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University of Minnesota, Jan. 21, 2011

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Equity Derivatives

Dave Engebretson

Quantitative Analyst

Citigroup Derivative Markets, Inc.

January 21, 2011

University of Minnesota, Jan. 21, 2011

Contents

- Vanilla Options
- Terminology
- Pricing Methods
- Risk and Hedging
- Spreads

- Exotic Options
- Questions/Discussion

University of Minnesota, Jan. 21, 2011

Options Terminology

Call/put: a call permits the holder to buy a share of stock for the strike price; a put permits the holder to sell a share of stock for the strike price

Spot price (S): the price of the underlying stock

Strike price (K): the price for which a share of stock may be bought/sold

Expiration date: the final date on which an option may be exercised

American/european: european-style options may only be exercised on their expiration dates; american-style options may be exercised on any date through (and including) their expiration dates

Example: an IBM $100 american call expiring on 20-Jan-2012 permits its holder to buy a share of IBM for $100 on any business day up to, and including, 20-Jan-2012

University of Minnesota, Jan. 21, 2011

Volatility

Volatility (s) is a measure of how random a product is, usually defining a one-year standard deviation

The left picture shows low volatility - the path is very predictable

The right picture shows high volatility - the path cannot be well predicted

University of Minnesota, Jan. 21, 2011

T = 0

T > 0

Options 101

Call payoff

Put payoff

Spot price

Spot price

Put payoff

Call payoff

Spot price

Spot price

University of Minnesota, Jan. 21, 2011

s ~ 0

Options 102

Convolving the probability distribution with the final payoff gives today's fair price for the option.

Higher volatility gives higher value because, while it samples more lower spots, it also samples more higher spots.

Different strikes correspond to shifting the red payoff curve horizontally; different spot prices correspond to shifting the blue probability distribution

s >> 0

Probability

Probability

Spot price

Spot price

University of Minnesota, Jan. 21, 2011

T = 0

Limiting Values

Call payoff

Put payoff

Spot price

Spot price

American call

>= 0

>= S â€“ K (intrinsic)

American put

>= 0

>= K â€“ S (intrinsic)

Why?

European options can be worth less than intrinsic value

University of Minnesota, Jan. 21, 2011

Limiting Values

If S << K and s = 0 then a european put will be worth K â€“ S at expiration

Consider the following scenario:

Buy the european put for K e-rt - S, buy a share of stock for S, pay interest on the borrowed difference of K e-rt

At expiration exercise the put, receiving K and closing my position, and use the K to repay the loan of K e-rt

Net profit: 0

European call

>= 0

>= S â€“ K e-rt (intrinsic)

European put

>= 0

>= K e-rt â€“ S (intrinsic)

University of Minnesota, Jan. 21, 2011

Limiting Values

Why must american calls be worth at least S â€“ K?

If an american call is worth less than S â€“ K, I could do the following:

Buy the call for C < S â€“ K

Sell a share simultaneously for S

Immediately exercise the call (american), paying K to receive a share

I then have no net shares (sold one, exercised into one) and my total cash intake is -C + S â€“ K

Is this advantageous?

-C + S â€“ K > 0

S â€“ K > C

This was our initial assumption, so we have an arbitrage

?

?

University of Minnesota, Jan. 21, 2011

Pricing Methods

Closed-form solutions for option prices apply only in certain cases (european options without dividends, etc.)

Iterative solutions can handle far more types of derivatives, but cost more in calculation time

Monte-Carlo pricing for some very exotic derivatives â€“ this converges very slowly and introduces randomness into pricing

Closed-form

Iterative

Monte-Carlo

Speed

Modern computing and parallel processing mean fewer resources devoted to building faster iterative or closed-form solutions

Capability

University of Minnesota, Jan. 21, 2011

Black-Scholes

European options without dividends can be priced in closed form using this model

University of Minnesota, Jan. 21, 2011

(spot2a, time2)

(spot2b, time2)

val2a

val2b

pu

pd

val1

(spot1, time1)

Binomial Trees

Link the value at one unknown point (spot1, time1) with values at two known points (spot2a, time2) and (spot2b, time2)

Several choices of pu, pd, Su=spot2b/spot1, Sd=spot2a/spot1 exist, each with advantages and disadvantages

University of Minnesota, Jan. 21, 2011

T = 0

Pricing an Option with a Binomial Tree

1. Discretize the payoff at expiration, choose normal vs. log-normal evolution

2. Evolve the first timestep

3. Repeat step 2 to cover the entire lifetime of the option

1

Call payoff

Spot price

T = Dt

T = 2Dt

3

2

Call payoff

Call payoff

Spot price

Spot price

University of Minnesota, Jan. 21, 2011

Monte-Carlo

Generate a multitude of paths consistent with desired distribution and dynamics

For each path, compute the value of the option

Appropriately average values for all the paths

Greeks: best to compute with perturbations to existing paths. Why?

Slow convergence, but able to handle just about any type of option; may obtain slightly different results when recalculating the same option

University of Minnesota, Jan. 21, 2011

Risk

Greeks for call, plotted vs. K / S

Delta

Gamma,

Vega

Theta

Rho

University of Minnesota, Jan. 21, 2011

Risk

ATM greeks, plotted vs. time

University of Minnesota, Jan. 21, 2011

Hedging

Delta - shares of stock

Rho - interest rate futures

Gamma, Vega, Theta - other options

Gamma, Theta ~

Vega ~

Gamma,

Vega

Theta

University of Minnesota, Jan. 21, 2011

Spreads

A spread is a group of trades done together

Netting of risk

Often a cheaper way to take specific positions

Some spreads are listed on exchanges, many are OTC

All spreads have at least two legs, but can have many

Payoff

Spot price

University of Minnesota, Jan. 21, 2011

Combo

A combo is a long call with a short put at the same strike

The payoff replicates a forward

Combo payoff

Payoff

Spot price

Spot price

University of Minnesota, Jan. 21, 2011

Call Spread

A call spread is a long call of one strike with a short call of another strike

These can be bullish or bearish depending which strike is bought

Call spread payoff

Payoff

Spot price

Spot price

University of Minnesota, Jan. 21, 2011

Straddle

A straddle is a long call with a long put at the same strike

The payoff is a bet on volatility

Payoff

Straddle payoff

Spot price

Spot price

University of Minnesota, Jan. 21, 2011

Butterfly

A butterfly is a combination of three equally spaced strikes in 1/-2/1 ratios

Butterflies pay off when the stock ends near the middle strike, price is probability

Butterfly payoff

Payoff

Spot price

Spot price

University of Minnesota, Jan. 21, 2011

Put-Call Parity

Compare a comboâ€™s payoff with the payoff of a share of stock minus a bond

Call â€“ Put = S â€“ K e-rt

Combo payoff

Payoff

Spot price

Spot price

University of Minnesota, Jan. 21, 2011

Exotic Option Types

American - not solvable in closed form, so are they exotic?

Asian â€“ payoff depends not on terminal spot, but on average spot over defined time period

Bermudan â€“ can only be exercised on predetermined dates, so something between european and american

Binary (digital) â€“ all-or-nothing depending on a condition being met

Cliquet (compound) â€“ an option to deliver an option. Call on call, call on put, etc.

Knock-in/knock-out (barrier) â€“ options that come into/go out of existence when a condition is met, such as spot reaching a predetermined value

Variance/volatility/dividend swap â€“ an agreement to exchange money based on realized variance, volatility, or dividends

University of Minnesota, Jan. 21, 2011

American Options

Use a binomial tree, raise values to intrinsic at each time step

If an option is raised to intrinsic, exercise it

Non-dividend calls donâ€™t get exercised

Bermudan â€“ same, but only raise to intrinsic at exercise dates

Raise this point to intrinsic and exercise!

Call payoff

Call payoff

Spot price

Spot price

University of Minnesota, Jan. 21, 2011

Binary Options

Start with a call spread, bring the strikes closer together, and increase the number of units of call spread

Binary option payoff

Call spread payoff

Spot price

Spot price

University of Minnesota, Jan. 21, 2011

Questions?

University of Minnesota, Jan. 21, 2011