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t OptionsStefano Grazioli

Critical Thinking

Financial Engineering = Financial analytics

Lab Easy meter

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t OptionsAn introduction

(spans two lectures)

Risk

Managing RiskAuditing

Disaster planning

Insurance

Risk Mitigation

Diversification

Business continuity

Hedging & Options

Riskmanagement

Optionis a contract giving the buyer the right, but not the obligation, to buy or sell an underlying asset (for example a stock) at a specific price on or before a specified date

Options are derivatives.

CBOE CBOE trades options on 3,300 securities.

More than 50,000 series listed. 1/4 of US option trading Hybrid market: 97% total (68% volume) is electronic

Source: CBOE & OCC web site – 2013 - Table includes CBOE + C2 combined

Year 2013

Example Scenario You own 100,000 GOOGLE stocks. @ $1,200 -> $120,000,000. You are pretty happy. But you are also worried. What if the price drops to $1,000? You need some kind of insurance against that. Somebody is willing to commit to buying your GOOGLE stock at

$1,200 (if you want), two years from now. But she wants $10 per stock. Now. You decide that it is a good deal. So, you buy 100,000 contracts

that give you the choice to sell your stock at the agreed price two years from now.

You have bought 100,000 put options.

Put Options A put option gives to its holder the

right to sell the underlying security at a given price on or before a given date.

"Insurance" analogy

Types of Traders Speculators Arbitrageurs Hedgers (us)

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t WINITWhat Is New

In Technology?

Another Scenario You are an executive at the Coca Cola Company. You make $1,000,000 a year. You are pretty happy. The Board wants to make sure that you will do your best to keep the

price of the CocaCola stock up. Rather than giving you a well-deserved raise, they offer to you a

deal. They promise that in three years they will give you the chance to buy 200,000 stocks at $40.

Right now the stock is valued at $40. If the company does well, the stock price could go as up as $50. So you think: “In three years I could just get my 200,000 @ $40 and

then immediately sell them back to the market for $50....” You conclude that an extra $2,000,000 in your pocket is a good

thing. You have been given 200,000 call options.

Call Options A call option gives to its holder the

right to buy the underlying security at a given price on or by a given date

"security deposit" analogy

Nomenclature

IBM StockPrice: $185.00

underlier

“spot” (i.e., market) price

Call Optioncan buy 1 IBM stock@ $180.00on 5 Mar 2014

Put Optioncan sell 1 IBM stock@ $190.00on 18 Apr 2014

strike price

expiration:European vs. American

option price = premium

Nomenclature

IBM StockSpot Price: $185.00

Call Optioncan buy 1 IBM stock@ $180.00 today

Call Optioncan buy 1 IBM stock@ $185.00 today

Call Optioncan buy 1 IBM stock@ $190.00 today

In the money

At the money

Out of the

money

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Valuating Options

An introduction

Evaluating Options On expiration day, value is certain

and dependent on (= strike – spot) On any other day

value is not deterministic,because of uncertaintyabout the future.

Put Option:Can sell IBM for $200

Evaluating PUT Options The current value of a Put Option depends on:

1) the current price of the underlier -2) the strike price +3) the underlier volatility +4) the time to expiration +5) the risk-free interest rate -

IBM’s price is $205

NOW EXPIRATIONPAST

Bought a put option on IBM for $1x = $200

a) IBM’s market price is $190

b) IBM’s market price is $210

Question:what is the value

of the optionright now?

Solving the Option Evaluation Problem

The Black-Scholes Formulas

P = –S[N(–d1)] + Xe-rt[N(–d2)]

d1 = {ln(S/X) + (r + s 2/2)t} st d2 = d1 - stP = value of a European put option,S = current spot price,X = option “strike” or “exercise” price,t = time to option expiration (in years),r = riskless rate of interest (per annum),s = spot return volatility (per annum),N(z) = probability that a standardized normal variable will be less than z. In Excel, this can be calculated using NORMSDIST(d). Delta for a Call = N(d1) Delta for a Put = N(d1) -1

NORMSDIST(z)

0

0.2

0.4

0.6

0.8

1

1.2

-3 -2 -1 0 1 2 3

d

N (z)

z

Formulas Example:

S = $ 42, X = $40t = 0.5r = 0.10 (10% p.a.)s = 0.2 (20% p.a.)

Output:d1 = 0.7693d2 = 0.6278N(d1) = 0.7791N(d2) = 0.7349C = $4.76 and P=$0.81

BS Assumptions Unlimited borrowing and lending at a constant risk-

free interest rate. The stock price follows a geometric Brownian motion

with constant drift and volatility. There are no transaction costs. The stock does not pay a dividend. All securities are perfectly divisible (i.e. it is possible

to buy any fraction of a share). There are no restrictions on short selling. The model treats only European-style options.

Black Scholeswas so much fun…Let’s do it again!

Evaluating Call Options The current value of a call Option depends on:

1) the current price of the underlier +2) the strike price -3) the underlier volatility +4) the time to expiration +5) the risk-free interest rate +

CocaCola’s price is $40

NOW EXPIRATION

Call Option:Can buy CocaColafor $40

PAST

Bought a call option for $2.00, x=40

a) CocaCola’s price is $45

b) CocaCola’s price is $35

Question:what is the value

of the optionright now?

The Black-Scholes Formulas

C = S[N(d1)] – Xe-rt[N(d2)]

d1 = {ln(S/X) + (r + s 2/2)t} st d2 = d1 - stC = value of a European call optionS = current spot price,X = option “strike” or “exercise” price,t = time to option expiration (in years),r = riskless rate of interest (per annum),s = spot return volatility (per annum),N(z) = probability that a standardized normal variable will be less than d. In Excel, this can be calculated using NORMSDIST(z). Delta for a Call = N(d1) Delta for a Put = N(d1) -1

Market Mechanics Market listed: bid & ask Buyer & seller: holder & writer Long & short positions Blocks of 100 – NOT FOR THE TOURNAMENT Option class: defined by the underlier and type Option series: defined by an expiration date & strike

example: APPL May Call 290 Expiration: Sat after the 3rd Friday of the month

America vs European (TOURNAMENT)

Transaction costs: commissions on trading and exercising.