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 Option Finally, we compute the cost of this replicating portfolio. Delta
Option Pricing Step 1 is simple: the call will be exercised if state u occurs, and using not otherwise: Risk-Neutral Probabilities Call payoff if uS = 10. The Black-Scholes Call payoff if dS = 0. Model
Implied Volatility The Call Pricing Problem
Option Pricing: A Review
Rangarajan K. Sundaram
Introduction 110 1.02 10 Pricing Options by Replication 100 1 C The Option Delta 99 1.02 0 Option Pricing using Risk-Neutral Stock Cash Call Probabilities
The Black-Scholes Model
Implied Volatility The Replicating Portfolio for the Call
Option Pricing: A Review
Rangarajan K. Sundaram To replicate the call, consider the following portfolio: Introduction Pricing ∆c units of stock. Options by Replication B units of lending/borrowing.
The Option Delta Note that ∆c and B can be positive or negative.
Option Pricing using ∆c > 0: we are buying the stock. Risk-Neutral ∆ < 0: we are selling the stock. Probabilities c B > 0: we are investing. The Black-Scholes B < 0: we are borrowing. Model
Implied Volatility The Replicating Portfolio for the Call
Option Pricing: A Review For the portfolio to replicate the call, we must have: Rangarajan K. Sundaram ∆c · (110) + B · (1.02) = 10 Introduction
Pricing Options by ∆c · (90) + B · (1.02) = 0 Replication
The Option Delta Solving, we obtain:
Option Pricing using ∆C = 0.50 B = −44.12. Risk-Neutral Probabilities In words, the following portfolio perfectly replicates the call The Black-Scholes option: Model Implied A long position in 0.50 units of the stock. Volatility Borrowing of 44.12. Pricing the Call (Cont’d)
Option Pricing: A Review
Rangarajan K. Sundaram Cost of this portfolio: (0.50) · (100) − (44.12)(1) = 5.88. Introduction
Pricing Since the portfolio perfectly replicates the call, we must have Options by Replication C = 5.88. The Option Any other price leads to arbitrage: Delta Option Pricing If C > 5.88, we can sell the call and buy the replicating using Risk-Neutral portfolio. Probabilities If C < 5.88, we can buy the call and sell the replicating The Black-Scholes portfolio. Model
Implied Volatility Pricing the Put Option
Option Pricing: A Review
Rangarajan K. Sundaram To replicate the put, consider the following portfolio:
Introduction ∆p units of stock. Pricing B units of bond. Options by Replication It can be shown that the replicating portfolio now involves a The Option Delta short position in the stock and investment:
Option Pricing using ∆p = −0.50: a short position in 0.50 units of the stock. Risk-Neutral B = +53.92: investment of 53.92. Probabilities The As a consequence, the arbitrage-free price of the put is Black-Scholes Model
Implied (−0.50)(100) + 53.92 = 3.92. Volatility Summary
Option Pricing: A Review Option prices depend on volatility. Rangarajan K. Sundaram Thus, option pricing models begin with a description of volatility, or, more generally, of how the prices of the underlying Introduction evolves over time. Pricing Options by Replication Given a model of price evolution, options may be priced by
The Option replication. Delta Replicating a call involves a long position in the underlying Option Pricing using and borrowing. Risk-Neutral Probabilities Replicating a put involves a short position in the
The underlying and investment. Black-Scholes Model A key step in the replication process is identification of the Implied option delta, i.e., the size of the underlying position in the Volatility replicating portfolio. We turn to a more detailed examination of the delta now. The Option Delta
Option Pricing: A Review
Rangarajan K. Sundaram The delta of an option is the number of units of the underlying Introduction security that must be used to replicate the option. Pricing Options by Replication As such, the delta measures the “riskiness” of the option in
The Option terms of the underlying. Delta For example: if the delta of an option is (say) +0.60, then, Option Pricing using roughly speaking, the risk in the option position is the same as Risk-Neutral Probabilities the risk in being long 0.60 units of the underlying security.
The Black-Scholes Why “roughly speaking?” Model
Implied Volatility The Delta in Hedging Option Risk
Option Pricing: A Review Rangarajan K. The delta is central to pricing options by replication. Sundaram As a consequence, it is also central to hedging written option Introduction positions. Pricing Options by Replication For example, suppose we have written a call whose delta is
The Option currently +0.70. Delta Then, the risk in the call is the same as the risk in a long Option Pricing using position in 0.70 units of the underlying. Risk-Neutral Since we are short the call, we are essentially short 0.70 Probabilities units of the underlying. The Black-Scholes Thus, to hedge the position we simply buy 0.70 units of Model the underlying asset. Implied This is delta hedging. Volatility The Delta in Aggregating Option Risk
Option Pricing: A Review Rangarajan K. The delta enables us to aggregate option risk (on a given Sundaram underlying) and express it in terms of the underlying. Introduction For example, suppose we have a portfolio of stocks on IBM Pricing Options by stock with possibly different strikes and maturities: Replication
The Option Long 2000 calls (strike K1, maturity T1), each with a delta Delta of +0.48. Option Pricing using Long 1000 puts (strike K2, maturity T2), each with a delta Risk-Neutral of −0.55. Probabilities Short 1700 calls (strike K , maturity T ), each with a The 3 3 Black-Scholes delta of +0.63. Model Implied What is the aggregate risk in this portfolio? Volatility The Delta in Aggregating Option Risk
Option Pricing: A Review Each of the first group of options (the strike K1, maturity T1
Rangarajan K. calls) is like being long 0.48 units of the stock. Sundaram Since the portfolio is long 2,000 of these calls, the aggregate Introduction position is akin to being long 2000 × 0.48 = 960 units of the Pricing stock. Options by Replication Similarly, the second group of options (the strike K2, maturity The Option Delta T2 puts) is akin to being short 1000 × 0.55 = 550 units of the
Option Pricing stock. using Risk-Neutral The third group of options (the strike K3, maturity T3 calls) is Probabilities akin to being short 1700 × 0.63 = 1071 units of the stock. The Black-Scholes Model Thus, the aggregate position is: +960 − 550 − 1071 = −661 or
Implied a short position in 661 units of the stock. Volatility This can be delta hedged by taking an offsetting long position in the stock. The Delta as a Sensitivity measure
Option Pricing: A Review
Rangarajan K. Sundaram
Introduction The delta is also a sensitivity measure: it provides a snapshot
Pricing estimate of the dollar change in the value of a call for a given Options by change in the price of the underlying. Replication The Option For example, suppose the delta of a call is +0.50. Delta
Option Pricing Then, holding the call is “like” holding +0.50 units of the stock. using Risk-Neutral Probabilities Thus, a change of $1 in the price of the stock will lead to a change of +0.50 in the value of the call. The Black-Scholes Model
Implied Volatility The Sign of the Delta
Option Pricing: A Review
Rangarajan K. Sundaram In the binomial examples, the delta of the call was positive, Introduction while that of the put was negative. Pricing Options by These properties must always hold. That is: Replication The Option A long call option position is qualitatively like a long Delta position in the underlying security. Option Pricing using A long put option position is qualitatively like a short Risk-Neutral Probabilities position in the underlying security.
The Black-Scholes Why is this the case? Model
Implied Volatility Maximum Value of the Delta
Option Pricing: A Review
Rangarajan K. Sundaram Moreover, the delta of a call must always be less than +1. Introduction Pricing The maximum gain in the call’s payoff per dollar increase Options by Replication in the price of the underlying is $1. The Option Thus, we never need more than one unit of the underlying Delta in the replicating portfolio. Option Pricing using Risk-Neutral Similarly, the delta of a put must always be greater than −1 Probabilities since the maximum loss on the put for a $1 increase in the The stock price is $1. Black-Scholes Model
Implied Volatility Depth-in-the-Money (“Moneyness”) and the Delta
Option Pricing: A Review
Rangarajan K. Sundaram The delta of an option depends in a central way on the option’s Introduction depth-in-the-money. Pricing Options by Consider a call: Replication The Option If S K (i.e., is very high relative to K), the delta of a Delta call will be close to +1. Option Pricing using If S K (i.e., is very small relative to K), the delta of the Risk-Neutral Probabilities call will be close to zero.
The In general, as the stock price increases, the delta of the Black-Scholes call will increase from 0 to +1. Model
Implied Volatility Moneyness and Put Deltas
Option Pricing: A Review
Rangarajan K. Sundaram
Introduction Now, consider a put. Pricing Options by If S K, the put is deep out-of-the-money. Its delta will Replication be close to zero. The Option Delta If S K, the put is deep in-the-money. Its delta will be Option Pricing close to −1. using Risk-Neutral In general, as the stock price increases, the delta of the Probabilities put increases from −1 to 0. The Black-Scholes Model
Implied Volatility Moneyness and Option Deltas
Option Pricing: A Review
Rangarajan K. Sundaram To summarize the dependence on moneyness: Introduction The delta of a deep out-of-the-money option is close to Pricing Options by zero. Replication The absolute value of the delta of a deep in-the-money The Option option is close to 1. Delta As the option moves from out-of-the-money to Option Pricing using in-the-money, the absolute value of the delta increases Risk-Neutral Probabilities from 0 towards 1.
The Black-Scholes The behavior of the call and put deltas are illustrated in the Model figure on the next page. Implied Volatility The Option Deltas
Option Pricing: A Review
Rangarajan K. 1 Sundaram 0.8
Introduction 0.6 Call Delta Pricing Put Delta Options by 0.4 Replication 0.2 The Option Stock Prices Delta 0 72 80 88 96 104 112 120 128 Option Pricing -0.2 using Option Deltas Risk-Neutral -0.4 Probabilities
The -0.6 Black-Scholes Model -0.8
Implied -1 Volatility Implications for Replication/Hedging
Option Pricing: A Review
Rangarajan K. Sundaram The option delta’s behavior has an important implication for
Introduction option replication. Pricing In pricing forwards, “buy-and-hold” strategies in the spot asset Options by Replication suffice to replicate the outcome of the forward contract.
The Option Delta In contrast, since an option’s depth-in-the-money changes with
Option Pricing time, so will its delta. Thus, a static buy-and-hold strategy will using not suffice to replicate an option. Risk-Neutral Probabilities Rather, one must use a dynamic replication strategy in which The Black-Scholes the holding of the underlying security is constantly adjusted to Model reflect the option’s changing delta. Implied Volatility Summary
Option Pricing: A Review Rangarajan K. The option delta measures the number of units of the Sundaram underlying that must be held in a replicating portfolio. Introduction As such, the option delta plays many roles: Pricing Options by Replication Replication.
The Option (Delta-)Hedging. Delta As a sensitivity measure. Option Pricing using The option delta depends on depth-in-the-money of the option: Risk-Neutral Probabilities It is close to unity for deep in-the-money options. The Black-Scholes It is close to zero for deep out-of-the-money options. Model Thus, it offers an intuitive feel for the probability the Implied option will finish in-the-money. Volatility Risk-Neutral Pricing
Option Pricing: A Review Rangarajan K. An alternative approach to identifying the fair value of an option Sundaram is to use the model’s risk-neutral (or risk-adjusted) probabilities. Introduction This approach is guaranteed to give the same answer as Pricing Options by replication, but is computationally a lot simpler. Replication
The Option Pricing follows a three-step procedure. Delta
Option Pricing Identify the model’s risk-neutral probability. using Take expectations of the option’s payoffs under the Risk-Neutral Probabilities risk-neutral probability.
The Discount these expectations back to the present at the Black-Scholes Model risk-free rate.
Implied Volatility The number obtained in Step 3 is the fair value of the option. The Risk-Neutral Probability
Option Pricing: A Review
Rangarajan K. The risk-neutral probability is that probability under which Sundaram expected returns on all the model’s assets are the same.
Introduction For example, the binomial model has two assets: the stock Pricing which returns u or d, and the risk-free asset which returns R. Options by Replication (R = 1+ the risk-free interest rate.)
The Option Delta If q and 1 − q denote, respectively, the risk-neutral probabilities
Option Pricing of state u and state d, then q must satisfy using Risk-Neutral Probabilities q · u + (1 − q) · d = R.
The Black-Scholes This means the risk-neutral probability in the binomial model is Model R − d Implied q = . Volatility u − d Risk-Neutral Pricing: An Example
Option Pricing: A Consider the one-period binomial example in which u = 1.10, Review d = 0.90, and R = 1.02. Rangarajan K. Sundaram In this case, Introduction 1.02 − 0.90 Pricing q = = 0.60. Options by 1.10 − 0.90 Replication
The Option Therefore, the expected payoffs of the call under the risk-neutral Delta probability is Option Pricing using Risk-Neutral (0.60)(10) + (0.40)(0) = 6. Probabilities
The Discounting these payoffs back to the present at the risk-free Black-Scholes rate results in Model
Implied 6 = 5.88, Volatility 1.02 which is the same price for the option obtained by replication. Risk-Neutral Pricing: A Second Example
Option Pricing: A Review Rangarajan K. Now consider pricing the put with strike K = 100. Sundaram The expected payoff from the put at maturity is Introduction Pricing (0.60)(0) + (0.40)(10) = 4. Options by Replication Discounting this back to the present at the risk-free rate, we The Option Delta obtain Option Pricing using 4 Risk-Neutral = 3.92, Probabilities 1.02 The which is the same price obtained via replication. Black-Scholes Model As these examples show, risk-neutral pricing is computationally Implied Volatility much simpler than replication. The Balck-Scholes Model
Option Pricing: A Review
Rangarajan K. Sundaram The Black–Scholes model is unambiguously the best known model of option pricing. Introduction Pricing Also one of the most widely used: it is the benchmark model for Options by Replication Equities. The Option Delta Stock indices.
Option Pricing Currencies. using Futures. Risk-Neutral Probabilities Moreover, it forms the basis of the Black model that is The Black-Scholes commonly used to price some interest-rate derivatives such as Model caps and floors. Implied Volatility The Black-Scholes Model (Cont’d)
Option Pricing: A Review
Rangarajan K. Technically, the Black-Scholes model is much more complex Sundaram than discrete models like the binomial. Introduction In particular, it assumes continuous evolution of Pricing Options by uncertainty. Replication Pricing options in this framework requires the use of very The Option sophisticated mathematics. Delta Option Pricing What is gained by all this sophistication? using Risk-Neutral Probabilities Option prices in the Black-Scholes model can be expressed The in closed-form, i.e., as particular explicit functions of the Black-Scholes Model parameters.
Implied This makes computing option prices and option Volatility sensitivities very easy. Assumptions of the Model
Option Pricing: A Review
Rangarajan K. Sundaram The main assumption of the Black-Scholes model pertains to Introduction the evolution of the stock price. Pricing Options by This price is taken to evolve according to a geometric Replication Brownian motion. The Option Delta Shorn of technical details, this says essentially that two Option Pricing conditions must be met: using Risk-Neutral Probabilities Returns on the stock have a lognormal distribution with
The constant volatility. Black-Scholes Stock prices cannot jump (the market cannot “gap”). Model
Implied Volatility The Log-Normal Assumption
Option Pricing: A Review
Rangarajan K. The log-normal assumption says that the (natural) log of Sundaram returns is normally distributed: if S denotes the current price,
Introduction and St the price t years from the present, then Pricing S Options by ln t ∼ N(µt, σ2t). Replication S The Option Delta Mathematically, log-returns and continuously-compounded Option Pricing using returns represent the same thing: Risk-Neutral Probabilities S S ln t = x ⇔ t = ex ⇔ S = Sex . The S S t Black-Scholes Model Thus, log-normality says that returns on the stock, expressed in Implied Volatility continuously-compounded terms, are normally distributed. Volatility
Option Pricing: A Review
Rangarajan K. Sundaram The number σ is called the volatility of the stock. Thus, Introduction volatility in the Black-Scholes model refers to the standard
Pricing deviation of annual log-returns. Options by Replication The Black-Scholes model takes this volatility to be a constant. The Option In principle, this volatility can be estimated in two ways: Delta Option Pricing From historical data. (This is called historical volatility.) using Risk-Neutral From options prices. (This is called implied volatility.) Probabilities The We discuss the issue of volatility estimation in the last part of Black-Scholes Model this segment. For now, assume the level of volatility is known.
Implied Volatility Is GBM a Reasonable Assumption?
Option Pricing: A Review
Rangarajan K. Sundaram The log-normality and no-jumps conditions appear unreasonably restrictive: Introduction Pricing Volatility of markets is typically not constant over time. Options by Replication Market prices do sometimes “jump.” The Option In particular, the no-jumps assumption appears to rule out Delta dividends. Option Pricing using Risk-Neutral Dividends (and similar predictable jumps) are actually easily Probabilities handled by the model. The Black-Scholes The other issues are more problematic, and not easily resolved. Model We discuss them in the last part of this presentation. Implied Volatility Option Prices in the Black-Scholes Model
Option Pricing: A Review
Rangarajan K. Sundaram
Introduction Option prices in the Black–Scholes model may be recovered Pricing using a replicating portfolio argument. Options by Replication Of course, the construction—and maintenence—of a replicating The Option Delta portfolio is significantly more technically complex here than in
Option Pricing the binomial model. using Risk-Neutral We focus here instead on the final option prices that result and Probabilities the intuitive content of these prices. The Black-Scholes Model
Implied Volatility Notation
Option Pricing: A Review
Rangarajan K. Sundaram t: current time.
Introduction T : Horizon of the model. (So time-left-to-maturity: T − t.) Pricing Options by K: strike price of option. Replication
The Option St : current price of stock. Delta
Option Pricing ST : stock price at T . using Risk-Neutral µ, σ: Expected return and volatility of stock (annualized). Probabilities
The r: risk-free rate of interest. Black-Scholes Model C, P: Prices of call and put (European only). Implied Volatility Call Option Prices in the Black-Scholes Model
Option Pricing: A Review
Rangarajan K. The call-pricing formula in the Black-Scholes model is Sundaram C = S · N(d ) − e−r(T −t)K · N(d ) Introduction t 1 2
Pricing Options by where N(·) is the cumulative standard normal distribution [N(x) Replication is the probability under a standard normal distribution of an The Option Delta observation less than or equal to x], and Option Pricing using 1 St 1 2 Risk-Neutral d1 = √ ln + (r + σ )(T − t) Probabilities σ T − t K 2 The Black-Scholes √ Model d2 = d1 − σ T − t
Implied Volatility This formula has a surprisingly simple interpretation. Call Prices and Replication
Option Pricing: A Review
Rangarajan K. Sundaram To replicate a call in general, we must
Introduction Take a long position in ∆c units of the underlying, and Pricing Borrow B at the risk-free rate. Options by c Replication
The Option The cost of this replicating portfolio—which is the call price—is Delta
Option Pricing C = St · ∆c − Bc . (1) using Risk-Neutral Probabilities The Black-Scholes formula has an identical structure: it too is
The of the form Black-Scholes Model C = St × [Term 1] − [Term 2]. (2) Implied Volatility Call Prices and Replication (Cont’d)
Option Pricing: A Review
Rangarajan K. Sundaram Comparing these structures suggests that Introduction
Pricing ∆c = N(d1). (3) Options by Replication B = e−r(T −t)K · N(d ). (4) The Option c 2 Delta Option Pricing This is exactly correct! The Black-Scholes formula is obtained using Risk-Neutral precisely by showing that the composition of the replicating Probabilities portfolio is (3)–(4) and substituting this into (1). The Black-Scholes In particular, N(d1) is just the delta of the call option. Model
Implied Volatility Put Option Prices in the Black–Scholes Model
Option Pricing: A Review Rangarajan K. Recall that to replicate a put in general, we must Sundaram Take a short position in |∆ | units of the underlying, and Introduction p Lend Bp at the risk-free rate. Pricing Options by Replication Thus, in general, we can write the price of the put as The Option Delta P = Bp + St , ∆p (5) Option Pricing using Risk-Neutral The Black-Scholes formula identifies the exact composition of Probabilities ∆p and Bp: The Black-Scholes ∆ = −N(−d ) B = PV (K) N(−d ) Model p 1 p 2
Implied Volatility where N(·), d1, and d2 are all as defined above. Put Prices in the Black–Scholes Model (Cont’d)
Option Pricing: A Review
Rangarajan K. Sundaram Therefore, the price of the put is given by Introduction
Pricing P = PV (K) · N(−d2) − St · N(−d1). (6) Options by Replication Expression (6) is the Black-Scholes formula for a European put The Option Delta option.
Option Pricing using Equivalently, since N(x) + N(−x) = 1 for any x, we can also Risk-Neutral write Probabilities
The Black-Scholes P = St · [N(d1) − 1] + PV (K) · [1 − N(d2)] (7) Model
Implied Volatility Black-Scholes via Risk-Neutral Probabilities
Option Pricing: A Review Alternative way to derive Black–Scholes formulae: use Rangarajan K. Sundaram risk-neutral (or risk-adjusted) probabilities.
Introduction To identify option prices in this approach: take expectation of
Pricing terminal payoffs under the risk-neutral probability measure and Options by discount at the risk-free rate. Replication The Option The terminal payofffs of a call option with strike K are given by Delta Option Pricing max{S − K, 0}. using T Risk-Neutral Probabilities Therefore, the arbitrage-free price of the call option is given by The Black-Scholes −rT ∗ Model C = e Et [max{ST − K, 0}]
Implied ∗ Volatility where Et [·] denotes time–t expectations under the risk-neutral measure. Black-Scholes via Risk-Neutral Probs (Cont’d)
Option Pricing: A Review
Rangarajan K. Sundaram Equivalently, this expression may be written as Introduction −rT ∗ Pricing C = e Et [ST − K || ST ≥ K] Options by Replication Splitting up the expectation, we have The Option Delta −rT ∗ −rT ∗ Option Pricing C = e Et [ST || ST ≥ K] − e Et [K || ST ≥ K] . using Risk-Neutral Probabilities From this to the Black–Scholes formula is simply a matter of grinding through the expectations, which are tedious, but not The Black-Scholes otherwise difficult. Model
Implied Volatility Black-Scholes via Risk-Neutral Probs (Cont’d)
Option Pricing: A Review
Rangarajan K. Sundaram
Introduction Specifically, it can be shown that Pricing Options by −rT ∗ Replication e Et [ST || ST ≥ K] = St N(d1)
The Option −rT ∗ −rT Delta e Et [K || ST ≥ K] = e KN(d2) Option Pricing using Risk-Neutral In particular, N(d2) is the risk-neutral probability that the Probabilities option finishes in-the-money (i.e., that ST ≥ K). The Black-Scholes Model
Implied Volatility Remarks on the Black-Scholes Formulae
Option Pricing: A Review
Rangarajan K. Sundaram
Introduction Two remarkable features of the Black–Scholes formulae: Pricing Options by Replication Option prices only depend on five variables: S, K, r, T ,
The Option and σ. Delta Of these five variables, only one—the volatility σ—is not Option Pricing directly observable. using Risk-Neutral Probabilities This makes the model easy to implement in practice. The Black-Scholes Model
Implied Volatility Remarks (Cont’d)
Option Pricing: A Review Rangarajan K. It must also be stressed that these are arbitrage-free prices. Sundaram
Introduction That is, they are based on construction of replicating
Pricing portfolios that perfectly mimic option payoffs at maturity. Options by Thus, if prices differ from these predicted levels, the Replication replicating portfolios can be used to create riskless profits. The Option Delta The formulae can also be used to delta-hedge option positions. Option Pricing using Risk-Neutral For example, suppose we have written a call option whose Probabilities current delta, using the Black-Scholes formula, is N(d1). The Black-Scholes To hedge this position, we take a long position in N(d1) Model units of the underlying. Implied Of course, dynamic hedging is required. Volatility Remarks (Cont’d)
Option Pricing: A Review
Rangarajan K. Finally, it must be stressed again that closed-form expressions of Sundaram this sort for option prices is a rare occurence.
Introduction The particular advantage of having closed forms is that Pricing Options by they make computation of option sensitivities (or option Replication “greeks”) a simple task. The Option Delta Nonetheless, such closed-form expressions exist in the Option Pricing Black-Scholes framework only for European-style options. using Risk-Neutral Probabilities For example, closed-forms do not exist for American put The options. Black-Scholes Model However, it is possible to obtain closed-form solutions for
Implied certain classes of exotic options (such as compound Volatility options or barrier options). Plotting Black-Scholes Option Prices
Option Pricing: A Review Rangarajan K. Closed-forms make it easy to compute option prices in the Sundaram Black-Scholes model: Introduction 1 Input values for S , K, r, T −t, and σ. Pricing t √ 2 Options by 2 Compute d1 = [ln(St /K) + (r + σ /2)(T − t)]/[σ T − t]. Replication √ 3 Compute d2 = d1 − σ T − t. The Option Delta 4 Compute N(d1). Option Pricing 5 Compute N(d2). using 6 Risk-Neutral Compute option prices. Probabilities −r(T −t) The C = St N(d1) − e KN(d2) Black-Scholes Model −r(T −t) Implied P = e K [1 − N(d2)] − St [1 − N(d1)] Volatility Plotting Option Prices (Cont’d)
Option Pricing: A Review Rangarajan K. The following figure illustrates this procedure. Sundaram Four parameters are held fixed in the figure: Introduction
Pricing Options by K = 100, T − t = 0.50, σ = 0.20, r = 0.05 Replication
The Option The figures plot call and put prices as S varies from 72 to 128. Delta
Option Pricing Observe non-linear reaction of option prices to changes in stock using price. Risk-Neutral Probabilities For deep OTM options, slope ≈ 0. The Black-Scholes For deep ITM options, slope ≈1. Model Implied Of course, this slope is precisely the option delta! Volatility Plotting Option Prices
Option Pricing: A Review
Rangarajan K. Sundaram 25
Introduction Call Pricing 20 Options by Put Replication
The Option 15 Delta
Option Pricing Option Values 10 using Risk-Neutral Probabilities 5 The Black-Scholes Model Stock Prices 0 Implied 70 80 90 100 110 120 130 Volatility Implied Voaltility
Option Pricing: A Review
Rangarajan K. Sundaram
Introduction Given an option price, one can ask the question: what level of Pricing Options by volatility is implied by the observed price? Replication
The Option This level is the implied volatility. Delta Formally, implied volatility is the volatility level that would make Option Pricing using observed option prices consistent with the Black-Scholes Risk-Neutral Probabilities formula, given values for the other parameters.
The Black-Scholes Model
Implied Volatility Implied Volatility (Cont’d)
Option Pricing: A Review Rangarajan K. For example, suppose we are looking at a call on a Sundaram non-dividend-paying stock. Introduction Let K and T −t denote the call’s strike and time-to-maturity, Pricing Options by and let Cb be the call’s price. Replication Let S be the stock price and r the interest rate. The Option t Delta Then, the implied volatility is the unique level σ for which Option Pricing using bs Risk-Neutral C (S, K, T −t, r, σ) = Cb, Probabilities The where C bs is the Black-Scholes call option pricing formula. Black-Scholes Model Note that implied volatility is uniquely defined since C bs is Implied Volatility strictly increasing in σ. Implied Volatility (Cont’d)
Option Pricing: A Review
Rangarajan K. Sundaram
Introduction Pricing Implied volatility represents the “market’s” perception of Options by Replication volatility anticipated over the option’s lifetime. The Option Delta Implied volatility is thus forward looking.
Option Pricing using In contrast, historical volatility is backward looking. Risk-Neutral Probabilities
The Black-Scholes Model
Implied Volatility The Volatility Smile/Skew
Option Pricing: A Review In theory, any option (any K or T ) may be used for measuring Rangarajan K. Sundaram implied volatility.
Introduction Thus, if we fix maturity and plot implied volatilities against
Pricing strike prices, the plot should be a flat line. Options by Replication In practice, in equity markets, implied volatilities for “low” The Option strikes (corresponding to out-of-the-money puts) are typically Delta much higher than implied volatilties for ATM options. This is Option Pricing using the volatility skew. Risk-Neutral Probabilities In currency markets (and for many individual equities), the The picture is more symmetric with way-from-the-money options Black-Scholes Model having higher implied volatilities than at-the-money options.
Implied This is the volatility smile. Volatility See the screenshots on the next 4 slides. S&P 500 Implied Voaltility Plot
Option Pricing: A Review
Rangarajan K. Sundaram
Introduction
Pricing Options by Replication
The Option Delta
Option Pricing using Risk-Neutral Probabilities
The Black-Scholes Model
Implied Volatility S&P 500 Implied Voaltility Plot
Option Pricing: A Review
Rangarajan K. Sundaram
Introduction
Pricing Options by Replication
The Option Delta
Option Pricing using Risk-Neutral Probabilities
The Black-Scholes Model
Implied Volatility USD-GBP Implied Volatility Plot
Option Pricing: A Review
Rangarajan K. Sundaram
Introduction
Pricing Options by Replication
The Option Delta
Option Pricing using Risk-Neutral Probabilities
The Black-Scholes Model
Implied Volatility USD-GBP Implied Volatility Plot
Option Pricing: A Review
Rangarajan K. Sundaram
Introduction
Pricing Options by Replication
The Option Delta
Option Pricing using Risk-Neutral Probabilities
The Black-Scholes Model
Implied Volatility The Source of the Volatility Skew
Option Pricing: A Review
Rangarajan K. Sundaram Two sources are normally ascribed for the volatility skew. Introduction One is the returns distribution. The Black-Scholes model Pricing assumes log-returns are normally distributed. Options by Replication However, in every financial market, extreme observations are far The Option Delta more likely than predicted by the log-normal distribution.
Option Pricing using Extreme observations = observations in the tail of the Risk-Neutral Probabilities distribution.
The Empirical distributions exhibit “fat tails” or leptokurtosis. Black-Scholes Model Empirical log-returns distributions are often also skewed. Implied Volatility Source of the Volatility Skew (Cont’d)
Option Pricing: A Review Fat tails =⇒ Black-Scholes model with a constant volatility will Rangarajan K. Sundaram underprice out-of-the-money puts relative to those at-the-money.
Introduction Put differently, “correctly” priced out-of-the-money puts will reflect a higher implied volatility than at-themoney options. Pricing Options by Replication This is exactly the volatility skew! That is, the volatility skew is The Option evidence not only that the lognormal model is not a fully Delta accurate description of reality but also that the market Option Pricing using recognizes this shortcoming. Risk-Neutral Probabilities As such, there is valuable information in the smile/skew The concerning the “actual” (more accurately, the market’s Black-Scholes Model expectation) of the return distribution. For instance:
Implied Volatility More symmetric smile =⇒ Less skewed distribution. Flatter smile/skew =⇒ Smaller kurtosis. Source of the Volatility Skew (Cont’d)
Option Pricing: A Review Rangarajan K. The other reason commonly given for the volatility skew is that Sundaram the world as a whole is net long equities, and so there is a
Introduction positive net demand for protection in the form of puts.
Pricing Options by This demand for protection, coupled with market frictions, Replication raises the price of out-of-the-money puts relative to those The Option at-the-money, and results in the volatility skew. Delta
Option Pricing With currencies, on the other hand, there is greater symmetry using Risk-Neutral since the world is net long both currencies, so there is two-sided Probabilities demand for protection. The Black-Scholes Implicit in this argument is Rubinstein’s notion of Model “crash-o-phobia,” fear of a sudden large downward jump in Implied Volatility prices. Generalizing Black-Scholes
Option Pricing: A Review
Rangarajan K. Sundaram Obvious question: why not generalize the log-normal distribution? Introduction
Pricing Indeed, there may even be a “natural” generalization. Options by Replication The Black-Scholes model makes two uncomfortable The Option assumptions: Delta Option Pricing No jumps. using Risk-Neutral Constant volatility. Probabilities The If jumps are added to the log-normal model or if volatility is Black-Scholes Model allowed to be stochastic, the model will exhibit fat tails and
Implied even skewness. Volatility Generalizing Black-Scholes (Cont’d)
Option Pricing: A Review So why don’t we just do this? Rangarajan K. Sundaram We can, but complexity increases (how many new parameters do we need to estimate?). Introduction Jumps & SV have very different dynamic implications. Pricing Options by Reality is more complex than either model (indeed, Replication substantially so). The Option Delta Ultimately: better to use a simple model with known Option Pricing using shortcomings? Risk-Neutral Probabilities On jumps vs stochastic volatility models, see The Black-Scholes Das, S.R. and R.K. Sundaram (1999) “Of Smiles and Model Smirks: A Term-Structure Perspective,” Journal of Implied Volatility Financial and Quantitative Analysis 34(2), pp.211-239.