550.444 Modeling and Analysis Securities and Markets

1
550.444Modeling and AnalysisSecurities and
Markets

• April 28, 2008
• The Greeks and Sources of Option Risk

2
Where we are

• Last Week Applying Black-Scholes-Merton Theory
  for Option Analysis of Stock Indices, Currencies,
  and Futures (Chapter 14, OFOD)
• This Week The Greeks and Sources of Option
  Risk (Chapter 15, OFOD)
• Final Exam Thoughts
• Final Exam May 13th (Tuesday, 9am WH 304)

3
Assignment

• For Apr 29th May 1st (This Week)
• Read Hull Chapter 15
• Problems
• Chapter 14 4,13,23,30,3640,44

4
Plan for Today

• Brief Review of some Recent Material
• Black-Scholes-Merton Differential Equation
  Risk-Neutral Valuation
• Black-Scholes Formula
• Options on Dividend-Paying Stock, Index, and
  Currency
• Application of Black-Scholes-Merton Theory to
  Futures Options
• Sources of Risk In Option Positions
• Price of Underlying, Time, Volatility, and
  Interest Rate
• Delta, Gamma, Theta, Gamma, and Vega
• Discuss Exam, Course, Administrative Items, etc.

5
Previous Discussion

• Black-Scholes-Merton Differential Equation
• Derived from the Stock Price process and Itos
  lemma, knowing the derivative is a function of
  the Stock Price process
• Form riskless portfolio short one derivative
  long ? stock
• Gives the differential equation
• With boundary conditions, its solution describes
  the derivative
• Risk-Neutral Valuation applies as equation is
  independent of variable affected by risk
  preference only S, t, s, and r
• Principle of Risk-Neutral Valuation
• Assume the expected return of the underlying
  asset is r , i.e. µ r
• Calculate the expected payoff from the derivative
• Discount the expected payoff at the risk-free
  rate, r

6
Previously The Black-Scholes Formulas

• Black-Scholes Formulas for the Present Value of a
  European Call, c (Put, p ) with expiration T and
  strike K on a non-dividend paying stock with
  price S0
• where
• and

7
Last Week Option Results for a Stock paying a
known Dividend Yield

• Extend earlier results to a stock paying a
  dividend yield q
• Lower Bound for European calls
• Lower Bound for European puts
• Put-Call Parity
• Extending Black-Scholes formulas to a stock
  paying a dividend yield q (replace S0 by S0 e-qT
  )
• where

8
Last Week Black-Scholes Differential Equation
w/Dividend Yield

• The Black-Scholes-Merton differential equation
  for a stock with dividend yield
• The Black-Scholes equation is independent of all
  variables affected by risk preference
• Only variables are S, t, s, and r no expected
  return, µ
• Allows use of the principle of Risk-Neutral
  Valuation
• Assume the expected total return of the
  underlying asset is r
• The dividends provide a return of q , the stock
  price growth rate is r-q
• Calculate the expected payoff from the derivative
• When the growth rate is r-q , the expected stock
  price at T is S0 e(r-q)T
• Discount the expected payoff at the risk-free
  rate, r

9
Last Week The Binomial Model for a stock paying
a Known Dividend Yield

• Applicable to a stock paying a dividend yield, as
  covered earlier in Chapter 11
• To match stock price volatility, set
• Risk-neutral probability of an up move is chosen
  so the expected return is r-q over a time step
  of ?t and
• So
• With the derivative value

S0u u
p
S0
S0d d
(1 p )
10
Currency Options

• Currency options (both Euro- and American-style)
  trade on the Philadelphia Exchange (PHLX)
• There is also exists an active OTC market
• Currency options are used by corporations to buy
  insurance when they have an FX exposure
• Denote the foreign interest rate by rf
• When a U.S. company buys one unit of the foreign
  currency it has an investment of S0 dollars
• The return from investing at the foreign rate is
  rf S0 dollars
• This shows that the foreign currency provides a
  dividend yield at rate rf

11
European-Style Currency Options

• Foreign currency an asset with a dividend
  yield of rf
• Use the formula for an option on a stock paying a
  dividend yield with S0, the current exchange
  rate, and q r
• Black-Scholes gives
• where

12
European-Style Currency Options

• Alternatively, we can simplify the Black-Scholes
  formulas by using the forward rate, F0 , for
  maturity T
• Now Black-Scholes gives
• where

13
Mechanics of Futures Options

• Futures Options are American-Style
• When a Call futures option is exercised the
  holder acquires
• Long position in the futures
• Cash equal to the excess of the futures price
  over the strike price
• When a Put futures option is exercised the holder
  acquires
• Short position in the futures
• Cash equal to the excess of the strike price over
  the futures price
• If the futures position is closed out immediately
  at F0
• Payoff from Call F0 K
• Payoff from Put K F0

14
Futures Option Valuation from the Binomial
Approach

• Form the portfolio short one derivative and long
  ? futures
• The value at the end of one time period is
• when
• The value of the PF today is
• as the long future has no value at inception
• Substituting ? and simplifying gives
• where p (1 d)/(u d) as asserted back in
  Chapter 9

F0u u
F0
F0d d
15
Futures Prices Drift in a Risk-Neutral World

• Define Ft as the futures price at time t
• If we enter into a futures contract today its
  value is zero
• After a short increment, ?t , it provides a
  payoff
• If r is the ?t risk-free rate at time 0 ,
  risk-neutral valuation gives , as
  the contract has no value
• where denotes expectations in the
  risk-neutral world
• Thus we have similarly for
• so concatenating these results for any T
• The drift of the futures price in a risk-neutral
  world is zero
• From the stock price equation with dividend yield
  q equal to r

16
Valuing European Futures Options

• Black-Scholes formula for an option on a stock
  paying a dividend yield
• Set S0 current futures price, F0
• Set q risk-free rate, r , ensures the
  expected growth of F in is 0
• Results in
• where

17
Summary of Key Results

• We can treat stock indices, currencies, and
  futures like a stock paying a dividend yield of
  q
• For stock indices, q average dividend yield on
  the index over the option life
• For currencies, q r
• For futures, q r

18
Example

• A bank has sold for 300,000 a European call
  option on 100,000 shares of a nondividend paying
  stock
• S0 49, K 50, r 5, s 20,
• T 20 weeks, m 13
• The Black-Scholes value of the option is 240,000
• How does the bank hedge its risk to lock in a
  60,000 profit?

19
Naked Covered Positions

• Naked position
• Take no action
• Covered position
• Buy 100,000 shares today
• Both strategies leave the bank exposed to
  significant risk

20
Stop-Loss Strategy

• This involves
• Buying 100,000 shares as soon as price reaches
  50
• Selling 100,000 shares as soon as price falls
  below 50
• This deceptively simple hedging strategy does
  not work well

21
Delta (Underlying Price) Delta Hedging

• Delta (D) is the rate of change of the option
  price with respect to the underlying
• A delta hedge involves taking a position in size
  of -D of the underlying netting a neutral
  position

22
Delta (Underlying Price) Delta Hedging

• From the Black-Scholes formula we can find a
  closed form expression for the delta of a
  Euro-style Call
• Black-Scholes
• where
• By definition
• where and
• Since
• and
• Then

23
Delta (Underlying Price) Delta Hedging

• Similar to the derivation on the previous slide,
  for a Euro-style Put ? N(d1)-1
• The delta of a European call on a stock paying
  dividends at rate q is ? N (d 1)e qT
• The delta of a European put is ? e qT N (d
  1) 1
• In practice, the ? -hedged position must be
  frequently rebalanced as delta varies over a
  range of price movement
• Delta hedging a short option position involves a
  buy high, sell low trading rule
• Rarely a profitable investment strategy
• Lets look at an example (Table 15.1 15.2 on
  pages 350-351)

24
Delta (Underlying Price) Delta Hedging A
Simulation (15.2)
25
Using Futures for Delta Hedging

• The futures price for a contract on a
  non-dividend paying stock is FTS0 erT
• So as the price of the stock changes by ?S the
  futures price changes by ?S erT
• The delta of a futures contract is erT due to
  daily mark-to-market
• Contrast to the delta of a forward contract which
  is 1
• For a futures on an asset paying a dividend yield
  we can similarly see that the delta is e(r-q)T
• The delta of a futures contract is e(r-q)T times
  the delta of a spot contract of the asset, HF
  e(r-q)T HA
• The position required in futures for delta
  hedging is therefore e-(r-q)T times the position
  required in the corresponding spot contract

26
Theta (Time)

• Theta (Q) of a derivative (or portfolio of
  derivatives) is the rate of change of the value
  with respect to the passage of time
• The theta of a call or put is usually negative.
  This means that, if time passes with the price of
  the underlying asset and its volatility remaining
  the same, the value of the option declines
• We can develop a closed form for theta of a Euro
  Call as we did for delta from the Black-Scholes
  equation
• Use a slightly more general form to explicitly
  affirm today
• where

27
Theta (Time)

• Theta is the change in value due to a change in
  the passage of time
• as before
• so
• since
• Finally

28
Theta (Time)

• For a European Call

29
Gamma (Underlying Price)

• Gamma (G) is the rate of change of delta (D) with
  respect to a change in the price of the
  underlying asset
• Gamma is greatest for options that are close to
  the money
• Again we can develop a closed form for a
  Euro-Call
• Form for gamma is dominated by the form of the
  normal density function
• Maximized around the strike price for a given t

30
Gamma Addresses Delta Hedging Errors Caused By
Curvature

Call price
C''
C'
C
Stock price
S
S'
31
Interpretation of Gamma

• For a delta neutral portfolio,
• DP Q Dt ½GDS 2

DP
DP
DS
DS
Positive Gamma
Negative Gamma
32
Gamma (Underlying Price)

33
Relationship Between Delta, Gamma, and Theta

• The price of a single derivative on a
  non-dividend paying stock must satisfy the
  Black-Scholes differential equation
• A portfolio ? of such derivatives must also
  satisfy that differential equation
• When we use the risk notation we have
• For a delta hedged portfolio (?0) so
• When theta is large, gamma is large and negative
  ( vice versa)

34
Vega (Volatility)

• Vega (n) is the rate of change of the value of a
  derivatives portfolio with respect to volatility
• Vega tends to be greatest for options that are
  close to the money
• Again, a closed form for a Euro-Call
• is dominated by the normal distribution around
  the strike

35
Managing Delta, Gamma, Vega

• D can be changed by taking a position in the
  underlying
• To adjust G n it is necessary to take a
  position in an option or other derivative

36
Rho (Interest Rate)

• Rho is the rate of change of the value of a
  derivative with respect to the interest rate
• For currency options there are 2 rhos
• For the Euro-Call, a closed form results as

37
Hedging in Practice

• Traders usually ensure that their portfolios are
  delta-neutral at least once a day
• Whenever the opportunity arises, they improve
  gamma and vega
• As portfolio becomes larger hedging becomes less
  expensive

38
The End

• Questions?
• Final Exam
• Course