Binnenlandse Franqui leerstoel –VUB December, 2004 Opties André Farber Lecture 2: Options and investments

1. Introduction binomial pricing – Review 1-period binomial option pricing formulas: u = e σ ∆t d = 1/u -r∆t f = [ p fu + (1-p) fd ] e p = (er∆t – d)/(u - d) 2. Black Scholes formula (European option on non dividend paying stock) -rT European call: C = S N(d1) – K e N(d2)

European put: P = K N(-d2) – S N(-d1) S ln( ) Ke −rT d1 = + .5σ T σ T S ln( ) Ke −rT d 2 = − .5σ T = d1 −σ T σ T Note: Using put call parity: P = C – S + K e-rT -rT -rT = S N(d1) – K e N(d2) – S + K e -rT = K e [1 – N(d2)] – S [1 – N(d1)] -rT = K e N(-d2)] – S N(-d1) If stock pays continuous dividend yield q: replace S by S e-qT Illustration: Excel Lecture 2 worksheet B&S formula 3. American options Non dividend paying stock Call: Black Scholes formula (no early ) Put: No closed form solution – use binomial model -r∆t f = Max(K-S, [ p fu + (1-p) fd ] e ) Dividend paying stock (assume constant dividend yield q) No closed form solution – Use binomial model Put-Call parity: C + PV(K) = P + Se-qT Risk neutral probability of up: p = (e(r-q)∆t – d)/(u - d) Illustration: Excel Lecture 2 Binomial model 3. The Greek letters ∂f Slope Delta : δ = = f ' ∂S S -qT Black Scholes: Delta Call = N(d1) e -q∆t Binomial model: Delta = (fu – fd) / [(u-d)Se ]

1 ∂δ ∂²f Convexity Gamma: Γ = = = f " ∂S ∂S ² SS ∂f Time Theta: Θ = = f ' ∂T T ∂f Vega: = ∂σ ∂f Interest rate Rho: = ∂r Illustration: Excel Lecture 2 worksheet B&S Formula 4. Strategies involving options (list not exhaustive) Single option + stock or index : S + P = PV(K)+C convertible bond, ELN Example: Lehman Brother Equity Linked Notes: An Introduction Illustration: Excell worksheet Equity-LinkedNote Synthetic protective put: portfolio insurance V = S + P same value as with put V = n S + B n shares + bond

1 + δPut = n Dynamic hedging LOR and the crash of October 19, 1987: see Rubinstein 1999 Illustration: Excell worksheet PorfolioInsurance : S – C = PV(K) Examples Discount certificate (Boom time for structured product? Risk Feb 2004) Reverse convertible (Robeco Reverse Convertible op beleggingsfond) Illustration: Excel worksheet Reverse convertible Combinations : C + P Application: What is the market value of perfect forecasting? Perfect forecaster:

Long shares if expects V0(1+R)>V(1+rf)

V1 = Max(V0+V0R,V0+V0rf)

= V0(1+rf) + Max(0, V0(R-rf) Exotic (source: www.riskglossary.com) For pricing exotic options: see Rubinstein’s website: www.inthemoney.com Cliquet options A is an advance purchase of a put or that will become active at some specified future time. A premium is paid in advance, and the underlier and time to are specified at that time. The is determined when the option becomes active. Typically, it is set at-the-money based upon the underlier value at that time. Alternatively, it can be set a pre- determined percentage in-the-money or out-of-the-money. A ratchet option (or ) is a series of consecutive forward start options. The first is active immediately. The second becomes active when the

2 first expires, etc. Each option is struck at-the-money when it becomes active. The effect of the entire instrument is of an option that periodically "locks in" profits. Ratchet features can be incorporated into other structures. For example, there are ratchet caps or ratchet floors.

Barrier options A is a path dependent option that has one of two features: A knockout feature causes the option to immediately terminate if the underlier reaches a specified barrier level, or A knock-in feature causes the option to become effective only if the underlier first reaches a specified barrier level. Premiums are paid in advance. Due to the contingent nature of the option, they tend to be lower than for a corresponding vanilla option. Consider a knock-in call option with a strike price of EUR 100 and a knock-in barrier at EUR 110. Suppose the option was purchased when the underlier was at EUR 90. If the option expired with the underlier at EUR 103, but the underlier never reached the barrier level of EUR 110 during the life of the option, the option would expire worthless. On the other hand, if the underlier first rose to the EUR 110 barrier, this would cause the option to knock-in. It would then be worth EUR 3 when it expired with the underlier at EUR103. This is illustrated in Exhibit 1: Example: Up-And-In Barrier Call Option E xhibit 1

An up-and-in barrier call option expires worthless unless the underlier value hits the barrier at some time during the life of the option.

The particular option in this example is known as an "up-and-in" option because the underlier must first go "up" to the barrier before the option knocks "in." In all, there are eight flavors of barrier options comprising European puts or calls having barriers that are: • up-and-in, • down-and-in, • up-and-out, or • down-and-out. Of the eight, four either knock-in or knockout when they are in-the-money. These are called reverse barrier options. They can pose significant hedging challenges for the issuer.

3 Lookback A lookback option is a path dependent option settles based upon the maximum or minimum underlier value achieved during the entire life of the option. Essentially, at expiration, the holder can "look back" over the life of the option and exercise based upon the optimal underlier value achieved during that period. Lookbacks can be structured as puts or calls and come in two basic forms: • A fixed strike lookback option is cash settled and has a strike set in advance. It is exercised based upon the optimal underlier value achieved during the life of the option. In the case of a call, this is the highest underlier value achieved, so the call has a payoff equal to the greater of: zero or the difference between that highest value and the fixed strike. In the case of a put, the optimal value is the lowest underlier value achieved, and the payoff is the greater of: zero or the difference between the strike and that lowest value. • A floating strike lookback option can have cash or physical settled. It settles based upon a strike that is set equal to the optimal value achieved by the underlier over the life of the option. In the case of a call, that optimal value is the lowest value achieved by the underlier, so the call has a payoff equal to the difference between the value of the underlier at expiration and the lowest value achieved by the underlier over the life of the option. In the case of a put, the payoff is the difference between the highest value achieved by the underlier and the value of the underlier at expiration. Lookback options have obvious appeal, but they are expensive. Their structure doesn't mimic typical business liabilities, so they are largely a speculative device.

7. Market timing as an option

8. Hedge funds

References Lehman-Brother, Equity-Linked Notes – An Introduction, Rubinstein, M., The Real-World Pitfalls of Portfolio Insurance Strategy, September 1999

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