Transcript Financial Derivatives Futures and Swaps

Financial Options & Option Valuation
Session 4 – Binomial Model & Black Scholes
CORP FINC Shanghai 2015 - ANS
What determines option value?
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Stock Price (S)
Exercise Price (Strike Price) (X)
Volatility (σ)
Time to expiration (T)
Interest rates (Rf)
Dividend Payouts (D)
Try to guestimate…for a call option price… (5 min)
Stock Price ↑
Then call premium will?
Exercise Price ↑
Then…..?
Volatility ↑
Then…..?
Time to expiration↑
Then…..?
Interest rate ↑
Then…..?
Dividend payout ↑
Then…..?
Answer Try to guestimate…for a call option price… (5 min)
Stock Price ↑
Then call premium will? Go up
Then…..? Go down.
Exercise Price ↑
Volatility ↑
Then…..? Go up.
Time to expiration↑
Then…..? Go up.
Interest rate ↑
Then…..? Go up.
Dividend payout ↑
Then…..? Go down.
Your answer should be:
Call premium
Put premium
So up
up
down
X up
down
up
Rf up
up
down
D up
down
up
Time up
up
up
STDEV up
up
up
Binomial Option Pricing
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Assume a stock price can only take two possible values at expiration
Up (u=2) or down (d=0.5)
Suppose the stock now sells at $100 so at expiration u=$200 d=$50
If we buy a call with strike $125 on this stock this call option also has only
two possible results
up=$75 or down=$ 0
Replication means:
Compare this to buying 1 share and borrow $46.30 at Rf=8%
The pay off of this are:
Strategy
Today CF
Future CF if St>X Future CF if
(200)
ST<X(50)
Buy Stock
-$100
+$200
+$50
Write 2 Calls
+2C
- $150
$0
Borrow PV(50)
+$50/1.08
- $50
- $50
TOTAL
+2C-$53.70(=$0)
$0 (fair game)
$0 (fair game)
Binomial model
• Key to this analysis is the creation of a perfect hedge…
• The hedge ratio for a two state option like this is:
• H= (Cu-Cd)/(Su-Sd)=($75-$0)/($200-$50)=0.5
• Portfolio with 0.5 shares and 1 written option (strike $125)
will have a pay off of $25 with certainty….
• So now solve:
• Hedged portfolio value=present value certain pay off
• 0.5shares-1call (written)=$ 23.15
• With the value of 1 share = $100
• $50-1call=$23.15 so 1 call=$26.85
What if the option is overpriced?
Say $30 instead of $ 26.85
• Then you can make
arbitrage profits:
• Risk free $6.80…no
matter what happens
to share price!
Cash
flow
At
S=$50
At
S=$200
Write 2
options
$60
$0
-$150
Buy 1
share
-$100
$50
$200
Borrow
$40 at
8%
$40
-$43.20
-$43.20
Pay off
$0
$ 6.80
$ 6.80
Class assignment:
What if the option is under-priced?
Say $25 instead of $ 26.85 (5 min)
Cash
At
• Then you can make
flow
S=$50
arbitrage profits:
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• Risk free …no matter …….2
options
what happens to
….. 1
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share price!
share
At
S=$200
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Borrow/ ?
Lend
$ ? at
8%
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Pay off
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Answer…
• Then you can make
arbitrage profits:
• Risk free $4 no matter
what happens to share
price!
• The PV of $4=$3.70
• Or $ 1.85 per option
(exactly the amount by
which the option was
under priced!: $26.85$25=$1.85)
Cash
flow
At $50
At $200
Buy 2
options
-$50
$0
+$150
sell 1
share
$100
-$50
-$200
Lending
$50 at
8%
-$50
+$54
+$54
Pay off
$0
$4
$4
Breaking Up in smaller periods
• Lets say a stock can go up/down every half
year ;if up +10% if down -5%
• If you invest $100 today
• After half year it is u1=$110 or d1=$95
• After the next half year we can now have:
• U1u2=$121 u1d2=$104.50 d1u2= $104.50 or
d1d2=$90.25…
• We are creating a distribution of possible
outcomes with $104.50 more probable than
$121 or $90.25….
Class assignment:
Binomial model…(5 min)
• If up=+5% and down=-3% calculate how many
outcomes there can be if we invest 3 periods
(two outcomes only per period) starting with
$100….
• Give the probability for each outcome…
• Imagine we would do this for 365 (daily)
outcomes…what kind of output would you get?
• What kind of statistical distribution evolves?
Answer…
Probability
Calculation
Result
3 up moves
1/8=12.5%
$100*(1.05)^3
$ 115.76
2 up and 1
down moves
3/8=37.5%
$100*1.05^2*0.97
$ 106.94
1 up and 2
down moves
3/8=37.5%
$100*1.05*0.97^2
$98.79
3 down moves
1/8=12.5%
$100*0.97^3
$91.27
Black-Scholes Option Valuation
• Assuming that the risk free rate stays the
same over the life of the option
• Assuming that the volatility of the
underlying asset stays the same over the
life of the option σ
• Assuming Option held to
maturity…(European style option)
Without doing the math…
• Black-Scholes: value call=
• Current stock price*probability – present
value of strike price*probability
• Note that if dividend=0 that:
• Co=So-Xe-rt*N(d2)=The adjusted intrinsic
value= So-PV(X)
Class assignment: Black Scholes
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Assume the BS option model:
Call= Se-dt(N(d1))-Xe-rt(N(d2))
d1=(ln(S/X)+(r-d+σ2/2)t)/ (σ√t)
d2=d1- σ√t
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If you use EXCEL for N(d1) and N(d2) use NORMSDIST function!
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stock price (S) $100
Strike price (X) $95
Rf ( r)=10%
Dividend yield (d)=0
Time to expiration (t)= 1 quarter of a year
Standard deviation =0.50
A)Calculate the theoretical value of a call option with strike price $95 maturity 0.25
year…
B) if the volatility increases to 0.60 what happens to the value of the call? (calculate it)
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answer
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A) Calculate: d1= ln(100/95)+(0.10-0+0.5^2/2)0.25/(0.5*(0.25^0.5))=0.43
Calculate d2= 0.43-0.5*(0.25^0.5)=0.18
From the normal distribution find:
N(0.43)=0.6664 (interpolate)
N(0.18)=0.5714
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Co=$100*0.6664-$95*e -.10*0.25 *0.5714=$13.70
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B) If the volatility is 0.6 then :
D1= ln(100/95)+(0.10+0.36/2)0.25/(0.6*(0.25^0.5))=0.4043
D2= 0.4043-0.6(0.25^0.5)=0.1043
N(d1)=0.6570
N(d2)=0.5415
Co=$100*0.6570-$ 95*e -.10*0.25 *0.5415=$15.53
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Higher volatility results in higher call premium!
Option valuation
BS Model
Then
C
INPUTS
Stock Price
Exercise Price
Interest Rate
Dividend Yield
Time to Expiration
Standard Deviation
$
$
decimal
decimal
decimal
decimal
100
95
0.10
0.00
0.25
0.50
PROCESS
d1
d2
NORM d1
NORM d2
CALL
PUT
0.43017318
0.18017318
0.66646516
0.57149169
$
$
13.70
6.35
IF
Up
Up
Up
Up
Up
Up
UP
DOWN
UP
DOWN
UP
UP
Homework assignment 9:
Black & Scholes
• Calculate the theoretical value of a call
option for your company using BS
• Now compare the market value of that
option
• How big is the difference?
• How can that difference be explained?
Implied Volatility…
• If we assume the market value is correct
we set the BS calculation equal to the
market price leaving open the volatility
• The volatility included in today’s market
price for the option is the so called implied
volatility
• Excel can help us to find the volatility
(sigma)
Implied Volatility
• Consider one option series of your
company in which there is enough volume
trading
• Use the BS model to calculate the implied
volatility (leave sigma open and calculate
back)
• Set the price of the option at the current
market level
Implied Volatility Index - VIX
Investor fear gauge…
Class assignment:
Black Scholes put option valuation
(10 min)
• P= Xe-rt(1-N(d2))-Se-dt(1-N(d1))
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Say strike price=$95
Stock price= $100
Rf=10%
T= one quarter
Dividend yield=0
A) Calculate the put value with BS? (use the normal
distribution in your book pp 516-517)
• B) Show that if you use the call-put parity:
• P=C+PV(X)-S where PV(X)= Xe-rt and C= $ 13.70 and
that the value of the put is the same!
Answer:
• BS European option:
• P= Xe-rt(1-N(d2))-Se-dt(1-N(d1))
• A) So: $95*e-.10*0.25*(1-0.5714) - $100(1-.6664)=
• B) Using call put parity:
• P=C+PV(X)-S= $13.70+$95e -.10*.25 -$100= $ 6.35
$ 6.35
The put-call parity…
• Relates prices of put and call options according to:
• P=C-So + PV(X) + PV(dividends)
• X= strike price of both call and put option
• PV(X)= present value of the claim to X dollars to be paid
at expiration of the options
• Buy a call and write a put with same strike price…then
set the Present Value of the pay off equal to C-P…
The put-call parity
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Assume:
S= Selling Price
P= Price of Put Option
C= Price of Call Option
X= strike price
R= risk less rate
T= Time then X*e^-rt= NPV of realizable risk less share price (P
and C converge)
• S+P-C= X*e^-rt
• So P= C +(X*e^-rt - S) is the relationship between the price of the
Put and the price of the Call
Class Assignment:
Testing Put-Call Parity
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Consider the following data for a stock:
Stock price: $110
Call price (t=0.5 X=$105): $14
Put price (t=0.5 X=$105) : $5
Risk free rate 5% (continuously compounded
rate)
• 1) Are these prices for the options violating the
parity rule? Calculate!
• 2) If violated how could you create an arbitrage
opportunity out of this?
Answer:
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1) Parity if: C-P=S-Xe-rT
So $14-$5= $110-$105*e -0.5*5
So $9= $ 7.59….this is a violation of parity
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2) Arbitrage: Buy the cheap position ($7.59) and sell the expensive position ($9) i.e. borrow the
PV of the exercise price X, Buy the stock, sell call and buy put:
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Buy the cheap position:
Borrow PV of X= Xe-rT= +$ 102.41 (cash in)
Buy stock - $110 (cash out)
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Sell the expensive position:
Sell Call: +$14 (cash in)
Buy Put: -$5 (cash out)
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Total $1.41
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If S<$105 the pay offs are S-$105-$ 0+($105-S)= $ 0
If S>$105 the pay offs are S-$105-(S-$105)-$0=$ 0
Black Scholes
• The Black-Scholes model is used to calculate a theoretical call price
(ignoring dividends paid during the life of the option) using the five
key determinants of an option's price: stock price, strike price,
volatility, time to expiration, and short-term (risk free) interest rate.
Myron Scholes and Fischer Black
Some spreadsheets will show you
the option Greeks;
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Delta (δ): Measures how much the
premium changes if the underlying share
price rises with $ 1.- (positive for Call
options and negative for Put options)
Black Scholes Option Pricing Model
INPUTS
Symbols
Stock Price Now
Standard Deviation Annual
Riskfree rate Annual
Exercise Price
Time to Maturity in Years
S
σ
r
E
T
$102.50
86.07%
5.47%
$100.00
0.3556
C
0.342635
-0.17062
0.634064
0.432262
$22.60
P
-0.34264
0.170619
0.365936
0.567738
$18.17
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Gamma (γ): Measures how sensitive
delta is for changes in the underlying
asset price (important for risk managers)
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Vega (ν): Measures how much the
premium changes if the volatility rises
with 1%; higher volatility usually means
higher option premia
d1
d2
N(d1)
N(d2)
Cal Price
Theta (θ): Measrures how much the
premium falls when the option draws one
day closer to expiry
-d1
-d2
N(-d1)
N(-d2)
Put Price
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OUTPUTS
Rho (ρ): Measrures how much the
premium changes if the riskless rate
rises with 1% (positive for call options
and negative for put options)
Example…
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Results
Price
P
Calc typeValue
0.25517
Price of the call option
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Delta
D
0.28144
Premium changes with $ 0.28144 if share price is up $1
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Gamma
G
0.21606
Sensitivity of delta for changes in price of share
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Vega
V
0.01757
Premium will go up with $ 0.01757 if volatility is up 1%
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Theta
T
-0.00419
1 day closer to expiry the premium will fall $ 0.00419
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Rho
R
0.00597
If the risk less rate is up 1% the premium will increase $ 0.00597
Class Assignment: Valuation Options
1) You are determining the Value of a Call Option with the Binomial model for Wal Mart with
X=$75
You know: So=$80 St(u)=$90 St(d)=$70
Maturity 1 year and Rf=2% per year
What is the theoretical value of this Call Option?
(Show your calculation!)
2) Somebody is telling you that you made a mistake and that St(u)=$95 and St(d)=$65 i.e.
the STDEV(return) is higher than in 1;
Recalculate the theoretical value of the Call Option with X=$75 under these new
assumptions with the Binomial model.
3) Assume that your answer under 2 is “correct”; now use the Black Scholes model ; if the
value of the Call is what you found in 2) what is the implied volatility of Wal Mart stock
according to BS? Copy the values of d1 and d2 as well as N(d1) and N(d2) to your answer
sheet! You can assume a dividend yield of 2.4%...
4) If somebody told you that the market value of this option is $10 how can you
execute ARBITRAGE and gain a RISK FREE profit? How much profit can you
realize? Show your detailed actions and COMPUTATIONS.
Solutions…
CF Today
St=$90
St=$70
Buy 3 Stocks
-$240
+$270
+$210
Write 4 Calls
+4C
-$60
$0
Borrow PV($210)
+$210/1.02=$205.88
-$210
-$210
TOTAL
$0
$0
$0
The hedge ratio: (Cu-Cd)/(Su-Sd)=$15/$20=3/4 (buy 3 stocks write 4 calls)
-$240+4C+$205.88 =$0 so C=$8.53
And if Su=$95 and Sd=$65
CF Today
St=$95
St=$65
Buy 2 Stocks
-$160
+$190
+$130
Write 3 Calls
+3C
-$60
$0
Borrow PV($130)
+$130/1.02=$127.45
-$130
-$130
TOTAL
$0
$0
$0
The hedge ratio: (Cu-Cd)/(Su-Sd)=$20/$30=2/3 (buy 2 stocks write 3 calls)
-$260+3C+$127.45 =$0 so C=$10.85
The volatility of the Wal Mart stock is higher so this should indeed result in a
higher value for the same option…
Black Scholes inputs..
Option valuation
BS Model
INPUTS
Stock Price
Exercise Price
Interest Rate
Dividend Yield
Time to Expiration
$
$
decimal
decimal
decimal
Standard Deviation
decimal
80
75
0.02
0.02
1
0.27730
PROCESS
d1
0.356964176
d2
0.079664176
NORM d1
0.639440688
NORM d2
0.531747824
CALL
PUT
$
$
10.85
6.26
Arbitrage
Action
NOW
St=$95
St=$65
Buy 3 Calls X=$75
(market price $10)
-$30
+$60
$0
Sell 2 stocks WMT
+$160
-$190
-$130
Lend/INVEST $130
- $130
+$130(1.02)
+$130(1.02)
TOTAL
$0
$ 2.60
+$2.60
Note that if the option is cheaper in the market than what it should be
according to the valuation you will buy it! You can realize a risk free
profit of $2.60 for 3 calls or $2.60/3 for every call the PV of this is exactly
the price difference of $0.85. ($10.85-$10)
Q.E.D.