Options, Derivatives, Risk Management

Options, Derivatives, Risk Management

(Welch, Chapter 27)

Ivo Welch

UCLA Anderson School, Corporate Finance, Winter 2014

January 13, 2015

Did you bring your calculator? Did you read these notes and the chapter ahead of time?

Maintained Assumptions

É We assume perfect markets, so we assume

1. No differences in opinion.

2. No taxes.

3. No transaction costs.

4. No big sellers/buyers—we have infinitely many clones that can buy

or sell.

É We assume uncertainty and risk-aversion, as in the portfolio

optimization and CAPM chapters (Chapters 8 and 9).

Definition of Derivative

A financial instrument whose payoff depends on some other asset.

É Often a (side-)bet between investors. Thus, for every long there is a short.

(Opposite=Stock is in positive net supply.)

É A priori, both should be better off—contracts are voluntary. But after the fact,

one will lose, one will win.

É Examples:

É Life Insurance. Car Insurance.

É A bet paying off $50 if the price of oil on December 31, 2020 will be > $100. É The price of oil on December 31, 2020, minus $100.

É The price of oil, squared, on December 31, 2020, minus $100, squared.

É The most prominent financial derivatives:

É Forward and Future — an agreement to buy or sell something for an agreed upon

time in the future. Often written so that no money exchanges hands today. Future = settled every day to make price zero (reduces run-away default).

É Option — an agreement that one party has the right but not the obligation to

buy or sell something to the other party for an agreed-upon time period.

É Many others, such as swaps.

Option

The right but not obligation to transact at a predetermined strike price K at a predetermined time (or time range) in the future. The final time is usually abbreviated by capital T.

É Call option: The right to buy 1 share. Its value at expiration is

C(K, T) = max(0, ST– K)

É Put option: The right to sell 1 share. Its value at expiration is

P(K, T) = max(0, K – ST)

É Jargon: “Out-of-the-money” = if it expired now, you would get nothing. “In

the money” = if it expired now, you would get something. “At the money” = right at the cusp. “Above-the-money” = S > K.

É American option = the option holder can exercise anytime before the final

moment. European option = exercise only at the final instant.

É Exchange-traded options have compensating rules for cases in which shares

split, but not for dividend payments, even liquidating ones.

É PS: (Usually) sold in lots of 100 only, called a “contract.”

Call Option Example: IBM on May 31, 2002

The expiration date T, July 20, 2002, was 0.1333 years away. The prevailing interest rates were 1.77% over 1 month, and 1.95% over 6 months.

Underlying Strike Call Put

Base Asset St Expiration T Price K Price Ct Price Pt

IBM $80.50 July 20, 2002 $85 $1.900 $6.200

CJul 20=max(0,SJul 20– $85)⇒ C@May 31= $1.90

PJul 20=max(0, $85–SJul 20)⇒ P@May 31= $6.20 Different Strike Prices

IBM $80.50 July 20, 2002 $75 $7.400 $1.725

IBM $80.50 July 20, 2002 $80 $4.150 $3.400

IBM $80.50 July 20, 2002 $90 $0.725 $10.100

Different Expiration Dates

IBM $80.50 Oct. 19, 2002 $85 $4.550 $8.700

IBM $80.50 Jan. 18, 2003 $85 $6.550 $10.200

Why would someone want to purchase a call option?

Why would someone want to sell a call option?

Does the seller of an option need to own the underlying

(IBM) stock?

Why would someone want to purchase a put option?

Why would someone want to sell a put option?

As a function of the stock price at expiration T, what is

the payoff table and payoff diagram of a call option with

strike price K=$90?

(Long and short. Final payoff only. Ignore upfront cost.)

As a function of the stock price at expiration T, what is

the payoff table and payoff diagram of a put option with

strike price K=$90?

(Long and short. Final payoff only. Ignore upfront cost.)

Is a put option the exact flip side of a call option? That is,

is it the same to buy 1 call option or to sell 1 put option?

Spread (Long and Short of Same Time): As a function of

the stock price at expiration T, what is the payoff table

and payoff diagram of a position with one put option long

with strike price K=$90 and one put option short with

strike K = $

70

?

Other Common Positions

É Spread: (Long Call, Short Call). Or (Long Put, Short Put). É Combination: (Long Call, Short Put). (Short Call, Long Put). É Straddle: (Long Call, Long Put). (Short Call, Short Put).

É Most popular position.

É Speculates on ...?

É Your brokerage has special buttons to purchase many such special

positions.

É (By smartly combining puts and calls, you can even construct

binomial options, e.g., which pay off $1 when the price is between $50 and $51, and $0 otherwise.)

No-Arbitrage Relationships

(All Options:) É C₀≤S₀. É C₀≥ 0. P₀≥ 0. É C₀(K_Low)≥C₀(K_High) É P₀(K_Low)≤P₀(K_High)

(American Options w/o Dividends, but usually w/ Divs, too:)

É C₀≥ max(0, S₀– K). É P₀≥ max(0, K – S₀). É P₀(T later)≥P₀(T earlier). É C₀(T later)≥C₀(T earlier).

(If I use subscript 0, it means any time before expiration.)

Say S = $

80

. C(K = $

100

) = $

30

. P(K = $

100

) = $

50

.

To expiration, r =

10

%. European. How do you get rich?

Put-Call Parity

For European call and put options without dividends

C₀(K) = P₀(K) + S₀– PV₀(K)

If there were no European put options, but you could

buy/sell European call options, could you manufacture a

European put option?

Assume zero dividends.

1. What is the value of immediate exercise for an

American call option?

2. Is an American call option worth more exercised or

unexercised?

3. What is the value of an American call option

compared to a European call option?

Assume zero dividends.

1. What is the value of immediate exercise for an

American put option?

2. Is an American put option worth more exercised or

unexercised?

3. What is the value of an American put option

compared to a European put option?

The American Put

Why is the Put’s American feature worthwhile, while the Call’s American feature is not?

É Take an expiration date of +1 million years.

É If the stock price is $100 and the call’s strike price is $50 (ITM),

does time work for you or against you?

É If the stock price is $50 and the put’s strike price is $100 (ITM),

does time work for you or against you?

What is the Price of the Call?

Our ultimate goal is still to price the following call:

Stock Price Now S₀ $80.50

Agreed-Upon Strike Price K $85.00

Time Remaining to Maturity t 0.1333 years Interest Rate on Risk-Free Bonds R_F 1.77% per year Volatility (Standard Deviation) σ 30% per year of the Underlying Stock

If you know the put option, you can tell me the price of the call. But, without the put, you are stuck.

The solution: binomial pricing.

Let’s say you buy (hold)

δ

=

0

.

5

shares at a stock price of

$120 and borrow $50 (flow b = (–$

50

)) in bonds.

É

What is the net cost of your position today?

É

If the price of shares will be S = $

150

and the

interest rate is 1% from now to next period, then

what will your position be worth?

Binomial Stock Price Movements: up = 5%. down = 4%.

rr= 0.1%. =

_⇒

u =

1

.

05

. d =

0

.

96

. r =

1

.

001

.

' & $ % ' & $ % ' & $ % ' & $ % ' & $ % ' & $ %   * HH HH_Hj   * HH HH_Hj   * HH HH_Hj S0= $50.00. B = $100 ST1= d· S0= $48.00 B = $100.1 ST1= u· S0= $52.50 B = $100.1 ST2= u2· S0= $55.125 B = $100.2 ST2= u·d·S0= $50.40 B = $100.2 ST2= d2· S0= $46.08 B = $100.2

Instant 0 = Now Instant 1 Instant 2

What is the expected rate of return on the stock?

Binomial Pricing

Price a call with a strike price $50, expiring at Instant 2.

' & $ % ' & $ % ' & $ % ' & $ % ' & $ % ' & $ %   * HH HH_Hj   * HH HH_Hj   * HH HH_Hj S0= $50.00 C0=? ST1= d· S0= $48.00 Cd_T1=? ST1= u· S0= $52.50 Cu T1=? ST2= u2 · S0= $55.125 Cuu_T2= $5.125 ST2= u·d·S0= $50.40 Cdu T2= $0.40 ST2= d2· S0= $46.08 Cdd T2= $0.00

Instant 0 = Now Instant 1 Instant 2

You can now forget about the contract—we have used all its information (t and K). We could actually price anything that is a payoff that depends only on ST2.

Binomial Pricing: Price?

' & $ % ' & $ % ' & $ %   * HH HH_Hj Ignore Ignore Ignore Ignore ST2= u·d·S0= $50.40 Cdu T2= $0.40 ST2= d2· S0= $46.08 Cdd T2= $0.00

Instant 0 = Now Instant 1 Instant 2 = Expiration

How many

δ

shares and how many bonds b do you have to

issue (i.e., borrow) today in order to receive next instant

$0.40 if S = $

50

.

40

and $0.00 if S = $

46

.

08

? What are

the equations?

What is the expected rate of return on the stock? Do we

need it?

How much do have to lay out to hold 0.0926 shares if you

borrow $4.262 and shares cost $48?

' & $ % ' & $ % ' & $ %   * HH HH_Hj

(you are here)

Ignore ST1= d· S0= $48.00 Cd T1=? Ignore Ignore

Instant 0 = Now Instant 1 Instant 2 = Expiration

Binomial Pricing: Working Backwards

' & $ % ' & $ % ' & $ %   * HH HH_Hj

(you are here)

Ignore ST1= d· S0= $48.00 Cd T1=? Ignore Ignore ST2= u·d·S0= $50.40 Cdu T2= $0.40 ST2= d2· S0= $46.08 Cdd T2= $0.00

Instant 0 = Now Instant 1 Instant 2 = Expiration

' & $ % ' & $ % ' & $ %   * HH HH_Hj   * HH HH_Hj

A portfolio consisting of 0.0926 shares and $4.262 borrowed from the bank costs $0.182 if S = $48. It is worth $0.40 if the stock goes to $50.40 and $0.00 if the stock goes to $46.08.

Can you do the upper branch?

' & $ % ' & $ % ' & $ % ' & $ % ' & $ % ' & $ %   * HH HH_Hj   * HH HH_Hj   * HH HH_Hj Ignore Ignore ST1= u· S0= $52.50 Cu T1=? ST2= u2· S0= $55.125 Cuu T2= $5.125 ST2= u·d·S0= $50.40 Cdu T2= $0.40 Ignore

Instant 0 = Now Instant 1 Instant 2

Can you do the left twig?

' & $ % ' & $ % ' & $ % ' & $ % ' & $ % ' & $ %   * HH HH_Hj   * HH HH_Hj   * HH HH_Hj S0= $50.00 C0=? ST1= d· S0= $48.00 Cd T1= $0.182 ST1= u· S0= $52.50 Cu T1= $2.550 Ignore Ignore Ignore

Instant 0 = Now Instant 1 Instant 2

The Solved Tree (and Binomial Price)

' & $ % ' & $ % ' & $ % ' & $ % ' & $ % ' & $ %   * HH HH_Hj   * HH HH_Hj   * HH HH_Hj S0= $50.00 C0= $1.26, δ≈ 0.5 ST1= d· S0= $48.00 Cd T1= $0.182, δ≈ 0.1 ST1= u· S0= $52.50 Cu T1= $2.550, δ = 1 ST2= u2· S0= $55.125 Cuu T2= $5.125 ST2= u·d·S0= $50.40 Cdu T2= $0.40 ST2= d2_{· S} 0= $46.08 Cdd T2= $0.00

Instant 0 = Now Instant 1 Instant 2

How often do you have to readjust your mimicking

stock+bond portfolio to have the same payoffs as the

option?

Intuition

You buy a fraction of the stock and borrow some money to create a portfolio that will respond just like a real option to a tiny change in the underlying stock market basis, at least over the next instant of time. By the law of one price, the two portfolios should cost the same amount of money.

To replicate the call, you need to buy the stock and borrow some money. So, what matters for your own replication ability is the interest that you have to pay for borrowing money from now to the expiration of the put. In an imperfect market, this may not be the same as your lending interest rate. Also, in an imperfect market, you may be able to lend out your stock (to hedge funds who want to short), which can earn you extra fees. Net in net, B-S type arbitrage is something that experts in the control of transaction costs should do, not you.

(Warning: day-end prices can be misleading. the NYSE closing prices occur before the CBOE closing prices.)

Do Stock Prices (and Returns) Seem Binomial?

1 Binomial Outcome 5 Levels

50 100 150 200 250 0.0 0.1 0.2 0.3 0.4 0.5

Stock Price (in $)

Probability 2 Levels 50 100 150 200 250 0.00 0.05 0.10 0.15 0.20 0.25 0.30

Stock Price (in $)

Probability 5 Levels 50 Levels 500 Levels 50 100 150 200 250 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Stock Price (in $)

Probability 50 Levels 50 100 150 200 250 0.00 0.01 0.02 0.03

Stock Price (in $)

Probability

500 Levels

Over an infinite number of periods, this would look “log-normal.”

What is the Price of the Call?

Get back to at our original question. How do we price:

Stock Price Now S₀ $80.50

Agreed-Upon Strike Price K $85.00

Time Remaining to Maturity t 0.1333 years Interest Rate on Risk-Free Bonds RF 1.77% per year Volatility (Standard Deviation) σ 30% per year of the Underlying Stock

Let’s say you have an infinite number of levels, not just 3 levels. Does this not make the problem even harder?

Nice Black-Scholes Formula

C0(S0, K, t, RF,σ) = S0· N(d1) – PV(K)· N(d2) d1= log_NS0/PV(K)  (σ ·pt) + 1 2·(σ · p t) d2= d1–σ · p t The five inputs are as follows:

S0 is the current stock price.

t is the time left to maturity. K is the strike price.

PV(K) is the present value of K. Thus, it depends on the risk-free rate,R_F.

σ is the standard deviation of the underlying stock’s continuously

com-pounded rate of return (i.e., oflog(1+rt)). It is often casually called

just “the stock volatility.”

N is the cumulative normal distribution (Excel normsdist()) The time units must be the same on PV(K), sigma, and t. Often all annualized. log_Nis the natural logarithm, not the base-10 logarithm. Calculators often use “ln” for this.

Insights

É S₀· N(·) – PV(K)· N(·) still looks like “buy some stock and

borrow (short some bonds).”

É “Hedge ratio”: Instead of δ, we now have N(d₁).

É N(d_[1,2]) is like a probability: a number between 0 and 1. É σ ·pt is the standard deviation of log-returns to expiration. σ

only enters this way.

É PV(K) is the discounted strike price. K enters only in this form. É t enters only to scale K and σ.

É S/PV(K) is how much your call is in the money. É The log thereof is positive iff S₀> PV(K).

É If you are out of the money, and volatility to expiration is 0,

d₁= –∞, so N(d₁) =0.

É If you are far in the money, and volatility to expiration is 0,

d₁= +∞, so N(d₁) =1.

Steps

É Calculate PV(K). É Calculate σ ·pt É Calculate d₁ É Calculate d₂ É Calculate the rest.

Let’s do the full calculation once by hand.

S

₀

= $

80

.

50

. K = $

85

. t =

0

.

133

y. R

_F

=

1

.

78

%/y.

σ

=

0

.

3

/y.

What is the PV(K)?

S

₀

= $

80

.

50

. PV(K) = $

84

.

80

. t =

0

.

133

y.

σ

=

0

.

3

/y.

What is d

₁

?

What is d

₂

?

What is

_N

(–

0

.

4204

)? What is

_N

(–

0

.

5300

)

The Cumulative Normal Distribution

−3 −2 −1 0 1 2 3 0.1 0.2 0.3 z n(z) Area =15.87% −3 −2 −1 0 1 2 3 0.0 0.2 0.4 0.6 0.8 1.0 z N(z) N(−1)=15.87%

The right-side figure plots normsdist() in Excel.

What is IBM’s Black-Scholes European Call Option Value?

Quasi-Binomial BS Pricing — Trust me

IBM = $80.50. Call Strike = $85. r= 1.77%/1 month, 1.95%/3 months. Time to Expiration = 0.1333 years.

É If IBM goes to $80.51, the call increases in value by 0.3371 cents.

É If IBM goes to $80.49, the call decreases in value by 0.3371 cents.

Thus, if you buy33.71%· $80.50 = $27.14 of IBM stock, then a 1 cent change

in IBM stock means your portfolio changes by 0.3371 cents. The $27.14 position and the option react the same way.

You also need to finance this purchase, though, borrowing $25.28. Your portfolio net cost is $1.86.

Using such mimicking portfolios and the law of one price, you can work back an infinite tree from the final instant before expiration to determine the share price today. (Or just use B-S.) The call price is thus determined by arbitrage. It is called dynamic arbitrage, because every instant, you may have to change your hedge portfolio a little.

If there were no options on a traded base asset, could you

manufacture the payoffs of an option?

If IBM’s European Call Option costs $1.86, what should be

IBM’s American Call Option value?

What is IBM’s European Put Option value?

What is IBM’s American Put Option value?

How do you get the Black-Scholes inputs?

Remaining Problem

How do you get the volatility??

Method 1: Historical Volatility

Use Historical Volatility.

Volatilities are easier to forecast than means. Why?

Often modeled with complex GARCH etc. models: volatility is both strongly auto-regressive and mean-reverting.

Method 2: Implied Volatility

S = $80.50, K = $85, t =0.133, r =1.78% 20 40 60 80 100 120 0 2 4 6 8 10 12 Sigma (in %)

Call Option Value (in $)

P=$1.90

sigma=30.38%

There is no closed-form solution. You must plot the B-S formula for all possible sigmas, and find the one that matches the actual price. (When not otherwise qualified, implied-vol refers to the B-S model.)

Table With Implied Vols

Underlying Expira- Strike Option Option Implied Option Option Implied Base Asset tion T Price K Type Price Volatility Type Price Volatility IBM $80.50 July 20, 2002 $85 Call $1.900 30.38% Put $6.200 29.82%

Different Strike Prices

IBM $80.50 July 20, 2002 $75 Call $7.400 34.89% Put $1.725 34.51% IBM $80.50 July 20, 2002 $80 Call $4.150 32.58% Put $3.400 31.67% IBM $80.50 July 20, 2002 $90 Call $0.725 29.24% Put $10.100 29.18%

Different Expiration Dates

IBM $80.50 Oct. 19, 2002 $85 Call $4.550 31.32% Put $8.700 31.61% IBM $80.50 Jan. 18, 2003 $85 Call $6.550 31.71% Put $10.200 31.40%

If the B-S model held, all implied vols should be identical. Note: the implied vol is called delta. there are also other “greeks”:

ÉVega: price sensitivity to changes in volatility. ÉTheta: price sensitivity to passage of time. ÉRho: price sensitivity to changes in interest rate.

ÉLambda, Omega: gearing (leverage). delta times S/V.

ÉMany other second derivatives.

Where BS works and where it fails.

É Below-of-the-money options have higher prices than BS suggests.

É This is probably due to sudden risk of catastrophic drops.

É It is also partly due to market imperfections—if you want to sell $1

million of puts that are far below the money, your counterparty will worry that you know something that (s)he does not.

É Above-of-the-money options have modestly higher prices than BS

suggests.

É This is called the “option smirk” when the strike price is on the X

axis and the imp vol is on the Y axis.

There are more complex models than B-S, e.g., Merton jump models. There are also models taking into account dividends and models that work with futures instead of stocks.

Vol of Vol?

É There is even an implied volatility index, the VIX. It is the implied

volatility of various S&P500 options.

É You can buy and sell options on the VIX itself, too! You are then

basically speculating whether the implied volatility will go up or down. You can speculate that the vol of the vol is lower than other people think! These are very popular (incl as hedges against increases in risk).

Comparative Statics: How does BS change with its

parameters?

Volatility Estimation

Volatility estimation is a big deal. There is a whole army of people on Wall Street (not just, but mostly quants) engaged in the business of forecasting it. If you can estimate volatility better than others, you can sell expensive options and buy cheap options! For example, if the market prices options at an implied volatility of 30% and you think it is 20%, then sell puts and calls!

Why is the expected rate of return on the stock not in the

BS price?

With the Price at Time 0, you can do other things

See next two figs.

BS Values Prior To Expiration

0 20 40 60 80 100 120 140 0 10 20 30 40 50 60

Current Stock Value (in $)

Current Call O

ption Value (in $)

1 Day Remaining

1 Year Remaining

5 Years Remaining

Note that the y-axis here is the value at any point in time, not just the value at the instant of expiration.

BS Riskier Purchases

0 50 100 150 -100 0 100 200 300

Final stock value (in $)

Call option rat

e of return ( in %)

Call(K=$0) (= buy the stock)

Call(K=$70) Call(K=$90) Call(K=$100)

Note that the y-axis here is rates of return, not payoffs.

The point is: calls with higher strike prices (and puts with lower strike prices) are riskier gambles. They don’t pay off anything more often, but when they do, it could be much more.

Have you seen options before this chapter?

Corporate Risk Management and Hedging

É Uses (synthetic) securities to offset risk. E.g., a gold mine may sell calls on

gold. or futures on gold.

É In a perfect market, investors can hedge and unhedge at will. So, hedging

is irrelevant.

É Eliminating unnecessary risk (that is not the strength of the company)

may reduce the probability of bankruptcy, and in a non-M&M world, may thus enhance value.

É BUT: Is it clear whether it is good or bad for Southwest Airlines if jet fuel

increases in price? Should SWA really hedge?

É Without good self-discipline and controls, hedging can quickly deteriorate

into gambling. Every few years, some 30-year old trader gets caught having gambled away billions of dollars. (Most of the time, they get caught only having gambled away a few million dollars, and are let go quietly.)

É Most large publicly-traded firms, not in the business of speculation, should

avoid risk-management, except in the most obvious of cases and with excellent controls.