Binnenlandse Francqui Leerstoel VUB 2004-2005 2. Options and investments

1
Binnenlandse Francqui Leerstoel VUB 2004-20052.
Options and investments

• Professor André Farber
• Solvay Business School
• Université Libre de Bruxelles

2
Lessons from the binomial model

• Need to model the stock price evolution
• Binomial model
• discrete time, discrete variable
• volatility captured by u and d
• Markov process
• Future movements in stock price depend only on
  where we are, not the history of how we got where
  we are
• Consistent with weak-form market efficiency
• Risk neutral valuation
• The value of a derivative is its expected payoff
  in a risk-neutral world discounted at the
  risk-free rate

3
Mutiperiod extension European option

• u²S
• uS
• S udS
• dS
• d²S

• Recursive method
• (European and American options)
• ?Value option at maturity
• ?Work backward through the tree. Apply 1-period
  binomial formula at each node
• Risk neutral discounting
• (European options only)
• ?Value option at maturity
• ?Discount expected future value (risk neutral) at
  the riskfree interest rate

4
Multiperiod valuation Example

• Data
• S 100
• Interest rate (cc) 5
• Volatility ? 30
• European call option
• Strike price X 100,
• Maturity 2 months
• Binomial model 2 steps
• Time step ?t 0.0833
• u 1.0905 d 0.9170
• p 0.5024

• 0 1 2 Risk neutral
  probability 118.91 p²
• 18.91 0.2524
• 109.05
• 9.46
• 100.00 100.00 2p(1-p)
• 4.73 0.00 0.5000
• 91.70
• 0.00
• 84.10 (1-p)²
• 0.00 0.2476
• Risk neutral expected value 4.77
• Call value 4.77 e-.05(.1667) 4.73

5
From binomial to Black Scholes

• Consider
• European option
• on non dividend paying stock
• constant volatility
• constant interest rate
• Limiting case of binomial model as ?t?0

6
Convergence of Binomial Model
7
Understanding the PDE

• Assume we are in a risk neutral world

Expected change of the value of derivative
security
Change of the value with respect to time
Change of the value with respect to the price of
the underlying asset
Change of the value with respect to volatility
8
Black Scholes PDE and the binomial model

• We have
• Binomial model p fu (1-p) fd er?t
• Use Taylor approximation
• fu f (u-1) S fS ½ (u1)² S² fSS ft ?t
• fd f (d-1) S fS ½ (d1)² S² fSS ft ?t
• u 1 ?v?t ½ ?²?t
• d 1 ?v?t ½ ?²?t
• er?t 1 r?t
• Substituting in the binomial option pricing model
  leads to the differential equation derived by
  Black and Scholes
• BS PDE ft rS fS ½ ?² fSS r f

9
And now, the Black Scholes formulas

• Closed form solutions for European options on non
  dividend paying stocks assuming
• Constant volatility
• Constant risk-free interest rate

Call option
Put option
N(x) cumulative probability distribution
function for a standardized normal variable
10
Understanding Black Scholes

• Remember the call valuation formula derived in
  the binomial model
• C ? S0 B
• Compare with the BS formula for a call option
• Same structure
• N(d1) is the delta of the option
• shares to buy to create a synthetic call
• The rate of change of the option price with
  respect to the price of the underlying asset (the
  partial derivative CS)
• K e-rT N(d2) is the amount to borrow to create a
  synthetic call

N(d2) risk-neutral probability that the option
will be exercised at maturity
11
A closer look at d1 and d2
2 elements determine d1 and d2
A measure of the moneyness of the option.The
distance between the exercise price and the stock
price
S0 / Ke-rt
Time adjusted volatility.The volatility of the
return on the underlying asset between now and
maturity.
12
Example
Stock price S0 100 Exercise price K 100 (at
the money option) Maturity T 1 year Interest
rate (continuous) r 5 Volatility ? 0.15
ln(S0 / K e-rT) ln(1.0513) 0.05
?vT 0.15
d1 (0.05)/(0.15) (0.5)(0.15) 0.4083
N(d1) 0.6585
European call 100 ? 0.6585 - 100 ? 0.95123 ?
0.6019 8.60
d2 0.4083 0.15 0.2583
N(d2) 0.6019
13
Relationship between call value and spot price
For call option, time value gt 0
14
European put option

• European call option C S0 N(d1) PV(K) N(d2)
• Put-Call Parity P C S0 PV(K)
• European put option P S0 N(d1)-1
  PV(K)1-N(d2)
• P - S0
  N(-d1) PV(K) N(-d2)

Risk-neutral probability of exercising the option
Proba(STgtX)
Delta of call option
Risk-neutral probability of exercising the option
Proba(STltX)
Delta of put option
(Remember N(x) 1 N(-x)
15
Example

• Stock price S0 100
• Exercise price K 100 (at the money option)
• Maturity T 1 year
• Interest rate (continuous) r 5
• Volatility ? 0.15

N(-d1) 1 N(d1) 1 0.6585 0.3415
N(-d2) 1 N(d2) 1 0.6019 0.3981
European put option - 100 x 0.3415 95.123 x
0.3981 3.72
16
Relationship between Put Value and Spot Price
For put option, time value gt0 or lt0
17
Dividend paying stock

• If the underlying asset pays a dividend,
  substract the present value of future dividends
  from the stock price before using Black Scholes.
• If stock pays a continuous dividend yield q,
  replace stock price S0 by S0e-qT.
• Three important applications
• Options on stock indices (q is the continuous
  dividend yield)
• Currency options (q is the foreign risk-free
  interest rate)
• Options on futures contracts (q is the risk-free
  interest rate)

18
Black Scholes Merton with constant dividend yield
The partial differential equation(See Hull 5th
ed. Appendix 13A)
Expected growth rate of stock
Call option
Put option
19
Options on stock indices

• Option contracts are on a multiple times the
  index (100 in US)
• The most popular underlying US indices are
• the Dow Jones Industrial (European) DJX
• the SP 100 (American) OEX
• the SP 500 (European) SPX
• Contracts are settled in cash
• Example July 2, 2002 SP 500 968.65
• SPX September
• Strike Call Put
• 900 - 15.601,005 30 53.501,025 21.40 59.80
• Source Wall Street Journal

20
Fundamental determinants of option value
Call value Put Value
Current asset price S Delta ? 0 lt Delta lt 1 ? - 1 lt Delta lt 0
Striking price K ? ?
Interest rate r Rho ? ?
Dividend yield q ? ?
Time-to-maturity T Theta ? ?
Volatility Vega ? ?
21
Example
22
The Greeks

• Delta
• Gamma
• Theta
• Vega (not a Greek)
• Rho

23
Delta

• Sensitivity of derivative value to changes in
  price of underlying asset
• Delta ?f / ?S
• As a first approximation ?f Delta x ?S
• In example, for call option f 10.451 Delta
  0.637
• If ?S 1 ?f 0.637 ? f 11.088
• If S 101 f 11.097
  error because of convexity

Binomial model Delta (fu fd) / (uS
dS) European optionsDelta call e-qT
N(d1)Delta put Delta call - 1
Forward Delta 1 Call 0 lt Delta lt 1 Put
-1 lt Delta lt 0
24
Calculation of delta
25
Variation of delta with the stock price for a call
26
Delta and maturity
27
Delta hedging

• Suppose that you have sold 1 call option (you are
  short 1 call)
• How many shares should you buy to hedge you
  position?
• The value of your portfolio is
• V n S C
• If the stock price changes, the value of your
  portfolio will also change.
• ?V n ?S - ?C
• You want to compensate any change in the value of
  the shorted option by a equal change in the value
  of your stocks.
• For small ?S ?C Delta ?S
• ?V 0 ? n Delta

28
Effectiveness of Delta hedging
29
Gamma

• A measure of convexity
• Gamma ?Delta / ?S ?²f / ?S²
• Taylor df fS dS ½ fSS dS²
• Translated into derivative language
• ?f Delta ?S ½ Gamma ?S²
• In example, for call f 10.451 Delta
  0.637 Gamma 0.019
• If ?S 1 ?f 0.637 ½ 0.019 ? f 11.097
• If S 101 f 11.097

30
Variation of Gamma with the stock price
31
Gamma and maturity
32
Gamma hedging

• Back to previous example.
• We have a delta neutral portfolio
• Short 1 call option
• Long Delta 0.637 shares
• The Gamma of this portfolio is equal to the gamma
  of the call option
• V n S C ??V²/?S² - Gammacall
• To make the position gamma neutral we have to
  include a traded option with a positive gamma. To
  keep delta neutrality we have to solve
  simultaneously 2 equations
• Delta neutrality
• Gamma neutrality

33
Theta

• Measure time evolution of asset
• Theta - ?f / ?T
• (the minus sign means maturity decreases with the
  passage of time)
• In example, Theta of call option - 6.41
• Expressed per day Theta - 6.41 / 365 -0.018
  (in example)
• Theta -6.41 /
  252 - 0.025 (as in Hull)

34
Variation of Theta with the stock price
35
Relation between delta, gamma, theta

• Remember PDE

Gamma
Theta
Delta
36
Trading strategies

• A single option and a stock covered call,
  protective put
• Covered call S-C
• Protective put SP
• Spreads bull, bear, butterfly, calendar
• Bull C(X1) C(X2) X1ltX2
• Bear C(X1) C(X2) X1gtX2
• Butterfly C(X1) C(X3) 2C(X2)
  X1ltX2ltX3
• Calendar C(T1)-C(T2) T1gtT2
• Combinations straddle, strips and straps,
  strangle
• Straddle CP
• Strip C 2P
• Strap 2CP
• Strangle C(X2)P(X1)
  X1ltX2

37
Protective Put
38
Equity Linked Note

• (See Lehman Brother Equity Linked Note An
  Introduction)

Equity Linked Note
Capital garantee
Bond




Equity Participation
Call option
39
Equity Linked Note Example

• 5-year 100 principal protected ELN with 100
  participation in the upside of the SP 500 index.
• See Excel file.

40
Covered Call
Profit
At maturity
Immediate
Stock Price
41
Reverse Convertible

• Robeco Eerste Reverse Convertible op
  beleggingsfonds
• Van 17 februari tot 6 maart 2003 uur is het
  mogelijk in te schrijven op de Robeco Reverse
  Convertible op Robeco N.V. mrt 03/04 (Robeco
  Reverse Convertible), uitgebracht door Rabo
  Securities in samenwerking met Robeco.
• De Robeco Reverse Convertible is een
  obligatielening met een looptijd van één jaar
  waarop een couponrente van 9 wordt gegeven,
  hoger dan een gewone éénjaarslening. De
  uitgevende instelling, Rabo Securities N.V.,
  heeft aan het einde van de looptijd de keuze om
  de obligatie af te lossen in contanten of af te
  lossen in een van tevoren vastgesteld aantal
  aandelen in het beleggingsfonds Robeco. Dit is
  afhankelijk van de koers van het aandeel Robeco
  N.V. Bijzondere omstandigheden daargelaten, zal
  Rabo Securities kiezen voor een aflossing in
  aandelen als de koers aan het einde van de
  looptijd lager is dan die op 7 maart 2003. Het
  aantal aandelen is gelijk aan de nominale inleg
  gedeeld door de openingskoers van Robeco op 7
  maart 2003. Hierdoor bestaat het risico voor de
  belegger aan het einde van de looptijd aandelen
  Robeco te ontvangen, die een lagere waarde
  vertegenwoordigen dan de nominale inleg. Is de
  koers per saldo gelijk gebleven of gestegen, dan
  wordt de nominale inleg in contanten
  teruggegeven.
• .

42
Portfolio insurance

• Use synthetic put option with dynamic hedging
• V S P same value as with put
• ?V ?S ?P same sensitivity to underlying
  asset
• (1 dPut) ?S
• V n S B n shares bond
• 1 dPut n
• Dynamic hedging
• LOR and the crash of October 19, 1987 see
  Rubinstein 1999
• Illustration Excell worksheet PorfolioInsurance

43
Bull Call Spread
44
Bear Call Spread
45
Butterfly
46
Straddle
47
Strip
48
Strap
49
Strangle