BLACK-SCHOLES-MERTON FIN501 Asset Pricing Lecture 08 Option Pricing (2)

FIN501 Asset Pricing Lecture 08 Option Pricing (1)

Markus K. Brunnermeier LECTURE 08: MULTI-PERIOD MODEL OPTIONS: BLACK-SCHOLES-MERTON FIN501 Asset Pricing Lecture 08 Option Pricing (2)

specify observe/specify Preferences & existing Technology Asset Prices

• evolution of states NAC/LOOP NAC/LOOP • risk preferences • aggregation

State Prices q absolute (or stochastic discount relative asset pricing factor/Martingale measure) asset pricing

LOOP

derive derive Asset Prices Price for (new) asset

Only works as long as market completeness doesn’t change FIN501 Asset Pricing Lecture 08 Option Pricing (3)

• Price claim at 푡 = 0 with many final payoffs at 푇 o 푆 states/histories • Dynamic replication with only 2 assets o Dynamic trading strategy FIN501 Asset Pricing Lecture 08 Option Pricing (4) Option Pricing

• European call option maturing at time 푇 with strike 퐾 ⇒ + 퐶푇 = 푆푇 − 퐾 , no cash flows in between • Why multi-period problem o Not able to statically replicate this payoff using just the stock and risk-free bond o Need to dynamically hedge – required stock position changes for each period until maturity • Recall static hedge for forward, using put-call parity • Replication strategy depends on specified random process need to know how stock price evolves over time. o Binomial (Cox-Ross-Rubinstein) model is canonical FIN501 Asset Pricing Lecture 08 Option Pricing (5) Binominal Option Pricing

• Assumptions: o Bond: 푓 푓 • Constant risk-free rate 푅 = 푅푡 ∀푡 푓 푟푇/푛 푇 푅 = 푒 for period with length 푛 o Stock: • which pays no dividend • stock price moves from 푆 to either 푢푆 or 푑푆, 푢푆

푆 푑푆 • i.i.d. over time ⇒ final distribution of 푆푇 is binomial • No arbitrage: 푢 > 푅푓 > 푑 FIN501 Asset Pricing Lecture 08 Option Pricing (6) A One-Period Binomial Tree

• Example of a single-period model o 푆 = 50, 푢 = 2, 푑 = 0.5, 푅푓 = 1.25 100

50 25 o What is value of a European call option with 퐾 = 50? + o Option payoff: 푆푇 − 퐾 50

퐶 =? o Use replication to price 0 FIN501 Asset Pricing Lecture 08 Option Pricing (7) Single-period Replication

• Long ∆ stocks and 퐵 dollars in bond • Payoff from portfolio: Δ푢푆 + 푅푓퐵 = 100Δ + 1.25퐵 Δ푆 + 퐵 = 50Δ + 퐵 Δ푑푆 + 푅푓퐵 = 25Δ + 1.25퐵 • 퐶푢 option payoff in up state and 퐶푑 as option payoff in down state 퐶푢 = 50, 퐶푑 = 0 • Replicating strategy must match payoffs: 푓 퐶푢 = Δ푢푆 + 푅 퐵 푓 퐶푑 = Δ푑푆 + 푅 퐵 FIN501 Asset Pricing Lecture 08 Option Pricing (8) Single-period Replication

• Solving these equations yields: 퐶 − 퐶 Δ = 푢 푑 푆 푢 − 푑 푢퐶 − 푑퐶 퐵 = 푑 푢 푅퐹 푢 − 푑

2 • In previous example, Δ = and 퐵 = −13.33, so the option value is 3 퐶 = Δ푆 + 퐵 = 20

• Interpretation of Δ: sensitivity of call price to a change in the stock price. Equivalently, tells you how to hedge risk of option o To hedge a long position in call, short Δ shares of stock FIN501 Asset Pricing Lecture 08 Option Pricing (9) Risk-neutral Probabilities

• Substituting Δ and 퐵 from into formula 퐶 −퐶 푢퐶 −푑퐶 1 푅푓−푑 푢−푅푓 퐶 = 푢 푑 푆 + 푑 푢 = 퐶 + 퐶 푆(푢−푑) 푅푓 푢−푑 푅푓 푢−푑 푢 푢−푑 푑 푄 푄 휋푢 1−휋푢 where 휋푄 is risk neutral prob. (EMM) 1 • Hence, like any asset 푝푗 = 퐸푄 푥푗 푅푓 1 1 퐶 = 휋푄퐶 + 1 − 휋푄 퐶 = 퐸푄[퐶 ] 푅푓 푢 푢 푢 푑 푅푓 푡+1 FIN501 Asset Pricing Lecture 08 Option Pricing (10) Risk-neutral Probabilities

• Note that 휋푄is the probability that would justify the current stock price 푆 in a risk-neutral world: 1 푆 = 휋푄푢푆 + 1 − 휋푄 푑푆 푅푓 푢 푢 푅푓 − 푑 휋푄 = 푢 푢 − 푑

• No arbitrage requires 푢 > 푅푓 > 푑 • Note: relative asset pricing o we don’t need to know objective probability (푃-measure). o 푄-measure is sufficient FIN501 Asset Pricing Lecture 08 Option Pricing (11) The Binomial Formula in a Graph FIN501 Asset Pricing Lecture 08 Option Pricing (12) Two-period Binomial Tree

• Concatenation of single-period trees:

푢2푆

푢푆 푢푑푆 푆 푑푆 푑2푆 FIN501 Asset Pricing Lecture 08 Option Pricing (13) Two-period Binomial Tree

• Example: 푆 = 50, 푢 = 2, 푑 = 0.5, 푅푓 = 1.25

200

100 50 50 25 12.5

• Option payoff: 150

퐶푢 0 퐶 퐶푑 0 FIN501 Asset Pricing Lecture 08 Option Pricing (14) Two-period Binomial Tree

• To price the option, work backwards from final period. 200 150

100 Cu 50 0

• We know how to price this from before: 푅푓 − 푑 1.25 − 0.5 휋푄 = = = 0.5 푢 푢 − 푑 2 − 0.5 1 퐶 = 휋푄퐶 + 1 − 휋푄 퐶 = 60 푢 푅푓 푢 푢푢 푢 푢푑 • Three-step procedure: 푄 1. Compute risk-neutral probability, 휋푢 2. Plug into formula for 퐶 at each node for prices, going backwards from final node. 3. Plug into formula for Δ and 퐵 at each node for replicating strategy, going backwards from the final node.. FIN501 Asset Pricing Lecture 08 Option Pricing (15) Two-period Binomial Tree

퐶푢푢 • General formulas for two-period tree: 푄 푄 푄 휋푢 퐶푢푢 + 1 − 휋 퐶푢푑 • 휋푢 = (푅 − 푑)/(푢 − 푑) 퐶 = 푢 푢 푅푓 퐶푢푢 − 퐶푢푑 ∆푢 = 2 푄 푄 푢 푆 − 푢푑푆 휋푢 퐶푢 + 1 − 휋 퐶푑 퐶 = 푢 퐵푢 = 퐶푢 − ∆푢푆 푅푓 2 휋푄2퐶 + 2휋푄 1 − 휋푄 퐶 + 1 − 휋푄 퐶 퐶푢푑 = 푢 푢푢 푢 푢 푢푑 푢 푢푑 푅푓 2 퐶 − 퐶 푄 푄 푢 푑 휋푢 퐶푢푑 + 1 − 휋 퐶푑푑 Δ = , 퐵 = 퐶 − ∆푆 퐶 = 푢 푢푆 − 푑푆 푑 푅푓 퐶 − 퐶 ∆ = 푢푑 푑푑 푑 푢푑푆 − 푑2푆 퐵푑 = 퐶푑 − ∆푑푆 퐶푑푑 • Synthetic option requires dynamic hedging o Must change the portfolio as stock price moves FIN501 Asset Pricing Lecture 08 Option Pricing (16) Arbitraging a Mispriced Option

• 3-period tree: o 푆 = 80, 퐾 + 80, 푢 = 1.5, 푑 = 0.5, 푅 = 1.1 푅푓−푑 • Implies 휋푄 = = 0.6 푢 푢−푑 • Cost of dynamic replication strategy $34.08 • If the call is selling for $36, how to arbitrage? o Sell the real call o Buy the synthetic call • Up-front profit: o 퐶 − ∆푆 + 퐵 = 36– 34.08 = 1.92 FIN501 Asset Pricing Lecture 08 Option Pricing (18) Towards Black-Scholes

• General binomial formula for a European call on non-dividend paying stock 푛 periods from expiration: 푛 1 푛! 푗 푛−푗 + 퐶 = 휋푄 1 − 휋푄 푢푗푑푛−푗푆 − 퐾 (푅푓)푛 푗! 푛 − 푗 ! 푢 푢 푗=0 • Take parameters: 푇 푇 휎 1 −휎 푢 = 푒 푛, 푑 = = 푒 푛 푢 • Where: o 푛 = number of periods in tree o 푇 = time to expiration (e.g., measured in years) o 휎 = standard deviation of continuously compounded return 푟푓푇 o Also take 푅푓 = 푒 푛 • As 푛 → ∞ o Binominal tree → geometric Brownian motion o Binominal formula → Black Scholes Merton FIN501 Asset Pricing Lecture 08 Option Pricing (19) Black-Scholes

−푟푓푇 퐶 = 푆풩 푑1 − 퐾푒 풩 푑2 2 1 푆 푓 휎 푑1 = ln + 푟 + 푇 휎 푇 퐾 2 2 1 푆 푓 휎 푑2 = ln + 푟 − 푇 = 푑1 − 휎 푇 휎 푇 퐾 2 o 푆 = current stock price o 퐾 = strike o 푇 = time to maturity o 푟 = annualized continuously compounded risk-free rate, o 휎 = annualized standard dev. of cont. comp. rate of return on underlying

• Price of a put-option by put-call parity 푃 = 퐶 − 푆 + 퐾푒−푟푇 −푟푇 = 퐾푒 풩 −푑2 − 푆풩 −푑1 FIN501 Asset Pricing Lecture 08 Option Pricing (20) Interpreting Black-Scholes

• Option has intrinsic value 푆 − 퐾 + and time-value 퐶 − 푆 − 퐾 +

25 Time Value 20 Intrinsic Value 15

3 6 9

72 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 75 78 0.001 S FIN501 Asset Pricing Lecture 08 Option Pricing (21) Delta

• Δ is the sensitivity of option price to a small change in the stock price o Number of shares needed to make a synthetic call o Also measures riskiness of an option position

1 NB: For 푆 = 퐾 and 푇 → 0, Δ = 풩 푑 = Δ푐 = 풩 푑1 1 2 푟푇 퐵푐 = −퐾푒 풩 푑2 • Delta of o Call: Δ푐 ∈ [0,1] Put: Δ푐 ∈ [−1,0] o Stock: 1 Bond: 0 푗 o portfolio: 푗 ℎ Δ푖 FIN501 Asset Pricing Lecture 08 Option Pricing (22) Option Greeks

• What happens to option price when one input changes? o Delta (Δ): change in option price when stock increases by $1 o Gamma (Γ): change in delta when option price increases by $1 o Vega: change in option price when volatility increases by 1% o Theta (휃): change in option price when time to maturity decreases by 1 day o Rho (휌): change in option price when interest rate rises by 1% • Greek measures for portfolios o The Greek measure of a portfolio is weighted average of Greeks of individual portfolio components: FIN501 Asset Pricing Lecture 08 Option Pricing (23) Delta-Hedging

• Portfolio ℎ is delta-neutral if 푗 ℎ푗Δ푗 = 0

• Delta-neutral portfolios are a way to hedge out the risk of an option (or portfolio of options) o Example: suppose you write 1 European call whose delta is 0.61. How can you trade to be delta-neutral? ℎ푐Δ푐 + ℎ푠Δ푠 = −1 0.61 + ℎ푠 1 = 0

o So we need to hold 0.61 shares of the stock.

• Delta hedging makes you directionally neutral on the position. o But only linearly! - Γ FIN501 Asset Pricing Lecture 08 Option Pricing (24) Portfolio Insurance & 1987 Crash

• Δ-replication strategy leads to inverse demand o Sell when price goes down o Buy when price goes up

• ⇒ Destabilizes market

• 1987 – (uninformed) portfolio insurance trading was interpreted as “informed” sellers. FIN501 Asset Pricing Lecture 08 Option Pricing (25) Notes on Black-Scholes

• Delta-hedging is not a perfect hedge if you do not trade continuously o Delta-hedging is a linear approximation to the option value o But convexity implies second-order derivatives matter o Hedge is more effective for smaller price changes • Delta-Gamma hedging reduces the basis risk of the hedge. • B-S model assumes that volatility is constant over time and returns are normal – no fat tails. o Volatility “smile” o BS underprices out-of-the-money puts (and thus in-the-money calls) o BS overprices out-of-the-money calls (and thus in-the-money puts) o Ways forward: stochastic volatility • Other issues: stochastic interest rates, bid-ask transaction costs, etc. FIN501 Asset Pricing Lecture 08 Option Pricing (26) Implied Vol., Smiles and Smirks

• Implied volatility o Use current option price and assume B-S model holds o Back out volatility o VIX versus implied volatility of 500 stocks • Smile/Smirk o Implied volatility across various strike prices • BS implies horizontal line s2 o Smile/Smirk after 1987 Smile/smirk K FIN501 Asset Pricing Lecture 08 Option Pricing (27) Merton Model

• Firm’s balance sheet

A L Put option payoff (short) Assets Debt Follow Geometric Equity Brownian M. Call option payoff (long)

o Call on equity is essential a “call on a call” • Merton: Observe equity prices (including vol) and back out o firm value o debt value FIN501 Asset Pricing Lecture 08 Option Pricing (28) Collateral Debt Obligations (CDO)

• Collateralized Debt Obligation- repackage cash flows from a set of assets A L • Tranches: Assets Senior Tranche o Senior tranche paid out first, e.g. mortgages Junior Tranche o Mezzanine second, (correlation) o junior tranche is paid out last • Can adapt option pricing theory, useful in pricing CDOs: o Tranches can be priced using analogues from option pricing formulas o Estimate “implied default correlations” that price the tranches correctly FIN501 Asset Pricing Lecture 08 Option Pricing (29) Pricing of Any Non-linear Payoff

• Method can be used to price any non-linear payoff