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Option Pricing: a Review Option Pricing: A Review Rangarajan K. Sundaram Introduction Option Pricing: A Review Pricing Options by Replication The Option Rangarajan K. Sundaram Delta Option Pricing Stern School of Business using Risk-Neutral New York University Probabilities The Invesco Great Wall Fund Management Co. Black-Scholes Model Shenzhen: June 14, 2008 Implied Volatility Outline Option Pricing: A Review Rangarajan K. 1 Introduction Sundaram Introduction 2 Pricing Options by Replication Pricing Options by Replication 3 The Option Delta The Option Delta Option Pricing 4 Option Pricing using Risk-Neutral Probabilities using Risk-Neutral Probabilities The 5 The Black-Scholes Model Black-Scholes Model Implied 6 Implied Volatility Volatility Introduction Option Pricing: A Review Rangarajan K. Sundaram These notes review the principles underlying option pricing and Introduction some of the key concepts. Pricing Options by One objective is to highlight the factors that affect option Replication prices, and to see how and why they matter. The Option Delta We also discuss important concepts such as the option delta Option Pricing and its properties, implied volatility and the volatility skew. using Risk-Neutral Probabilities For the most part, we focus on the Black-Scholes model, but as The motivation and illustration, we also briefly examine the binomial Black-Scholes Model model. Implied Volatility Outline of Presentation Option Pricing: A Review Rangarajan K. Sundaram The material that follows is divided into six (unequal) parts: Introduction Pricing Options: Definitions, importance of volatility. Options by Replication Pricing of options by replication: Main ideas, a binomial The Option example. Delta The option delta: Definition, importance, behavior. Option Pricing Pricing of options using risk-neutral probabilities. using Risk-Neutral The Black-Scholes model: Assumptions, the formulae, Probabilities some intuition. The Black-Scholes Implied Volatility and the volatility skew/smile. Model Implied Volatility Definitions and Preliminaries Option Pricing: A Review Rangarajan K. Sundaram An option is a financial security that gives the holder the right Introduction to buy or sell a specified quantity of a specified asset at a Pricing Options by specified price on or before a specified date. Replication The Option Buy = Call option. Sell = Put option Delta On/before: American. Only on: European Option Pricing Specified price = Strike or exercise price using Risk-Neutral Specified date = Maturity or expiration date Probabilities Buyer = holder = long position The Black-Scholes Seller = writer = short position Model Implied Volatility Options as Insurance Option Pricing: A Review Rangarajan K. Options provide financial insurance. Sundaram The option holder has the right, bit not the obligation, to Introduction participate in the specified trade. Pricing Options by Replication Example: Consider holding a put option on Cisco stock with a The Option strike of $25. (Cisco's current price: $26.75.) Delta Option Pricing The put provides a holder of the stock with protection using Risk-Neutral against the price falling below $25. Probabilities The What about a call with a strike of (say) $27.50? Black-Scholes Model The call provides a buyer with protection against the price Implied Volatility increasing above $25. The Option Premium Option Pricing: A Review Rangarajan K. Sundaram Introduction Pricing The writer of the option provides this insurance to the holder. Options by Replication In exchange, writer receives an upfront fee called the option The Option price or the option premium. Delta Option Pricing Key question we examine: How is this price determined? What using Risk-Neutral factors matter? Probabilities The Black-Scholes Model Implied Volatility The Importance of Volatility: A Simple Example Option Pricing: A Review Suppose current stock price is S = 100. Rangarajan K. Sundaram Consider two possible distributions for ST . In each case, suppose that the \up" and "down" moves each have probability 1=2. Introduction Pricing Options by 120 Replication 110 The Option Delta 100 Option Pricing 100 using Risk-Neutral 90 Probabilities The 80 Black-Scholes Model Implied Case 1: Low Vol Case 2: High Vol Volatility Same mean but second distribution is more volatile. Call Payoffs and Volatility Option Pricing: A Review Consider a call with a strike of K = 100. Rangarajan K. Sundaram Payoffs from the call at maturity: Introduction Pricing 20 Options by Replication The Option 10 Delta Option Pricing C C using Risk-Neutral Probabilities 0 0 The Black-Scholes Model Call Payoffs: Low Vol Call Payoffs: High Vol Implied Volatility The second distribution for ST clearly yields superior payoffs. Put Payoffs and Volatility Option Pricing: A Review Puts similarly benefit from volatility. Consider a put with a Rangarajan K. strike of K = 100. Sundaram Payoffs at maturity: Introduction Pricing Options by 0 0 Replication The Option Delta P P Option Pricing using 10 Risk-Neutral Probabilities 20 The Black-Scholes Put Payoffs: Low Vol Put Payoffs: High Vol Model Implied Volatility Once again, the second distribution for ST clearly yields superior payoffs. Options and Volatility (Cont'd) Option Pricing: A Review In both cases, the superior payoffs from high volatility are a Rangarajan K. Sundaram consequence of \optionality." Introduction A forward with a delivery price of K = 100 does not Pricing similarly benefit from volatility. Options by Replication Thus, all long option positions are also long volatility positions. The Option Delta That is, long option positions increase in value when Option Pricing volatility goes up and decrease in value when volatility using Risk-Neutral goes down. Probabilities Of course, this means that all written option positions are The Black-Scholes short volatility positions. Model Implied Thus, the amount of volatility anticipated over an option's life is Volatility a central determinant of option values. Put{Call Parity Option Pricing: A Review Rangarajan K. One of the most important results in all of option pricing theory. Sundaram It relates the prices of otherwise identical European puts and Introduction calls: Pricing Options by Replication P + S = C + PV (K): The Option Delta Put-call parity is proved by comparing two portfolios and Option Pricing using showing that they have the same payoffs at maturity. Risk-Neutral Probabilities Portfolio A Long stock, long put with strike K and The Black-Scholes maturity T . Model Portfolio B Long call with strike K and maturity T , Implied investment of PV (K) for maturity at T . Volatility Options and Replication Option Pricing: A Review Rangarajan K. Sundaram As with all derivatives, the basic idea behind pricing options is Introduction replication: we look to create identical payoffs to the option's Pricing Options by using Replication The Option Long/short positions in the underlying secutiy. Delta Default-risk-free investment/borrowing. Option Pricing using Risk-Neutral Once we have a portfolio that replicates the option, the cost of Probabilities the option must be equal to the cost of replicating (or The \synthesizing") it. Black-Scholes Model Implied Volatility Pricing Options by Replication (Cont'd) Option Pricing: A Review Rangarajan K. As we have just seen, volatility is a primary determinant of Sundaram option value, so we cannot price options without first modelling Introduction volatility. Pricing Options by More generally, we need to model uncertainty in the evolution of Replication the price of the underlying security. The Option Delta It is this dimension that makes option pricing more complex Option Pricing than forward pricing. using Risk-Neutral Probabilities It is also on this dimension that different \option pricing" The models make different assumptions: Black-Scholes Model Discrete (\lattice") models: e.g., the binomial. Implied Continuous models: e.g., Black-Scholes. Volatility Pricing Options by Replication (Cont'd) Option Pricing: A Review Rangarajan K. Once we have a model of prices evolution, options can be priced Sundaram by replication: Introduction Identify option payoffs at maturity. Pricing Set up a portfolio to replicate these payoffs. Options by Replication Value the portfolio and hence price the option. The Option Delta The replication process can be technically involved; we illustrate Option Pricing it using a simple example|a one-period binomial model. using Risk-Neutral Probabilities From the example, we draw inferences about the replication The process in general, and, in particular, about the behavior of the Black-Scholes option delta. Model Implied Using the intuition gained here, we examine the Black-Scholes Volatility model. A Binomial Example Option Pricing: A Review Rangarajan K. Sundaram Consider a stock that is currently trading at S = 100. Introduction Pricing Suppose that one period from now, it will have one of two Options by Replication possible prices: either uS = 110 or dS = 90. The Option Suppose further that it is possible to borrow or lend over this Delta period at an interest rate of 2%. Option Pricing using Risk-Neutral What should be the price of a call option that expires in one Probabilities period and has a strike price of K = 100? What about a similar The Black-Scholes put option? Model Implied Volatility Pricing the Call Option Option Pricing: A Review Rangarajan K. Sundaram We price the call in three steps: Introduction Pricing First, we identify its possible payoffs at maturity. Options by Replication Then, we set up a portfolio to replicate these payoffs. The
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