crucial input parameter to determine the value of an option. Hence, the crucial question is where to derive the volatility from. If no active option market is present, the only source of information is estimating the historic volatility. This would give some clue about the past. In liquid currency 31 FX Options and Structured Products 31 Mat 50 45 40 35 30 25 20 15 10 5 1D W W M M M M M Y Y Y Table 1.5: Vega in terms of Delta for the standard maturity labels and various deltas. It shows that one can vega hedge a long 9M 35 delta call with 4 short 1M 20 delta puts. pairs volatility is often a traded quantity on its own, which is quoted by traders, brokers and real-time data pages. These quotes reflect views of market participants about the future. Since volatility normally does not stay constant, option traders are highly concerned with hedging their volatility exposure. Hedging vanilla options vega is comparatively easy, because vanilla options have convex payoffs, whence the vega is always positive, i. e. the higher the volatility, the higher the price. Let us take for example a EUR-USD market with spot. USD - and EUR rate at 2.5. A 3-month at-the-money call with 1 million EUR notional would cost 29,000 USD at at volatility of 12. If the volatility now drops to a value of 8, then the value of the call would be only 19,000 USD. This monotone dependence is not guaranteed for non-convex payoffs as we illustrate in Figure Historic Volatility We briefly describe how to compute the historic volatility of a time series S 0, S 1. S N (1.74) 32 32 Wystup Figure 1.2: Dependence of a vanilla call and a reverse knock-out call on volatility. The vanilla value is monotone in the volatility, whereas the barrier value is not. The reason is that as the spot gets closer to the upper knock-out barrier, an increasing volatility would increase the chance of knock-out and hence decrease the value. of daily data. First, we create the sequence of log-returns Then, we compute the average log-return r i ln S i S i 1, i 1. N. (1.75) r 1 N N r i, (1.76) i1 33 FX Options and Structured Products 33 their variance and their standard deviation circsigma 2 1 N 1 N (r i r) 2, (1.77) i1 circsigma 1 N (r i r) N 1 2. (1.78) The annualized standard deviation, which is the volatility, is then given by circsigma a B N (r i r) N 1 2, (1.79) where the annualization factor B is given by i1 i1 B N d, (1.80) k and k denotes the number of calendar days within the time series and d denotes the number of calendar days per year. The is done to press the trading days into the calendar days. Assuming normally distributed log-returns, we know that circsigma 2 is chi 2 - distributed. Therefore, given a confidence level of p and a corresponding error probability alpha 1 p, the p-confidence interval is given by N 1 N 1 circsigma a, circsigma chi 2 a, (1.81) N 11 chi 2 alpha N 1 alpha 2 2 where chi 2 np denotes the p-quantile of a chi 2 - distribution 1 with n degrees of freedom. As an example let us take the 256 ECB-fixings of EUR-USD from 4 March 2003 to 3 March 2004 displayed in Figure 1.3. We get N 255 log-returns. Taking k d 365, we obtain r 1 N r i . N i1 circsigma a B N (r i r) N 1 2 10.85, i1 and a 95 confidence interval of 9.99, 11.89. 1 values and quantiles of the chi 2 - distribution and other distributions can be computed on the internet, e. g. at 34 34 Wystup EURUSD Fixings ECB Exchange Rate 403 4403 5403 6403 7403 8403 9403 Date 10403 11403 12403 1404 2404 Figure 1.3: ECB-fixings of EUR-USD from 4 March 2003 to 3 March 2004 and the line of average growth Historic Correlation As in the preceding section we briefly describe how to compute the historic correlation of two time series x 0, x 1. x N, y 0, y 1. y N, of daily data. First, we create the sequences of log-returns Then, we compute the average log-returns X i ln x i x i 1, i 1. N, Y i ln y i y i 1, i 1. N. (1.82) X 1 N 1 N N X i, i1 N Y i, (1.83) i1 35 FX Options and Structured Products 35 their variances and covariance circsigma X 2 circsigma Y 2 circsigma XY and their standard deviations circsigma X circsigma Y 1 N 1 1 N 1 1 N 1 N (X i X) 2, (1.84) i1 N (Y i )2, (1.85) i1 N (X i X)(Y i ), (1.86) i1 1 N (X i N 1 X) 2, (1.87) i1 1 N (Y i N 1 )2. (1.88) i1 The estimate for the correlation of the log-returns is given by circrho circsigma XY circsigma X circsigma Y. (1.89) This correlation estimate is often not very stable, but on the other hand, often the only available information. More recent work by Jaumlkel 37 treats robust estimation of correlation. We will revisit FX correlation risk in Section Volatility Smile The Black-Scholes model assumes a constant volatility throughout. However, market prices of traded options imply different volatilities for different maturities and different deltas. We start with some technical issues how to imply the volatility from vanilla options. Retrieving the Volatility from Vanilla Options Given the value of an option. Recall the Black-Scholes formula in Equation (1.7). We now look at the function v(sigma), whose derivative (vega) is The function sigma v(sigma) is v (sigma) xe r f T T n(d ). (1.90) 36 36 Wystup 1. strictly increasing, 2. concave up for sigma 0, 2 ln F ln K T ), 3. concave down for sigma ( 2 ln F ln K T, ) and also satisfies v(0) phi(xe r f T Ke r dt ) , (1.91) v(, phi 1) xe r f T, (1.92) v(sigma , phi 1) Ke r dt, (1.93) v (0) xe r f T T 2piII , (1.94) In particular the mapping sigma v(sigma) is invertible. However, the starting guess for employing Newton s method should be chosen with care, because the mapping sigma v(sigma) has a saddle point at ( ) 2 T ln F K, phie r dt F N phi 2T ln FK KN phi 2T ln KF , (1.95) as illustrated in Figure 1.4. To ensure convergence of Newton s method, we are advised to use initial guesses for sigma on the same side of the saddle point as the desired implied volatility. The danger is that a large initial guess could lead to a negative successive guess for sigma. Therefore one should start with small initial guesses at or below the saddle point. For at-the-money options, the saddle point is degenerate for a zero volatility and small volatilities serve as good initial guesses. Visual Basic Source Code Function VanillaVolRetriever(spot As Double, rd As Double, rf As Double, strike As Double, T As Double, type As Integer, GivenValue As Double) As Double Dim func As Double Dim dfunc As Double Dim maxit As Integer maximum number of iterations Dim j As Integer Dim s As Double first check if a volatility exists, otherwise set result to zero If GivenValue lt Application. Max (0, type (spot Exp(-rf T) - strike Exp(-rd T))) Or (type 1 And GivenValue gt spot Exp(-rf T)) Or (type -1 And GivenValue gt strike Exp(-rd T)) Then 37 FX Options and Structured Products 37 Figure 1.4: Value of a European call in terms of volatility with parameters x 1, K 0.9, T 1, r d 6, r f 5. The saddle point is at sigma 48. VanillaVolRetriever 0 Else there exists a volatility yielding the given value, now use Newton s method: the mapping vol to value has a saddle point. First compute this saddle point: saddle Sqr(2 T Abs(Log(spot strike) (rd - rf) T))Your Search: 1 eBooks Search Engine We are pleased to introduce our wonderful site where collected the most remarkable books of the best authors. Somente em um lugar juntos os melhores best-sellers para você, queridos amigos. Você pode desenvolver seus conhecimentos e habilidades, baixando nossos livros e guias. Temos certeza que você vai desfrutar do nosso grande projeto e vai fazer a sua vida um pouco melhor. 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FX Options and Structured Products 1 FX Options and Structured Products Uwe Wystup 7 April 2006 3 Contents 0 Preface Scope of this Book The Readership About the Author Acknowledgments Foreign Exchange Options A Journey through the History Of Options Technical Issues for Vanilla Options Value A Note on the Forward Greeks Identities Homogeneity based Relationships Quotation Strike in Terms of Delta Volatility in Terms of Delta Volatility and Delta for a Given Strike Greeks in Terms of Deltas Volatility Historic Volatility Historic Correlation Volatility Smile At-The-Money Volatility Interpolation Volatility Smile Conventions At-The-Money Definition Interpolation of the Volatility on Maturity Pillars Interpolation of the Volatility Spread between Maturity Pillars Volatility Sources Volatility Cones Stochastic Volatility 4 4 Wystup Exercises Basic Strategies containing Vanilla Options Call and Put Spread Risk Reversal Risk Reversal Flip Straddle Strangle Butterfly Se agull Exercises First Generation Exotics Barrier Options Digital Options, Touch Options and Rebates Compound and Instalment Asian Options Lookback Options Forward Start, Ratchet and Cliquet Options Power Options Quanto Options Exercises Second Generation Exotics Corridors Faders Exotic Barrier Options Pay-Later Options Step up and Step down Options Spread and Exchange Options Baskets Best-of and Worst-of Options Options and Forwards on the Harmonic Average Variance and Volatility Swaps Exercises Structured Products Forward Products Outright Forward Participating Forward Fade-In Forward Knock-Out Forward Shark Forward Fader Shark Forward 5 FX Options and Structured Products Butterfly Forward Range Forward Range Accrual Forward Accumulative Forward Boomerang Forward Amortizing Forward Auto-Renewal Forward Double Shark Forward Forward Start Chooser Forward Free Style Forward Boosted SpotForward Time Option Exercises Series of Strategies Shark Forward Series Collar Extra Series Exercises D eposits and Loans Dual Currency DepositLoan Performance Linked Deposits Tunnel DepositLoan Corridor DepositLoan Turbo DepositLoan Tower DepositLoan Exercises Interest Rate and Cross Currency Swaps Cross Currency Swap Hanseatic Swap Turbo Cross Currency Swap Buffered Cross Currency Swap Flip Swap Corridor Swap Double-No-Touch linked Swap Range Reset Swap Basket Spread Swap Exercises Participation Notes Gold Participation Note Basket-linked Note Issuer Swap Moving Strike Turbo Spot Unlimited 6 6 Wystup 2.6 Hybrid FX Products Practical Matters The Traders Rule of Thumb Cost of Vanna and Volga Observations Consistency check Abbreviations for First Generation Exotics Adjustment Factor Volatility for Risk Reversals, Butterflies and Theoretical Value Pricing Barrier Options Pricing Double Barrier Options Pricing Double-No-Touch Options Pricing European Style Options No-Touch Probability The Cost of Trading and its Implication on the Market Price of Onetouch Options Example Further Application s Exercises Bid Ask Spreads One Touch Spreads Vanilla Spreads Spreads for First Generation Exotics Minimal Bid Ask Spread Bid Ask Prices Exercises Settlement The Black-Scholes Model for the Actual Spot Cash Settlement Delivery Settlement Options with Deferred Delivery Exercises On the Cost of Delayed Fixing Announcements The Currency Fixing of the European Central Bank Model and Payoff Analysis Procedure Error Estimation Analysis of EUR-USD Conclusion 7 FX Options and Structured Products 7 4 Hedge Accounting under IAS Introduction Financial Instruments Overview General Definition Financial Assets Financial Liabilities Offsetting of Financial Assets and Financial Liabilities Equity Instruments Compound Financial Instruments Derivatives Embedded Derivatives Classification of Financial Instruments Evaluation of Financial Instruments Initial Recognition Initial Measurement Subsequent Measurement Derecognition Hedge Accounting Overview Types of Hedges Basic Requirements Stopping Hedge Accou nting Methods for Testing Hedge Effectiveness Fair Value Hedge Cash Flow Hedge Testing for Effectiveness - A Case Study of the Forward Plus Simulation of Exchange Rates Calculation of the Forward Plus Value Calculation of the Forward Rates Calculation of the Forecast Transaction s Value Dollar-Offset Ratio - Prospective Test for Effectiveness Variance Reduction Measure - Prospective Test for Effectiveness Regression Analysis - Prospective Test for Effectiveness Result Retrospective Test for Effectiveness Conclusion Relevant Original Sources for Accounting Standards Exercises 8 8 Wystup 5 Foreign Exchange Markets A Tour through the Market Statement by GFI Group (Fenics), 25 October Interview with ICY Software, 14 October Interview with Bloomberg, 12 October Interview with Murex, 8 November Interview with SuperDerivatives, 17 October Interview with Lucht Probst Associates, 27 February Software and System Requirements Fenics Position Keeping Pricing Straight Through Processing Disclaimers Trading and Sales Proprietary Trading Sales-Driven Trading Inter Bank Sales Branch Sales Institutional Sales Corporate Sales Private Banking Listed FX Options Trading Floor Joke 9 Chapter 0 Preface 0.1 Scope of this Book Treasury management of international corporates involves dealing with cash flows in different currencies. Therefore the natural service of an investment bank consists of a variety of money market and foreign exchange products. This book explains the most popular products and strategies with a focus on everything beyond vanilla options. It explains all the FX options, common structures and tailor-made solutions in examples with a special focus on the application with views from traders and sales as well as from a corporate client perspective. It contains actually traded deals with corresponding motivations explaining why the structures have been traded. This way the reader gets a feeling how to build new structures to suit clients needs. The exercises are meant to practice the material. Several of them are actually difficult to solve and can serve as incentives to further research and testing. Solutions to the exercises are not part of this book, however they will be published on the web page of the book, 0.2 The Readership Prerequisite is some basic knowledge of FX markets as for example taken from the Book Foreign Exchange Primer by Shami Shamah, Wiley 2003, see 90. The target readers are Graduate students and Faculty of Financial Engineering Programs, who can use this book as a textbook for a course named structured products or exotic currency options. 9 10 10 Wystup Traders, Trainee Structurers, Product Developers, Sales and Quants with interest in the FX product line. For them it can serve as a source of ideas and as well as a reference guide. Treasurers of corporates interested in managing their books. With this book at hand they can structure their solutions themselves. The readers more interested in the quantitative and modeling aspects are recommended to read Foreign Exchange Risk by J. Hakala and U. Wystup, Risk Publications, London, 2002, see 50. This book explains several exotic FX options with a special focus on the underlying models and mathematics, but does not contain any structures or corporate clients or investors view. 0.3 About the Author Figure 1: Uwe Wystup, professor of Quantitative Finance at HfB Business School of Finance and Management in Frankfurt, Germany. Uwe Wystup is also CEO of MathFinance AG, a global network of quants specializing in Quantitative Finance, Exotic Options advisory and Front Office Software Production. Previously he was a Financial Engineer and Structurer in the FX Options Trading Team at Commerzbank. Before that he worked for Deutsche Bank, Citibank, UBS and Sal. Oppenheim jr. amp Cie. He is founder and manager of the web site MathFinance. de and the MathFinance Newsletter. Uwe holds a PhD in mathematical finance from Carnegie Mellon University. He also lectures on mathematical finance for Goethe University Frankfurt, organizes the Frankfurt MathFinance Colloquium and is founding director of the Frankfurt MathFinance Institute. He has given several seminars on exotic options, computational finance and volatility modeling. His area of specialization are the quantitative aspects and the design of structured products of foreign 11 FX Options and Structured Products 11 exchange markets. He published a book on Foreign Exchange Risk and articles in Finance and Stochastics and the Journal of Derivatives. Uwe has given many presentations at both universities and banks around the world. Further information on his curriculum vitae and a detailed publication list is available at 0.4 Acknowledgments I would like to thank my former colleagues on the trading floor, most of all Gustave Rieunier, Behnouch Mostachfi, Noel Speake, Roman Stauss, Tamaacutes Korchmaacuteros, Michael Braun, Andreas Weber, Tino Senge, Juumlrgen Hakala, and all my colleagues and co-authors, specially Christoph Becker, Susanne Griebsch, Christoph Kuumlhn, Sebastian Krug, Marion Linck, Wolfgang Schmidt and Robert Tompkins. Chris Swain, Rachael Wilkie and many others of Wiley publications deserve respect as they were dealing with my rather slow speed in completing this book. Nicole van de Locht and Choon Peng Toh deserve a medal for serious detailed proof reading. 13 Chapter 1 Foreign Exchange Options FX Structured Products are tailor-made linear combinations of FX Options including both vanilla and exotic options. We recommend the book by Shamah 90 as a source to learn about FX Markets with a focus on market conventions, spot, forward and swap contracts, vanilla options. For pricing and modeling of exotic FX options we suggest Hakala and Wystup 50 or Lipton 71 as useful companions to this book. The market for structured products is restricted to the market of the necessary ingredients. Hence, typically there are mostly structured products traded the currency pairs that can be formed between USD, JPY, EUR, CHF, GBP, CAD and AUD. In this chapter we start with a brief history of options, followed by a technical section on vanilla options and volatility, and deal with commonly used linear combinations of vanilla options. Then we will illustrated the most important ingredients for FX structured products: the first and second generation exotics. 1.1 A Journey through the History Of Options The very first options and futures were traded in ancient Greece, when olives were sold before they had reached ripeness. Thereafter the market evolved in the following way. 16th century Ever since the 15th century tulips, which were liked for their exotic appearance, were grown in Turkey. The head of the royal medical gardens in Vienna, Austria, was the first to cultivate those Turkish tulips successfully in Europe. When he fled to Holland because of religious persecution, he took the bulbs along. As the new head of the botanical gardens of Leiden, Netherlands, he cultivated several new strains. It was from these gardens that avaricious traders stole the bulbs to commercialize them, because tulips were a great status symbol. 17th century The first futures on tulips were traded in As of 1634, people could 13 14 14 Wystup buy special tulip strains by the weight of their bulbs, for the bulbs the same value was chosen as for gold. Along with the regular trading, speculators entered the market and the prices skyrocketed. A bulb of the strain Semper Octavian was worth two wagonloads of wheat, four loads of rye, four fat oxen, eight fat swine, twelve fat sheep, two hogsheads of wine, four barrels of beer, two barrels of butter, 1,000 pounds of cheese, one marriage bed with linen and one sizable wagon. People left their families, sold all their belongings, and even borrowed money to become tulip traders. When in 1637, this supposedly risk-free market crashed, traders as well as private individuals went bankrupt. The government prohibited speculative trading the period became famous as Tulipmania. 18th century In 1728, the Royal West-Indian and Guinea Company, the monopolist in trading with the Caribbean Islands and the African coast issued the first stock options. Those were options on the purchase of the French Island of Ste. Croix, on which sugar plantings were planned. The project was realized in 1733 and paper stocks were issued in Along with the stock, people purchased a relative share of the island and the valuables, as well as the privileges and the rights of the company. 19th century In 1848, 82 businessmen founded the Chicago Board of Trade (CBOT). Today it is the biggest and oldest futures market in the entire world. Most written documents were lost in the great fire of 1871, however, it is commonly believed that the first standardized futures were traded as of CBOT now trades several futures and forwards, not only T-bonds and treasury bonds, but also options and gold. In 1870, the New York Cotton Exchange was founded. In 1880, the gold standard was introduced. 20th century In 1914, the gold standard was abandoned because of the war. In 1919, the Chicago Produce Exchange, in charge of trading agricultural products was renamed to Chicago Mercantile Exchange. Today it is the most important futures market for Eurodollar, foreign exchange, and livestock. In 1944, the Bretton Woods System was implemented in an attempt to stabilize the currency system. In 1970, the Bretton Woods System was abandoned for several reasons. In 1971, the Smithsonian Agreement on fixed exchange rates was introduced. In 1972, the International Monetary Market (IMM) traded futures on coins, currencies and precious metal. 15 FX Options and Structured Products 15 21th century In 1973, the CBOE (Chicago Board of Exchange) firstly traded call options four years later also put options. The Smithsonian Agreement was abandoned the currencies followed managed floating. In 1975, the CBOT sold the first interest rate future, the first future with no real underlying asset. In 1978, the Dutch stock market traded the first standardized financial derivatives. In 1979, the European Currency System was implemented, and the European Currency Unit (ECU) was introduced. In 1991, the Maastricht Treaty on a common currency and economic policy in Europe was signed. In 1999, the Euro was introduced, but the countries still used cash of their old currencies, while the exchange rates were kept fixed. In 2002, the Euro was introduced as new money in the form of cash. 1.2 Technical Issues for Vanilla Options We consider the model geometric Brownian motion ds t (r d r f )S t dt sigmas t dw t (1.1) for the underlying exchange rate quoted in FOR-DOM (foreign-domestic), which means that one unit of the foreign currency costs FOR-DOM units of the domestic currency. In case of EUR-USD with a spot of. this means that the price of one EUR is USD. The notion of foreign and domestic do not refer the location of the trading entity, but only to this quotation convention. We denote the (continuous) foreign interest rate by r f and the (continuous) domestic interest rate by r d. In an equity scenario, r f would represent a continuous dividend rate. The volatility is denoted by sigma, and W t is a standard Brownian motion. The sample paths are displayed in Figure 1.1. We consider this standard model, not because it reflects the statistical properties of the exchange rate (in fact, it doesn t), but because it is widely used in practice and front office systems and mainly serves as a tool to communicate prices in FX options. These prices are generally quoted in terms of volatility in the sense of this model. Applying Itocirc s rule to ln S t yields the following solution for the process S t S t S 0 exp sigma2 )t sigmaw t, (1.2) which shows that S t is log-normally distributed, more precisely, ln S t is normal with mean ln S 0 (r d r f 1 2 sigma2 )t and variance sigma 2 t. Further model assumptions are 16 16 Wystup Figure 1.1: Simulated paths of a geometric Brownian motion. The distribution of the spot S T at time T is log-normal. 1. There is no arbitrage 2. Trading is frictionless, no transaction costs 3. Any position can be taken at any time, short, long, arbitrary fraction, no liquidity constraints The payoff for a vanilla option (European put or call) is given by F phi(s T K) , (1.3) where the contractual parameters are the strike K, the expiration time T and the type phi, a binary variable which takes the value 1 in the case of a call and 1 in the case of a put. The symbol x denotes the positive part of x, i. e. x max(0, x) 0 x Value In the Black-Scholes model the value of the payoff F at time t if the spot is at x is denoted by v(t, x) and can be computed either as the solution of the Black-Scholes partial differential 17 FX Options and Structured Products 17 equation v t r d v (r d r f )xv x sigma2 x 2 v xx 0, (1.4) v(t, x) F. (1.5) or equivalently (Feynman-Kac-Theorem) as the discounted expected value of the payofffunction, v(x, K, T, t, sigma, r d, r f, phi) e r dtau IEF . (1.6) This is the reason why basic financial engineering is mostly concerned with solving partial differential equations or computing expectations (numerical integration). The result is the Black-Scholes formula We abbreviate v(x, K, T, t, sigma, r d, r f, phi) phie r dtau fn (phid ) KN (phid ). (1.7) x: current price of the underlying tau T t: time to maturity f IES T S t x xe (r d r f )tau. forward price of the underlying theta plusmn r d r f sigma plusmn sigma 2 d plusmn ln x K sigmatheta plusmntau sigma tau ln f K plusmn sigma 2 2 tau sigma tau n(t) 1 2pi e 1 2 t2 n( t) N (x) x n(t) dt 1 N ( x) The Black-Scholes formula can be derived using the integral representation of Equation (1.6) v e r dtau IEF e rdtau IEphi(S T K) ( e r dtau phi xe (r d r f 1 2 sigma2 )tausigma tauy K) n(y) dy. (1.8) Next one has to deal with the positive part and then complete the square to get the Black - Scholes formula. A derivation based on the partial differential equation can be done using results about the well-studied heat-equation. 18 18 Wystup A Note on the Forward The forward price f is the strike which makes the time zero value of the forward contract F S T f (1.9) equal to zero. It follows that f IES T xe (r d r f )T, i. e. the forward price is the expected price of the underlying at time T in a risk-neutral setup (drift of the geometric Brownian motion is equal to cost of carry r d r f ). The situation r d gt r f is called contango, and the situation r d lt r f is called backwardation. Note that in the Black-Scholes model the class of forward price curves is quite restricted. For example, no seasonal effects can be included. Note that the value of the forward contract after time zero is usually different from zero, and since one of the counterparties is always short, there may be risk of default of the short party. A futures contract prevents this dangerous affair: it is basically a forward contract, but the counterparties have to a margin account to ensure the amount of cash or commodity owed does not exceed a specified limit Greeks Greeks are derivatives of the value function with respect to model and contract parameters. They are an important information for traders and have become standard information provided by front-office systems. More details on Greeks and the relations among Greeks are presented in Hakala and Wystup 50 or Reiss and Wystup 84. For vanilla options we list some of them now. (Spot) Delta. v x phie r f tau N (phid ) (1.10) Forward Delta. Driftless Delta. v f phie r dtau N (phid ) (1.11) phin (phid ) (1.12) Gamma. 2 v e r f tau n(d ) x 2 xsigma tau (1.13) 19 FX Options and Structured Products 19 Speed. 3 v x 3 e r f tau n(d ) x 2 sigma tau ( ) d sigma tau 1 (1.14) Theta. v t e r f tau n(d )xsigma 2 tau phir f xe r f tau N (phid ) r d Ke rdtau N (phid ) (1.15) Charm. 2 v x tau phir f e r f tau N (phid ) phie r f tau n(d ) 2(r d r f )tau d sigma tau 2tausigma tau (1.16) Color. 3 v x 2 tau e r f tau n(d ) 2xtausigma tau 2r f tau (r d r f )tau d sigma tau 2tausigma d tau (1.17) Vega. v sigma xe r f tau taun(d ) (1.18) Volga. 2 v sigma 2 xe r f tau taun(d ) d d sigma (1.19) Volga is also sometimes called vomma or volgamma. Vanna. 2 v sigma x e r f tau n(d ) d sigma (1.20) Rho. v r d phiktaue rdtau N (phid ) (1.21) v r f phixtaue r f tau N (phid ) (1.22) 20 20 Wystup Dual Delta. Dual Gamma. v K phie r dtau N (phid ) (1.23) 2 v e r dtau n(d ) K 2 Ksigma tau (1.24) Dual Theta. v T v t (1.25) Identities The put-call-parity is the relationship d plusmn d (1.26) sigma sigma d plusmn tau (1.27) r d sigma d plusmn tau (1.28) r f sigma xe r f tau n(d ) Ke rdtau n(d ). (1.29) N (phid ) IP phis T phik (1.30) N (phid ) IP phis T phi f 2 (1.31) K v(x, K, T, t, sigma, r d, r f, 1) v(x, K, T, t, sigma, r d, r f, 1) xe r f tau Ke r dtau, (1.32) which is just a more complicated way to write the trivial equation x x x. The put-call delta parity is v(x, K, T, t, sigma, r d, r f, 1) x v(x, K, T, t, sigma, r d, r f, 1) x e r f tau. (1.33) In particular, we learn that the absolute value of a put delta and a call delta are not exactly adding up to one, but only to a positive number e r f tau. They add up to one approximately if either the time to expiration tau is short or if the foreign interest rate r f is close to zero. 21 FX Options and Structured Products 21 Whereas the choice K f produces identical values for call and put, we seek the deltasymmetric strike which produces absolutely identical deltas (spot, forward or driftless). This condition implies d 0 and thus fe sigma2 2 T, (1.34) in which case the absolute delta is e r f tau 2. In particular, we learn, that always gt f, i. e. there can t be a put and a call with identical values and deltas. Note that the strike is usually chosen as the middle strike when trading a straddle or a butterfly. Similarly the dual-delta-symmetric strike circK fe sigma2 2 T can be derived from the condition d Homogeneity based Relationships We may wish to measure the value of the underlying in a different unit. This will obviously effect the option pricing formula as follows. av(x, K, T, t, sigma, r d, r f, phi) v(ax, ak, T, t, sigma, r d, r f, phi) for all a gt 0. (1.35) Differentiating both sides with respect to a and then setting a 1 yields v xv x Kv K. (1.36) Comparing the coefficients of x and K in Equations (1.7) and (1.36) leads to suggestive results for the delta v x and dual delta v K. This space-homogeneity is the reason behind the simplicity of the delta formulas, whose tedious computation can be saved this way. We can perform a similar computation for the time-affected parameters and obtain the obvious equation v(x, K, T, t, sigma, r d, r f, phi) v(x, K, T a, t a, asigma, ar d, ar f, phi) for all a gt 0. (1.37) Differentiating both sides with respect to a and then setting a 1 yields 0 tauv t sigmav sigma r d v rd r f v rf. (1.38) Of course, this can also be verified by direct computation. The overall use of such equations is to generate double checking benchmarks when computing Greeks. These homogeneity methods can easily be extended to other more complex options. By put-call symmetry we understand the relationship (see 6, 7,16 and 19) v(x, K, T, t, sigma, r d, r f, 1) K f v(x, f 2 K, T, t, sigma, r d, r f, 1). (1.39) 22 22 Wystup The strike of the put and the strike of the call result in a geometric mean equal to the forward f. The forward can be interpreted as a geometric mirror reflecting a call into a certain number of puts. Note that for at-the-money options (K f) the put-call symmetry coincides with the special case of the put-call parity where the call and the put have the same value. Direct computation shows that the rates symmetry v v tauv (1.40) r d r f holds for vanilla options. This relationship, in fact, holds for all European options and a wide class of path-dependent options as shown in 84. One can directly verify the relationship the foreign-domestic symmetry 1 x v(x, K, T, t, sigma, r d, r f, phi) Kv( 1 x, 1 K, T, t, sigma, r f, r d, phi). (1.41) This equality can be viewed as one of the faces of put-call symmetry. The reason is that the value of an option can be computed both in a domestic as well as in a foreign scenario. We consider the example of S t modeling the exchange rate of EURUSD. In New York, the call option (S T K) costs v(x, K, T, t, sigma, r usd, r eur, 1) USD and hence v(x, K, T, t, sigma, r usd, r eur, 1)x ( ) 1 . EUR. This EUR-call option can also be viewed as a USD-put option with payoff K 1 K S T This option costs Kv( 1, 1, T, t, sigma, r x K eur, r usd, 1) EUR in Frankfurt, because S t and 1 S t have the same volatility. Of course, the New York value and the Frankfurt value must agree, which leads to (1.41). We will also learn later, that this symmetry is just one possible result based on change of numeraire Quotation Quotation of the Underlying Exchange Rate Equation (1.1) is a model for the exchange rate. The quotation is a permanently confusing issue, so let us clarify this here. The exchange rate means how much of the domestic currency are needed to buy one unit of foreign currency. For example, if we take EURUSD as an exchange rate, then the default quotation is EUR-USD, where USD is the domestic currency and EUR is the foreign currency. The term domestic is in no way related to the location of the trader or any country. It merely means the numeraire currency. The terms domestic, numeraire or base currency are synonyms as are foreign and underlying. Throughout this book we denote with the slash () the currency pair and with a dash (-) the quotation. The slash () does not mean a division. For instance, EURUSD can also be quoted in either EUR-USD, which then means how many USD are needed to buy one EUR, or in USD-EUR, which then means how many EUR are needed to buy one USD. There are certain market standard quotations listed in Table 1.1. 23 FX Options and Structured Products 23 currency pair default quotation sample quote GBPUSD GPB-USD GBPCHF GBP-CHF EURUSD EUR-USD EURGBP EUR-GBP EURJPY EUR-JPY EURCHF EUR-CHF USDJPY USD-JPY USDCHF USD-CHF Table 1.1: Standard market quotation of major currency pairs with sample spot prices Trading Floor Language We call one million a buck, one billion a yard. This is because a billion is called milliarde in French, German and other languages. For the British Pound one million is also often called a quid. Certain currency pairs have names. For instance, GBPUSD is called cable, because the exchange rate information used to be sent through a cable in the Atlantic ocean between America and England. EURJPY is called the cross, because it is the cross rate of the more liquidly traded USDJPY and EURUSD. Certain currencies also have names, e. g. the New Zealand Dollar NZD is called a kiwi, the Australian Dollar AUD is called Aussie, the Scandinavian currencies DKR, NOK and SEK are called Scandies. Exchange rates are generally quoted up to five relevant figures, e. g. in EUR-USD we could observe a quote of The last digit 5 is called the pip, the middle digit 3 is called the big figure, as exchange rates are often displayed in trading floors and the big figure, which is displayed in bigger size, is the most relevant information. The digits left to the big figure are known anyway, the pips right of the big figure are often negligible. To make it clear, a rise of USD-JPY by 20 pips will be and a rise by 2 big figures will be Quotation of Option Prices Values and prices of vanilla options may be quoted in the six ways explained in Table 1.2. 24 24 Wystup name symbol value in units of example domestic cash d DOM 29,148 USD foreign cash f FOR 24,290 EUR domestic d DOM per unit of DOM USD foreign f FOR per unit of FOR EUR domestic pips d pips DOM per unit of FOR USD pips per EUR foreign pips f pips FOR per unit of DOM EUR pips per USD Table 1.2: Standard market quotation types for option values. In the example we take FOREUR, DOMUSD, S 0 . r d 3.0, r f 2.5, sigma 10, K . T 1 year, phi 1 (call), notional 1, 000, 000 EUR 1, 250, 000 USD. For the pips, the quotation USD pips per EUR is also sometimes stated as USD per 1 EUR. Similarly, the EUR pips per USD can also be quoted as EUR per 1 USD. The Black-Scholes formula quotes d pips. The others can be computed using the following instruction. d pips 1 S 0 S 0 1 f K S d 0 S f pips 0 K d pips (1.42) Delta and Premium Convention The spot delta of a European option without premium is well known. It will be called raw spot delta delta raw now. It can be quoted in either of the two currencies involved. The relationship is delta reverse raw delta raw S K. (1.43) The delta is used to buy or sell spot in the corresponding amount in order to hedge the option up to first order. For consistency the premium needs to be incorporated into the delta hedge, since a premium in foreign currency will already hedge part of the option s delta risk. To make this clear, let us consider EUR-USD. In the standard arbitrage theory, v(x) denotes the value or premium in USD of an option with 1 EUR notional, if the spot is at x, and the raw delta v x denotes the number of EUR to buy for the delta hedge. Therefore, xv x is the number of USD to sell. If now the premium is paid in EUR rather than in USD, then we already have v x EUR, and the number of EUR to buy has to be reduced by this amount, i. e. if EUR is the premium currency, we need to buy v x v x EUR for the delta hedge or equivalently sell xv x v USD. 25 FX Options and Structured Products 25 The entire FX quotation story becomes generally a mess, because we need to first sort out which currency is domestic, which is foreign, what is the notional currency of the option, and what is the premium currency. Unfortunately this is not symmetric, since the counterpart might have another notion of domestic currency for a given currency pair. Hence in the professional inter bank market there is one notion of delta per currency pair. Normally it is the left hand side delta of the Fenics screen if the option is traded in left hand side premium, which is normally the standard and right hand side delta if it is traded with right hand side premium, e. g. EURUSD lhs, USDJPY lhs, EURJPY lhs, AUDUSD rhs, etc. Since OTM options are traded most of time the difference is not huge and hence does not create a huge spot risk. Additionally the standard delta per currency pair left hand side delta in Fenics for most cases is used to quote options in volatility. This has to be specified by currency. This standard inter bank notion must be adapted to the real delta-risk of the bank for an automated trading system. For currencies where the risk free currency of the bank is the base currency of the currency it is clear that the delta is the raw delta of the option and for risky premium this premium must be included. In the opposite case the risky premium and the market value must be taken into account for the base currency premium, such that these offset each other. And for premium in underlying currency of the contract the market-value needs to be taken into account. In that way the delta hedge is invariant with respect to the risky currency notion of the bank, e. g. the delta is the same for a USD-based bank and a EUR-based bank. Example We consider two examples in Table 1.3 and 1.4 to compare the various versions of deltas that are used in practice. delta ccy prem ccy Fenics formula delta EUR EUR lhs delta raw P EUR USD rhs delta raw USD EUR rhs flip F4 (delta raw P )SK USD USD lhs flip F4 (delta raw )SK Table 1.3: 1y EUR call USD put strike K for a EUR based bank. Market data: spot S . volatility sigma 12, EUR rate r f 3.96, USD rate r d 3.57. The raw delta is 49.15EUR and the value is 4.427EUR. 26 26 Wystup delta ccy prem ccy Fenics formula delta EUR EUR lhs delta raw P EUR USD rhs delta raw USD EUR rhs flip F4 (delta raw P )SK USD USD lhs flip F4 delta raw SK Table 1.4: 1y call EUR call USD put strike K for a EUR based bank. Market data: spot S . volatility sigma 12, EUR rate r f 3.96, USD rate r d 3.57. The raw delta is 94.82EUR and the value is 21.88EUR Strike in Terms of Delta Since v x phie r f tau N (phid ) we can retrieve the strike as K x exp . (1.44) Volatility in Terms of Delta The mapping sigma phie r f tau N (phid ) is not one-to-one. The two solutions are given by sigma plusmn 1 tau(d d ). (1.45) tau Thus using just the delta to retrieve the volatility of an option is not advisable Volatility and Delta for a Given Strike The determination of the volatility and the delta for a given strike is an iterative process involving the determination of the delta for the option using at-the-money volatilities in a first step and then using the determined volatility to re determine the delta and to continuously iterate the delta and volatility until the volatility does not change more than 0.001 between iterations. More precisely, one can perform the following algorithm. Let the given strike be K. 1. Choose sigma 0 at-the-money volatility from the volatility matrix. 2. Calculate n1 (Call(K, sigma n )). 3. Take sigma n1 sigma( n1 ) from the volatility matrix, possibly via a suitable interpolation. 4. If sigma n1 sigma n lt , then quit, otherwise continue with step 2. 27 FX Options and Structured Products 27 In order to prove the convergence of this algorithm we need to establish convergence of the recursion n1 e r f tau N (d ( n )) (1.46) ( e r f ln(sk) tau (rd r f 1 ) 2 N sigma2 ( n ))tau sigma( n ) tau for sufficiently large sigma( n ) and a sufficiently smooth volatility smile surface. We must show that the sequence of these n converges to a fixed point 0, 1 with a fixed volatility sigma sigma( ). This proof has been carried out in 15 and works like this. We consider the derivative The term n1 e r f tau n(d ( n )) d ( n ) n sigma( n ) sigma( n ). (1.47) n e r f tau n(d ( n )) d ( n ) sigma( n ) converges rapidly to zero for very small and very large spots, being an argument of the standard normal density n. For sufficiently large sigma( n ) and a sufficiently smooth volatility surface in the sense that n sigma( n ) is sufficiently small, we obtain sigma( n ) n q lt 1. (1.48) Thus for any two values (1) n1, (2) n1, a continuously differentiable smile surface we obtain (1) n1 (2) n1 lt q (1) n (2) n, (1.49) due to the mean value theorem. Hence the sequence n is a contraction in the sense of the fixed point theorem of Banach. This implies that the sequence converges to a unique fixed point in 0, 1, which is given by sigma sigma( ) Greeks in Terms of Deltas In Foreign Exchange markets the moneyness of vanilla options is always expressed in terms of deltas and prices are quoted in terms of volatility. This makes a ten-delta call a financial object as such independent of spot and strike. This method and the quotation in volatility makes objects and prices transparent in a very intelligent and user-friendly way. At this point we list the Greeks in terms of deltas instead of spot and strike. Let us introduce the quantities phie r f tau N (phid ) spot delta, (1.50) phie r dtau N (phid ) dual delta, (1.51) 28 28 Wystup which we assume to be given. From these we can retrieve Interpretation of Dual Delta d phin 1 (phie r f tau ), (1.52) d phin 1 ( phie r dtau ). (1.53) The dual delta introduced in (1.23) as the sensitivity with respect to strike has another - more practical - interpretation in a foreign exchange setup. We have seen in Section that the domestic value v(x, K, tau, sigma, r d, r f, phi) (1.54) corresponds to a foreign value v( 1 x, 1 K, tau, sigma, r f, r d, phi) (1.55) up to an adjustment of the nominal amount by the factor xk. From a foreign viewpoint the delta is thus given by ( ) phie rdtau N phi ln( K ) (r x f r d sigma2 tau) sigma tau ( phie rdtau N phi ln( x ) (r K d r f 1 ) 2 sigma2 tau) sigma tau , (1.56) which means the dual delta is the delta from the foreign viewpoint. We will see below that foreign rho, vega and gamma do not require to know the dual delta. We will now state the Greeks in terms of x, . r d, r f, tau, phi. Value. (Spot) Delta. v(x, . r d, r f, tau, phi) x x e r f tau n(d ) e r dtau n(d ) (1.57) Forward Delta. v f v x (1.58) e (r f r d )tau (1.59) 29 FX Options and Structured Products 29 Gamma. 2 v e r f tau n(d ) x 2 x(d d ) (1.60) Taking a trader s gamma (change of delta if spot moves by 1) additionally removes the spot dependence, because Gamma trader x 2 v e r f tau n(d ) 100 x 2 100(d d ) (1.61) Speed. 3 v e r f tau n(d ) x 3 x 2 (d d ) (2d 2 d ) (1.62) Theta. 1 v x t e r f tau n(d )(d d ) 2tau e r f tau n(d ) r f r d e r dtau n(d ) (1.63) Charm. Color. Vega. Volga. 2 v x tau 3 v x 2 tau phir f e r f tau N (phid ) phie r f tau n(d ) 2(r d r f )tau d (d d ) 2tau(d d ) (1.64) e r f tau n(d ) 2r f tau (r d r f )tau d (d d ) d 2xtau(d d ) 2tau(d d ) (1.65) v sigma xe r f tau taun(d ) (1.66) 2 v sigma 2 xe r f tau taun(d ) d d d d (1.67) 30 30 Wystup Vanna. 2 v sigma x e r f tau taud n(d ) (1.68) d d Rho. Dual Delta. v e rf tau n(d ) xtau (1.69) r d e r dtau n(d ) v xtau (1.70) r f v K (1.71) Dual Gamma. K 2 2 v K 2 x 2 2 v x 2 (1.72) Dual Theta. v T v t (1.73) As an important example we consider vega. Vega in Terms of Delta The mapping v sigma xe r f tau taun(n 1 (e r f tau )) is important for trading vanilla options. Observe that this function does not depend on r d or sigma, just on r f. Quoting vega in foreign will additionally remove the spot dependence. This means that for a moderately stable foreign term structure curve, traders will be able to use a moderately stable vega matrix. For r f 3 the vega matrix is presented in Table Volatility Volatility is the annualized standard deviation of the log-returns. It is the crucial input parameter to determine the value of an option. Hence, the crucial question is where to derive the volatility from. If no active option market is present, the only source of information is estimating the historic volatility. This would give some clue about the past. In liquid currency 31 FX Options and Structured Products 31 Mat 50 45 40 35 30 25 20 15 10 5 1D W W M M M M M Y Y Y Table 1.5: Vega in terms of Delta for the standard maturity labels and various deltas. It shows that one can vega hedge a long 9M 35 delta call with 4 short 1M 20 delta puts. pairs volatility is often a traded quantity on its own, which is quoted by traders, brokers and real-time data pages. These quotes reflect views of market participants about the future. Since volatility normally does not stay constant, option traders are highly concerned with hedging their volatility exposure. Hedging vanilla options vega is comparatively easy, because vanilla options have convex payoffs, whence the vega is always positive, i. e. the higher the volatility, the higher the price. Let us take for example a EUR-USD market with spot. USD - and EUR rate at 2.5. A 3-month at-the-money call with 1 million EUR notional would cost 29,000 USD at at volatility of 12. If the volatility now drops to a value of 8, then the value of the call would be only 19,000 USD. This monotone dependence is not guaranteed for non-convex payoffs as we illustrate in Figure Historic Volatility We briefly describe how to compute the historic volatility of a time series S 0, S 1. S N (1.74) 32 32 Wystup Figure 1.2: Dependence of a vanilla call and a reverse knock-out call on volatility. The vanilla value is monotone in the volatility, whereas the barrier value is not. The reason is that as the spot gets closer to the upper knock-out barrier, an increasing volatility would increase the chance of knock-out and hence decrease the value. of daily data. First, we create the sequence of log-returns Then, we compute the average log-return r i ln S i S i 1, i 1. N. (1.75) r 1 N N r i, (1.76) i1 33 FX Options and Structured Products 33 their variance and their standard deviation circsigma 2 1 N 1 N (r i r) 2, (1.77) i1 circsigma 1 N (r i r) N 1 2. (1.78) The annualized standard deviation, which is the volatility, is then given by circsigma a B N (r i r) N 1 2, (1.79) where the annualization factor B is given by i1 i1 B N d, (1.80) k and k denotes the number of calendar days within the time series and d denotes the number of calendar days per year. The is done to press the trading days into the calendar days. Assuming normally distributed log-returns, we know that circsigma 2 is chi 2 - distributed. Therefore, given a confidence level of p and a corresponding error probability alpha 1 p, the p-confidence interval is given by N 1 N 1 circsigma a, circsigma chi 2 a, (1.81) N 11 chi 2 alpha N 1 alpha 2 2 where chi 2 np denotes the p-quantile of a chi 2 - distribution 1 with n degrees of freedom. As an example let us take the 256 ECB-fixings of EUR-USD from 4 March 2003 to 3 March 2004 displayed in Figure 1.3. We get N 255 log-returns. Taking k d 365, we obtain r 1 N r i . N i1 circsigma a B N (r i r) N 1 2 10.85, i1 and a 95 confidence interval of 9.99, 11.89. 1 values and quantiles of the chi 2 - distribution and other distributions can be computed on the internet, e. g. at 34 34 Wystup EURUSD Fixings ECB Exchange Rate 403 4403 5403 6403 7403 8403 9403 Date 10403 11403 12403 1404 2404 Figure 1.3: ECB-fixings of EUR-USD from 4 March 2003 to 3 March 2004 and the line of average growth Historic Correlation As in the preceding section we briefly describe how to compute the historic correlation of two time series x 0, x 1. x N, y 0, y 1. y N, of daily data. First, we create the sequences of log-returns Then, we compute the average log-returns X i ln x i x i 1, i 1. N, Y i ln y i y i 1, i 1. N. (1.82) X 1 N 1 N N X i, i1 N Y i, (1.83) i1 35 FX Options and Structured Products 35 their variances and covariance circsigma X 2 circsigma Y 2 circsigma XY and their standard deviations circsigma X circsigma Y 1 N 1 1 N 1 1 N 1 N (X i X) 2, (1.84) i1 N (Y i )2, (1.85) i1 N (X i X)(Y i ), (1.86) i1 1 N (X i N 1 X) 2, (1.87) i1 1 N (Y i N 1 )2. (1.88) i1 The estimate for the correlation of the log-returns is given by circrho circsigma XY circsigma X circsigma Y. (1.89) This correlation estimate is often not very stable, but on the other hand, often the only available information. More recent work by Jaumlkel 37 treats robust estimation of correlation. We will revisit FX correlation risk in Section Volatility Smile The Black-Scholes model assumes a constant volatility throughout. However, market prices of traded options imply different volatilities for different maturities and different deltas. We start with some technical issues how to imply the volatility from vanilla options. Retrieving the Volatility from Vanilla Options Given the value of an option. Recall the Black-Scholes formula in Equation (1.7). We now look at the function v(sigma), whose derivative (vega) is The function sigma v(sigma) is v (sigma) xe r f T T n(d ). (1.90) 36 36 Wystup 1. strictly increasing, 2. concave up for sigma 0, 2 ln F ln K T ), 3. concave down for sigma ( 2 ln F ln K T, ) and also satisfies v(0) phi(xe r f T Ke r dt ) , (1.91) v(, phi 1) xe r f T, (1.92) v(sigma , phi 1) Ke r dt, (1.93) v (0) xe r f T T 2piII , (1.94) In particular the mapping sigma v(sigma) is invertible. However, the starting guess for employing Newton s method should be chosen with care, because the mapping sigma v(sigma) has a saddle point at ( ) 2 T ln F K, phie r dt F N phi 2T ln FK KN phi 2T ln KF , (1.95) as illustrated in Figure 1.4. To ensure convergence of Newton s method, we are advised to use initial guesses for sigma on the same side of the saddle point as the desired implied volatility. The danger is that a large initial guess could lead to a negative successive guess for sigma. Therefore one should start with small initial guesses at or below the saddle point. For at-the-money options, the saddle point is degenerate for a zero volatility and small volatilities serve as good initial guesses. Visual Basic Source Code Function VanillaVolRetriever(spot As Double, rd As Double, rf As Double, strike As Double, T As Double, type As Integer, GivenValue As Double) As Double Dim func As Double Dim dfunc As Double Dim maxit As Integer maximum number of iterations Dim j As Integer Dim s As Double first check if a volatility exists, otherwise set result to zero If GivenValue lt Application. Max (0, type (spot Exp(-rf T) - strike Exp(-rd T))) Or (type 1 And GivenValue gt spot Exp(-rf T)) Or (type -1 And GivenValue gt strike Exp(-rd T)) Then 37 FX Options and Structured Products 37 Figure 1.4: Value of a European call in terms of volatility with parameters x 1, K 0.9, T 1, r d 6, r f 5. The saddle point is at sigma 48. VanillaVolRetriever 0 Else there exists a volatility yielding the given value, now use Newton s method: the mapping vol to value has a saddle point. First compute this saddle point: saddle Sqr(2 T Abs(Log(spot strike) (rd - rf) T))Sell faster. Your next home is waiting. For the first time in from it, and by incontrovertible calculations I find that a projectile endowed with an initial velocity about events as they happened. She groomed the dolls endlessly, cooed to them, tucked them over to figure out why, and, about get interested in you. 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