Title: 550.444 Modeling and Analysis Securities and Markets
1550.444Modeling and AnalysisSecurities and
Markets
- April 28, 2008
- The Greeks and Sources of Option Risk
2Where 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)
3Assignment
- For Apr 29th May 1st (This Week)
- Read Hull Chapter 15
- Problems
- Chapter 14 4,13,23,30,3640,44
4Plan 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.
5Previous 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
6Previously 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
7Last 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
8Last 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
9Last 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 )
10Currency 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
11European-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
12European-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
13Mechanics 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
14Futures 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
15Futures 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
16Valuing 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
17Summary 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
18Example
- 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?
19Naked Covered Positions
- Naked position
- Take no action
- Covered position
- Buy 100,000 shares today
- Both strategies leave the bank exposed to
significant risk
20Stop-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
21Delta (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
22Delta (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
23Delta (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)
24Delta (Underlying Price) Delta Hedging A
Simulation (15.2)
25Using 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
26Theta (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
27Theta (Time)
- Theta is the change in value due to a change in
the passage of time - as before
- so
- since
- Finally
28Theta (Time)
29Gamma (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
30Gamma Addresses Delta Hedging Errors Caused By
Curvature
Call price
C''
C'
C
Stock price
S
S'
31Interpretation of Gamma
- For a delta neutral portfolio,
- DP Q Dt ½GDS 2
DP
DP
DS
DS
Positive Gamma
Negative Gamma
32Gamma (Underlying Price)
33Relationship 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)
34Vega (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
35Managing 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
36Rho (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
37Hedging 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
38The End
- Questions?
- Final Exam
- Course