Equity Derivatives
Dave Engebretson
Quantitative Analyst
Citigroup Derivative Markets, Inc.
January 21, 2011
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Contents
Vanilla Options
 Terminology
 Pricing Methods
 Risk and Hedging
 Spreads
Exotic Options
Questions/Discussion
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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-Jan2012
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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
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Options 101
Put payoff
Call payoff
T=0
Spot price
Spot price
Put payoff
Call payoff
T>0
Spot price
Spot price
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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
s~0
Spot price
Spot price
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Limiting Values
Put payoff
Call payoff
T=0
Spot price
American call
>= 0
>= S – K (intrinsic)
Spot price
American put
>= 0
>= K – S (intrinsic)
European options can be worth less than intrinsic value
Why?
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Limiting Values
If S << K and s = 0 then a european put will be worth K – 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 – K e-rt (intrinsic)
European put
>= 0
>= K e-rt – S (intrinsic)
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Limiting Values
Why must american calls be worth at least S – K?
If an american call is worth less than S – K, I could do the following:
1. Buy the call for C < S – K
2. Sell a share simultaneously for S
3. 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 – K
Is this advantageous?
?
-C + S – K >? 0
S–K>C
This was our initial assumption, so we have an arbitrage
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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 – this converges very slowly
and introduces randomness into pricing
Speed
Capability
Closed-form
Iterative
Monte-Carlo
Modern computing and parallel processing mean fewer resources devoted to
building faster iterative or closed-form solutions
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Black-Scholes
European options without dividends can be priced in closed form using this model
C  S N d   Ke N d 
 rt
P  Ke
1 N d   1  N d S
ln S  r  s T
K
2
d 
 rt
2

s T
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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
(spot2a, time2)
(spot2b, time2)
val2a
val2b
pu
pd
val1
(spot1, time1)
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Pricing an Option with a Binomial Tree
2. Evolve the first timestep
T=0
Call payoff
1. Discretize the payoff at expiration,
choose normal vs. log-normal evolution
3. Repeat step 2 to cover the entire
lifetime of the option
1
Spot price
2
Spot price
T = 2Dt
Call payoff
Call payoff
T = Dt
3
Spot price
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Monte-Carlo
1.4
Generate a multitude of paths
consistent with desired
distribution and dynamics
1.3
1.2
For each path, compute the
value of the option
1.1
1
Appropriately average values
for all the paths
0.9
0.8
0.7
0
5
10
15
20
25
30
35
40
45
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
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Risk
Greeks for call, plotted vs. K / S
Delta
Gamma,
Vega
Theta
Rho
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Risk
ATM greeks, plotted vs. time
Call
Delta
Gamma
Vega
Theta
Rho
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Hedging
Delta - shares of stock
Rho - interest rate futures
Gamma,
Vega
Gamma, Vega, Theta - other options
Gamma, Theta ~ 1 T
Vega ~ T
Theta
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Spreads
A spread is a group of trades done together
Netting of risk
Often a cheaper way to take specific
positions
All spreads have at least two legs, but can
have many
Payoff
Some spreads are listed on exchanges,
many are OTC
Spot price
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Combo
A combo is a long call with a short put at the same strike
Payoff
Combo payoff
The payoff replicates a forward
Spot price
Spot price
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Call Spread
A call spread is a long call of one strike with a short call of another strike
Payoff
Call spread payoff
These can be bullish or bearish depending which strike is bought
Spot price
Spot price
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Straddle
A straddle is a long call with a long put at the same strike
Payoff
Straddle payoff
The payoff is a bet on volatility
Spot price
Spot price
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Butterfly
A butterfly is a combination of three equally spaced strikes in 1/-2/1 ratios
Payoff
Butterfly payoff
Butterflies pay off when the stock ends near the middle strike, price is probability
Spot price
Spot price
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Put-Call Parity
Compare a combo’s payoff with the payoff of a share of stock minus a bond
Payoff
Combo payoff
Call – Put = S – K e-rt
Spot price
Spot price
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Exotic Option Types
American - not solvable in closed form, so are they exotic?
Asian – payoff depends not on terminal spot, but on average spot over defined time
period
Bermudan – can only be exercised on predetermined dates, so something between
european and american
Binary (digital) – all-or-nothing depending on a condition being met
Cliquet (compound) – an option to deliver an option. Call on call, call on put, etc.
Knock-in/knock-out (barrier) – options that come into/go out of existence when a
condition is met, such as spot reaching a predetermined value
Variance/volatility/dividend swap – an agreement to exchange money based on
realized variance, volatility, or dividends
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American Options
Use a binomial tree, raise values to intrinsic at each time step
Raise this point to
intrinsic and
exercise!
If an option is raised to intrinsic, exercise it
Non-dividend calls don’t get exercised
Call payoff
Call payoff
Bermudan – same, but only raise to intrinsic at exercise dates
Spot price
Spot price
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Binary Options
Call spread payoff
Binary option payoff
Start with a call spread, bring the strikes closer together, and increase the number
of units of call spread
Spot price
Spot price
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Questions?
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