pricing cont’d & beginning greeks
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Pricing Cont’d & Beginning Greeks. Assumptions of the Black-Scholes Model. European exercise style Markets are efficient No transaction costs The stock pays no dividends during the option’s life Interest rates and volatility remain constant. - PowerPoint PPT PresentationTRANSCRIPT
Pricing Cont’d &
Beginning Greeks
Assumptions of the Black-Scholes Model
European exercise style Markets are efficient No transaction costs The stock pays no dividends during
the option’s life Interest rates and volatility remain
constant
The Stock Pays no Dividends During the Option’s Life
If you apply the BSOPM to two securities, one with no dividends and the other with a dividend yield, the model will predict the same call premium
Robert Merton developed a simple extension to the BSOPM to account for the payment of dividends
The Stock Pays Dividends During the Option’s Life (cont’d)
Adjust the BSOPM by following (=continuous dividend yield):
Tdd
T
TRXS
d
dNXedSNeC RTT
*1
*2
2
*1
*2
*1
*
and
2ln
where
)()(
Interest Rates and Volatility Remain Constant
There is no real “riskfree” interest rate
Often use the closest T-bill rate to expiry
Volatility expectations change constantly. That’s why option prices can change when everything else remains constant!
Calculating Black-Scholes Prices
from Historical Data: S, R, T that just was, and as standard
deviation of historical returns from some arbitrary past period
from Actual Data: S, R, T that just was, and implied from pricing of nearest “at-the-money” option (termed “implied volatility).
Implied Volatility Introduction Calculating implied volatility Volatility smiles
Introduction Instead of solving for the call
premium, assume the market-determined call premium is correct
Then solve for the volatility that makes the equation hold
This value is called the implied volatility
Calculating Implied Volatility Setup spreadsheet for pricing “at-the-
money” call option. Input actual price. Run SOLVER to equate actual and
calculated price by varying .
Volatility Smiles Volatility smiles are in contradiction
to the BSOPM, which assumes constant volatility across all strike prices
When you plot implied volatility against striking prices, the resulting graph often looks like a smile
Volatility Smiles (cont’d)Volatility Smile
Microsoft August 2000
0
10
20
30
40
50
60
40 45 50 55 60 65 70 75 80 85 90 95 100 105
Striking Price
Imp
lie
d V
ola
tili
ty (
%)
Current Stock Price
Problems Using the Black-Scholes Model
Does not work well with options that are deep-in-the-money or substantially out-of-the-money
Produces biased values for very low or very high volatility stocks Increases as the time until expiration
increases
May yield unreasonable values when an option has only a few days of life remaining
Beginning Greeks & Hedging Hedge Ratios Greeks (Option Price Sensitivities)
delta, gamma (Stock Price) rho (riskless rate) theta (time to expiration)vega (volatility)
Delta Hedging
Hedge Ratios Number of units of hedging security to
moderate value change in exposed position If trading options: Number of units of
underlying to hedge options portfolio If trading underlying: Number of options to
hedge underlying portfolio For now: we will act like trading European
Call Stock Options with no dividends on underlying stock.
Delta, Gamma Sensitivity of Call Option Price to
Stock Price change (Delta):
= N(d1) We calculated this to get option price. Gamma is change in Delta measure as
Stock Price changes….we’ll get to this later!
Delta Hedging If an option were on 1 share of stock,
then to delta hedge an option, we would take the overall position: +C - S = 0 (change)
This means whatever your position is in the option, take an opposite position in the stock (+ = bought option sell stock) (+ = sold option buy stock)
Recall the Pricing Example IBM is trading for $75. Historically, the volatility is 20% (A
call is available with an exercise of $70, an expiry of 6 months, and the risk free rate is 4%.
ln(75/70) + (.04 + (.2)2/2)(6/12)d1 = -------------------------------------------- = .70, N(d1) =.7580
.2 * (6/12)1/2
d2 = .70 - [ .2 * (6/12)1/2 ] = .56, N(d2) = .7123
C = $75 (.7580) - 70 e -.04(6/12) (.7123) = $7.98Intrinsic Value = $5, Time Value = $2.98
Hedge the IBM Option Say we bought (+) a one share IBM
option and want to hedge it:
+ C - S means
1 call option hedged with shares of IBM stock sold short (-).
= N(d1) = .758 shares sold short.
Overall position value:Call Option cost = -$
7.98 Stock (short) gave = +$ 56.85 (S = .758*75 = 56.85)
Overall account value: +$ 48.87
Hedge the IBM Option
Why a Hedge? Suppose IBM goes to $74.
ln(74/70) + (.04 + (.2)2/2)(6/12)d1 = -------------------------------------------- = 0.61, N(d1) =.7291
.2 * (6/12)1/2
d2 = 0.61 - [ .2 * (6/12)1/2 ] = 0.47, N(d2) = .6808
C = $74 (.7291) - 70 e -.04(6/12) (.6808) = $7.24
Results Call Option changed:
(7.24 - 7.98)/7.98 = -9.3% Stock Price changed:
(74 - 75)/75 = -1.3% Hedged Portfolio changed:
(Value now –7.24 + (.758*74) = $48.85) (48.85 - 48.87)/48.87 = -0.04%!
Now that’s a hedge!
Hedging Reality #1 Options are for 100 shares, not 1 share. You will rarely have one option to
hedge. Both these issues are just multiples!
+ C - S becomes + 100 C - 100 S for 1 actual option, or + X*100 C - X*100 S for X actual options
Hedging Reality #2 Hedging Stock more likely:
+ C - S = 0 becomes algebraically - (1/) C + S
So to hedge 100 shares of long stock (+), you would sell (-) 1/ options
For example, (1/.758) = 1.32 options
Hedging Reality #3 Convention does not hedge long stock
by selling call options (covered call). Convention hedges long stock with
bought put options (protective put).
Instead of - (1/) C + S- (1/P) P + S
Hedging Reality #3 cont’d P = [N(d1) - 1], so if
N(d1) < 1 (always), thenP < 0
This means- (1/P) P + S
actually has the same positions in stock and puts ( -(-) = + ).
This is what is expected, protective put is long put and long stock.
Reality #3 Example Remember IBM pricing:
ln(75/70) + (.04 + (.2)2/2)(6/12)d1 = -------------------------------------------- = .70, N(d1) =.7580
.2 * (6/12)1/2
d2 = .70 - [ .2 * (6/12)1/2 ] = .56, N(d2) = .7123C = $75 (.7580) - 70 e -.04(6/12) (.7123) = $7.98Put Price = Call Price + X e-rT - S
Put = $7.98 + 70 e -.04(6/12) - 75
= $1.59
Hedge 100 Shares of IBM - (1/P) P + S =
- 100 * (1/P) P + 100 * S P = N(d1) – 1 = .758 – 1 = -.242
- (1/P) = - (1/ -.242) = + 4.13 options Thus if “ + “ of + S means bought
stock, then “ + “ of +4.13 means bought put options!
That’s a protective put!
Hedge Setup Position in Stock: $75 * 100 = +$7500 Position in Put Options:
$1.59 * +4.13 * 100 = +$656.67 Total Initial Position =+$8156.67
IBM drops to $74 Remember call now worth $7.24
Puts now worth $1.85 * 4.13 * 100 = $ 764.05
Total Position = $7400 + 764.05 = $8164.05
Put Price = Call Price + X e-rT - S
Put = $7.24 + 70 e -.04(6/12) - 74
= $1.85
Results Stock Price changed:
(74 - 75)/75 = -1.3%
Portfolio changed: (8164.05 – 8156.67) / 8156.67 = +0.09%!!!!
Now that’s a hedge!