The Greek Letters

1
The Greek Letters
2
Pricing Options

• Both the Binomial Tree Approach and the Black
  Scholes approach produce the same option value as
  the number of steps in the Binomial tree becomes
  large.
• For this section we will concentrate on the
  Theoretical value of the option the black
  scholes solution.

3
Black Scholes

• Value of Call Option SN(d1)-Xe-rtN(d2)
• S Current value of underlying asset
• X Exercise price
• t life until expiration of option
• r riskless rate
• s2 variance
• N(d ) the cumulative normal distribution (the
  probability that a variable with a standard
  normal distribution will be less than d)

4
Black Scholes (Intuition)

• Value of Call Option
• SN(d1) - Xe-rt N(d2)
• The expected PV of cost Risk Neutral
• Value of S of investment Probability of
• if S X S X

5
Black Scholes

• Value of Call Option SN(d1)-Xe-rtN(d2)
• Where

6
Time Value of an Option

• The time value of an option is the difference in
  the theoretical price of the option and the
  intrinsic value.
• It represents the the possibility that the value
  of the option will increase over the time it is
  owned.

7
An Example1

• Assume that a financial institution has sold a
  European Call Option on a non dividend paying
  stock.
• S 49, X50, r 0.05, s0.20, t 20 weeks
  0.3846 years. Call option value 2.372
• Assume that the institution has sold the option
  for 3 a share or .628 more than its theoretical
  value.
• How can it hedge its risk?

8
Naked vs. Covered position

• The firm can do nothing and hold only the option
  (a naked position). It would then be forced to
  buy the shares if the owner of the option
  exercises it in 20 weeks. The profit diagram
  would look like the normal short call.
• The firm can buy the stock today and have a
  covered call. This introduces a downside risk,
  if the value of the stock decreases the firm
  looses due to the decline in the value of the
  share.

9
Profit Diagram Covered Call

• Profit

Long Spot
Covered Call
Short Call
10
Hedging with a Stop-Loss Strategy.

• One possible solution is to develop a dynamic
  buying strategy for the share. For example the
  firm could buy shares whenever the stock price is
  greater than the exercise price, It could then
  sell the shares if the stock price drops below
  the exercise price.
• It would then be hedged when the option will be
  exercised and unhedged when it will not be
  exercised.

11
Stop Loss Costs

• The problem is that there are substantial
  transaction costs associated with the strategy.
• Also there is uncertainty about the actual cost
  of the share. Therefore you are not buying and
  selling each time at the exercise price.
• A better approach is to use the delta of the
  option

12
Delta of an option

• The delta of the option shows how the theoretical
  price of the option will change with a small
  change in the underlying asset.

13
Time Value of the Option

• Plotting the value of the option compared to the
  profit and or payoff provides a starting point to
  explaining delta.
• Using the option above the following prices were
  obtained and graphed on the next slide.
• Stock Call Stock Call Stock Call
• 42 0.173 50 2.962 58 9.211
• 46 1.107 54 5.732 62 13.025

14
Time Value of Option
15
Call Option Value
16
Delta Graphically
17
Delta of an option

• Intuitively a higher stock price should lead to a
  higher call price. The relationship between
  changes in the call price and the stock price is
  expressed by a single variable, delta.
• The delta is the change in the call price for a
  very small change it the price of the underlying
  asset.

18
Calculating Delta

• Delta can be found from the call price equation
  as
• Using delta hedging for a short position in a
  European call option would require keeping a long
  position of N(d1) shares at any given time. (and
  vice versa).

19
Delta explanation

• Delta will be between 0 and 1.
• A 1 cent change in the price of the underlying
  asset leads to a change of delta cents in the
  price of the option.

20
Delta and the stock price

• For deep out-of-the money call options the delta
  will be close to zero. A small change in the
  stock price has little impact on the value of
  the option
• For deep in the money options delta will be close
  to 1. A small change in the stock price will
  have an almost one to one change in the option
  price.

21
Delta vs. Share Price
22
Delta and Time to MaturityX50 r0.05 s0.2
23
Delta X35 r0.05 s.2
24
Example of Delta Hedging

• Assume that we had sold the option in our example
  for 100,000 shares of stock.
• Using the information from before
• S 49, X50, r 0.05, s0.20, t 20 weeks
  0.3846 years. Call option value 2.3715
• Given 100,000 shares the value of the option is
  237,150
• Assuming a share price of 49, the delta of the
  option is .5828594

25
The hedged position

• The bank has a portfolio of delta shares for each
  share it has written an option on.
• This implies it owns 100,000(.5828594) 58,286
  shares.
• If the share price increases by 1 the value of
  the shares will increase by 58,286
• However the value of the option will decline.

26
Option value

• The value of the option at a price of 50 is
  2.926.
• Therefore the value of the option will decrease
  by (2.926 2.3715)100,000 55,450

27
Total position

• Gain on Spot position 58,286
• Loss on Option - 55,450
• Net change in portfolio 2,836
• They do not perfectly offset due to the size of
  the price change and rounding errors. The total
  value of the portfolio would change from
  3,093,161.06 to 3,095,997.00

28
Dynamic Hedging

• Since the value of delta changes at each stock
  price the amount of shares would need to be
  adjusted to keep the portfolio value hedged.
• The larger the price change the less successful
  the hedge.

29
Delta of a portfolio

• The delta of a portfolio of options is simply the
  weighted average of the individual deltas. Where
  the weight corresponds to the quantity of the
  option.
• It is therefore possible to adjust the delta of a
  portfolio quickly by adjusting one or more of the
  option positions.

30
Delta of a put option

• A long position in a put option should be hedged
  with a long position in the stock, (delta will be
  negative).
• Delta for the put is given by N(d1) 1
• Similar to call options, for deep in the money
  puts (Asset price is less than exercise price)
  the value of delta will be close to -1. For
  delta out of the money puts the delta will be
  close to zero.

31
Delta Hedging

• The delta neutral portfolio removes much but not
  all of the risk associated with the position.
• Looking at the value of the portfolio for a small
  range of prices changes provides a good
  indication of the ability of the hedge to remove
  the risk associated with a change in the stock
  price.
• The change hedge is not perfect because the value
  of the option is not a linear function with
  relation to changes in the stock price. Consider
  the previous portfolio.

32
Value of Delta Neutral Portfolio(1 Short Call
Delta Shares) x 100,000 shares
33
Gamma

• Gamma measures the curvature of the theoretical
  call option price line.

34
Gamma of an Option

• The change in delta for a small change in the
  stock price is called the options gamma
• Call gamma

35
Gamma Graphically
Gamma measures the amount of curvature In the
call price relationship, The reason the portfolio
Is not perfectly hedged is because delta provides
only a linear estimate of the call price change.
The hedge error is from the difference between
the estimate from delta and the actual
relationship
36
Gamma

• If gamma is small it implies that delta changes
  slowly which implies the cost to adjust the
  portfolio will be small.
• If gamma is large it implies that delta changes
  quickly and the cost to keep a portfolio delta
  neutral will be large.

37
Gamma Cont

• Gamma is the adjustment for the fact that the
  call option price dos not have a linear
  relationship with the spot price.
• Delta provides a linear approximation of the
  change in the value of the call option that is
  less accurate the large the change in the stock
  price.
• The impact of gamma is easy to see in our earlier
  example.
• The impact of gamma will be the largest when the
  stock price is close to the exercise price

38
Gamma

• The gamma of a non dividend paying stock option
  will always be positive (the larger the change in
  the stock price the larger the change in the
  value)

39
Gamma and Stock Price

• The impact of gamma will be the largest when the
  stock price is close to the exercise price.
• For deep in the money or deep out of the money
  call options gamma will be relatively small.

40
Gamma and Time to Maturity

• Gamma will be highest for at the money options
  close to maturity.
• Gamma will be low for both in the money and out
  of the money options that are close to maturity.

41
Gamma vs. Stock PriceX50, r 0.05, s.2, t.5
42
Gamma vs Time to MaturityX35, r0.05, s.2
43
Other Measures

• The sensitivity of the value of the option to a
  change in the expiration of the option is
  measured by theta

44
Theta

• Theta is generally negative for an option since
  as the time to maturity decreases the value of
  the option becomes less valuable. (Keeping
  everything else constant, as time passes the
  value of the option decreases).

45
ThetaX35, r0.05, t.5, s.2
46
Theta vs Time X35, s.2, r0.05
47
Relationship between Delta, Theta and Gamma

• From the derivation of the Black-Scholes Formula
  it can be shown that
• (We will show this soon)
• In a Delta Neutral portfolio delta 0 and the
  portfolio value remains relatively constant.
  This implies that if Theta is negative, Gamma
  needs to be of similar size and positive and vice
  versa. Therefore Theta is often considered as a
  proxy for Gamma.

48
Vega (or Kappa)

• The rate of change of the option value with
  respect to the volatility of the underlying
  asset is given by the Vega (also sometimes called
  kappa)
• The Black Scholes Model assumes that volatility
  is constant, so in theory this seems to be
  inconsistent with the model.
• However variations of the Black Scholes do allow
  for stochastic volatility and their estimates of
  Vega are very close to those form the Black
  Scholes model so it serves as an approximation.

49
Vega

• Vega will be highest for options that are at the
  money. As the option moves into or out of the
  money the impact of a volatility change is
  decreased.

50
Rho

• The final measure is the change in the value of
  the option with respect to the change in the
  interest rate. As we have discussed the interest
  rate has the smallest impact on the value of the
  option. Therefore this is not used often in
  trading.

51
Market Making and Delta Hedging

• Market Maker Individual who is ready to both
  sell and buy a given asset.
• Bid price Price market maker is willing to pay
  when buying the asset
• Ask price Price market maker is willing to
  accept to sell the asset
• A market maker can end up with an arbitrary
  position as a result of fullfilling orders this
  represents a risk that needs to be hedged. In
  the options market this is done with delta
  hedging.

52
Market Maker

• Assume that the market maker receives an order
  for a call option.
• The market maker can
• Leave the position unhedged
• Buy shares of the stock (a covered call) so tht
  if the option is exercised the firm will be able
  to provide the stock
• Use delta to hedge the risk

53
An Example

• Assume that a firm is writing the following
  option on 100 shares of stock.
• S 40
• X40
• s.30
• r.08
• t91/365
• This implies a call price of 2.7804 per share
• And a Delta of .5824

54
No Price Change

• If the price of the stock does not change the
  market maker realizes a profit of approximateloy
  1.7 cents.
• This is due to the time value of the option

55
The unhedged position with a price increase

• Assume that the stock price increases to 40.75
• At the new stock price the new value of the call
  is 3.2352
• This implies a loss of 2.7804-3.2352 .4548

56
Profit / Loss after holding one day
57
Price increase revisited

• Inhte case of the price increase to 40.75, the
  position decreased by 0.4548
• If the market maker had hedged using the delta of
  .5824, the value of the shares would have
  increased by
• 0.75(.5824) .4368
• The value of the change in the option is
  understated by approximately 0.018 due to the
  price increase. (similarly a decrease of the
  stock price would result in an overstatement of
  the change in the option price)

58
Delta Hedging for two days

• Assume that the market maker uses delta and buys
  58.24 shares to offset the option written on 100
  shares of stock.
• That represents a net investment of
• 58.24(40) - 278.04 2051.56
• Assume that the market maker borrowed the money
  and the interest charge for one day is then
• 2051.56e0.08/365-1 .45

59
Day 1

• Assume the stock price increases to 40.50

60
Rebalancing

• The new delta is .6142
• This implies the need to buy .6142-.5824(100)
  3.18 new shares of stock at 40.50
• This has a total cost of 128.79

61
Day 2

• Assume the stock now falls to 39.25 there is a
  gain on the options and a loss on the shares

62
Sources of cash flow

• Borrowing - limited by the market value of the
  securities in the portfolio.
• Purchase or sale of shares
• Interest

63
Delta hedging for several days

• Gain on loss on a daily basis depends upon 3
  things
• Gamma If there is a large change in the stock
  price the market maker becomes unhedged (dealt
  does not represent the actual change well).
• Theta If there is no change in the price of the
  share there is a gain from time vlaue
• Interest Cost there is a net carrying cost to
  purchasing the stock

64
A self financing portfolio

• The size of the stock change has a large impact
  on the delta neutral outcome.
• Assume that the share price increases or
  decreases by exactly one standard deviation each
  day.

65
Profit / Loss
66
Intuition

• If the stock price moves by one standard
  deviation each day in the binomial tree model, it
  would be approximately self financing!
• Next Relating the market makers profit or loss
  to the relationship between gamma, theta, and
  delta.

67
Adding Gamma

• From the example before we know that
• S 40 c2.7804 D.5824
• S40.75 c3.2352 D.6142
• Assuming we want to estimate the option price at
  40.75 one way to do this would be to use the
  delta, but this creates error since the
  sensitivity of the option changes as the price
  increases.
• c40.75 c40.75(D40.75) 2.7804.5824 3.362

68
Correcting for change in delta

• Another approach would be to average the two
  deltas. If the stock price change is small, the
  average of the two deltas should be an
  approximation of the actual price change.
• (D40D40.75)/2 (.5824.6142)/2 .5983
• The new call price would be
• C40.75 2.7804.75(.5983) 3.229 Which is
  closer to the actual price of c3.2352 than using
  delta alone

69
Second Approximation

• However if we are going to calculate Delta at the
  new price we might as well calculate the new
  price directly. Another approximation would be to
  use gamma of the option at 40.
• An approximation of D at 40.75 could be found by
  D40.75D40.75G40
• Then the new share price could be calculated by
  substituting D at 40.745 in the average delta
  equation above.

70
Delta Gamma Approximation

• Daverage (D40D40.75)/2
• D40.75D40.75G40
• Daverage (D40D40.75G40)/2
• C40.75 c40.75(Daverage)
• c40.75((D40D40.75G40)/2)
• c40.75((D40D40.75G40)/2)
• c40.75((D40(1/2)(.75)G40)
• c40.75D40(1/2)(.752) G40

71
Delat Gamma Approximation

• c40.75D40(1/2)(.752) G40
• Given a gamma .0652
• 2.7804.75(.5983)(1/2)(.0652)(.752)3.2355
• Compared to the actual price of c3.2352

72
Generalization

• Replacing the values for the share price with St
  for the initial price and Sth for the price
  after a small change of e Sth St
• Gamma can be approximated by the change in delta
  per dollar of stock price change or
• GSt(DSh-DS)/e
• Rearranging
• DSh eGStDS
• If gamma is constant (assuming a small change in
  price makes this assumption realistic) this
  approximation of delta will be exact.

73
Computing the option price change

• If gamma is constant, we can use the average
  delta to calculate a delta to use to approximate
  the new price
• DSh eGStDS
• Daverage(DShDS)/2
• Daverage(eGStDSDS)/2 DS(1/2)eGSt

74
The approximate share price

• CSthCSte DaverageCSte (DS(1/2)eGSt)
• CSte DSt(1/2)(e2)GSt
• Regardless of the direction of the price change
  the gamma correction adds back to the delta
  correction by itself. This makes sense given the
  graph of the value of the option.
• There is still an error since gamma changes as
  the stock price changes (the assumption of no
  change in gamma is close for small changes in
  stock price)

75
Adding Theta

• Theta attempts to account for the change in time
  as the option moves toward expiration.
• For a given period of h the change from the
  passing of time will be qh. Theta is a yearly
  number, so if we have a 1 day change it implies a
  q/365 change in the value of the option.
• Adding this to our earlier equation provides a
  new call price of
• CSte DSt(1/2)(e2)GStqh

76
The Market Makers Profit

• THE value of the Market Makers position is equal
  to being long delta shares of stock and short one
  call or
• DSt-cSt
• Assume that over time period h the stock price
  changes to Sth. The chang in the value of the
  portfolio will be given by

77
Change in Portfolio value

• Assume that over time period h the stock price
  changes to Sth. The change in the value of the
  portfolio will be given by
• D(Sth-St) - (CSth-Cs) - rh(DSt-CSt)

Change in Stock Value
Change in Value of Option
Interest Expense
CSte DSt(1/2)(e2)GStqh
78
Market Makers Profit

• CSthCSte DSt(1/2)(e2)GStqh
• Substitute
• D(Sth-St)-(CSth-Cs)-rh(DSt-CSt)
• After rearranging the market makers profit
  becomes
• -(1/2)(e2)GStqh -rh(DSt-CSt)

79
Impact of the change in the stock price

• (1/2)(e2)GStqh -rh(DSt-CSt)
• It is the magnitude, not the direction, of the
  change in the tock price (shown by e) in the
  profit equation that is important.
• Previously we showed that a one standard
  deviation change in the stock price produced a
  delta neutral portfolio (where there was no
  impact on the value of the portfolio)

80

• (1/2)(e2)GStqh -rh(DSt-CSt)
• If s is measured annually then a one-standard
  deviation move over a period of length h is equal
  to
• Therefore a squared move of one standard
  deviation is

81
Market Makers Profit

• substitute
• -(1/2)(e2)GStqh -rh(DSt-CSt)
• Market
• Makers -(1/2)(s2S2)GSt q -r(DSt-CSt)h
• Profit

82
Relation to Black Scholes

• Black and Scholes argued that the market maker
  should earn a risk free return if hedged with the
  option. This implies that the profit for each
  time period will be zero.
• Market
• Makers -(1/2)(s2S2)GSt q -r(DSt-CSt)h
  0
• Profit
• Dividing by h and rearranging produces
• (1/2)(s2S2)GSt q - rDStrCSt

83
Black Scholes Equation

• The equation on the last slide is known as the
  Black Scholes Partial Differential Equation and
  is a fundamental component of valuing financial
  assets, both riky and risk free.
• The equation holds for both American and European
  options (both calls and puts) assuming that
  volatility and the interest rate are constant and
  that the stock moves one standard deviation over
  small intervals of time and that there are no
  dividends paid on the stock.

84
Limits on the Equation

• While the equation holds in general, it fails if
  an American option is so deep in-the-money that
  it is optimal to early exercise the option.
• Think about an American Option that is deep in
  the money with an option price equal to X-S.
  Then D-1, G0, and q0
• (1/2)(s2S2)GSt q - rDStrCSt
• (1/2)(s2S2)(0) 0 - rDStr(X-St)
• 0 rX

85
Intuition

• If you are the owner of the option and have delta
  hedged it you loose interest ont eh strike
  price you could receive if it is exercised.
• If you have written the option and it is ot
  exercised, you are earning a risk free arbitrage
  profit of rX.
• In other words when it is deep-in-the money it
  should not earn the risk free rate and the
  equation should not hold.

86
One standard deivation change

• There is not reason to expect that the price will
  only change by one standard deviation.
• It shold change by one standard deviation or less
  approximately 68 of the time. If it is less
  than one standard deviation there is a light
  profit to the market maker.
• The other 32 of the time the market maker has a
  loss (the large the price change the larger the
  loss).
• However, the mean return on the position works
  out to being close to 0!

87
Re-hedging

• There is a cost to keeping the hedge in force.
• Often in practice the market maker will Re-Hedge
  infrequently.
• The cost to this is that there are less
  observations or chances to rehedge resulting in a
  mean return of 0

88
Delta Hedging in practice

• How can the risk of extreme price moves
  (resulting in a large loss on the market makers
  portfolio) be eliminated?
• There are many possible strategies that combine
  other options to hedge the risk of the large
  change in the stock price.

89
Strategy 1

• Buying out of the money options. An out of the
  money put or call (or both) could be bought so
  that they profit when there is a large decline or
  increase in the stock price. This offsets the
  loss in the portfolio.
• The downside to this is that there is a cost
  However, if the options are far out of the money
  they will be cheap.
• This will not work in aggregate since there still
  has to be a market maker willing to write the out
  of the money options as well.

90
Strategy 2 Static Option Replication

• By setting the bid and ask prices to help hedge.
• For example if you could buy a put with the same
  strike price of the written call (and also buy
  the underlying stock) you would be perfectly
  hedged.
• By adjusting the bid price so that anyone selling
  the call is willing to accept it, the market
  maker could hedge, but at a cost of buying both
  the put and stock.

91
Strategy 3 Gamma Neutral

• You can buy or sell options with a gamma that
  offsets the gamma of the original position.
• Assume you have sold the following call option
• X40, t.25, r.08, s.3
• Buying the following call option with the same
  gamma will hedge your position
• X45 t .33, r.08, s.3

92
Ratio of gammas

• To be gamma hedged you will need more than 1 of
  the X45 option for each of the X 40
• The ratio of their gammas provides the proper
  hedge ratio
• GX40,t.25 /GX40,t.25
• 0.0651063/.0524381.2408
• The resulting portfolio is shown in the next slide

93
(No Transcript)
94
Overnight Profit
Delta and gamma Neutral
Delta Neutral
95
Why not Gamma Hedge?

• The profits from writing the intital all will go
  to buyng the other call elimnating the market
  makers profit
• In aggregate it is not possible for all market
  makers to gamma hedge. Most end users buy puts
  and calls resulting in them having a positive
  gamma implying market makers are selling puts
  and calls resulting in a negative gamma.

96
Miscellaneous Topics

• Implied Volatility
• Volatility Smiles and Skews

97
Implied Volatility

• The one input in Black Scholes that cannot be
  observed is volatility
• Implied volatility is calculated as the
  volatility that would provide the observed option
  price when used in the Black Scholes equation
• The calculation needs to be done based upon an
  iterative process, since the volatility cannot be
  calculated directly.

98
Foreign Currency Options

• For foreign currency options the implied
  volatility is lowest for options at the money.
• As an option moves significantly out of the
  money or in the money the implied volatility
  increases.
• This creates a smile when graphing implied
  volatility

99
Implied Volatility
Strike Price
100
Implied distribution

• Given the volatility smile it is possible to
  calculate the risk neutral probability
  distribution for an asset as a future time
• Generally this distribution will be narrower than
  the assumed log normal distribution with the same
  mean and standard deviation and the implied
  distribution will have heavier tails (high
  probability of a larger rise or fall in the stock
  price.

101
Why not Log normal?

• The log normal distribution assume that
• Volatility of the asset price is constant
• The price changes smoothly with no jumps
• Neither of these conditions hold for exchange
  rates.
• The impact of this will be greater for longer
  maturity options

102
Volatility Skew

• For equity options the volatity decrease as the
  strike price increases.
• This implies that a lower volatility is used to
  price a deep out of the money call (or in the
  money put) option as opposed to a deep in the
  money option call (or out of the money put).
• The implied distribution is thus skewed to the
  left with a fatter tail to the left and a
  skinnier tail to the right. It is also narrower
  overall.

103
Why Equities have a Skew

• The skew was not apparent until after the stock
  market crash in October 1987
• Crashphobia?
• A decline in stock price decreases the market
  value of the firms equity, thus increasing its
  leverage. This might cause an increase in the
  volaitlity of the stock price.

104
Greeks and Smiles

• The greeks need to be adjusted if there is a
  smile or skew.
• Sticky strike Rule The greeks are correct if
  implied volatility was used in their calculation
  (assumes that implied volatility stays constant
  the next day)
• Better models are shown in chapter 24 (but not
  covered in the class).

105
Large Anticipated Jumps

• The final possibility is a bimodal distribution
  where large anticipated jumps in price are
  expected.
• In this case he implied volatility can take on
  the shape of a frown.

106
Stocks paying a known dividend yield

• In theory, the payment of a dividend at rate q
  will lower the growth rate of the stock compared
  to if it paid no dividend.
• The current value of the stock will decline by
  the rate q.
• If the stock does not pay a dividend it will
  grow to Sqert by the end of period t. Or is
  would grow to S at time t starting at S0e-qt
  today.
• The distribution of the prices of these two
  stocks should be the same.

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Black Scholes

• Given that the probability distributions are the
  same we can replace S in the black scholes
  equation by S0e-rt This makes the call price
  equal to
• Se-qtN(d1) -Xe-rtN(d2)

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109
Black Scholes Equation

• The differential equation can also be adjusted
  for the dividend growth rate.
• (1/2)(s2S2)GSt q - (r-q)DStrCSt
• Since the total return needs to be r in the risk
  neutral world this implies that the growth rate
  of the stock must be r-q.

(1/2)(s2S2)GSt q - rDStrCSt
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Currency Options

• Similarly, foreign currency options need to
  account for the rate of interest paid in the
  foregin economy. IN this case S is replaces by
• S0e-rf(t) where rf is the foreign risk free rate
  of interest. Again the equation for d1 and d2
  would be adjusted in a fashion similar to the
  dividined yield.

111
Futures Options

• For the futures option the futures price is
  effective at a future point in time. Therefore
  the PV of the futures price will replace S in the
  Black Scholes equation and F0 replaces S in the
  d1 equation
• call price Fe-rtN(d1) -Xe-rtN(d2)

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