VIP Independent Call Girls in Andheri 🌹 9920725232 ( Call Me ) Mumbai Escorts...
option pricing
1. Option Pricing.
• “When you want to estimate the price of fruit
salad, you average the prices of the fruit the
mixture contains. In a similar vein, I thought of
the option formula as a prescription for
estimating the price of a hybrid by averaging
the known market prices of its ingredients.”
Emanual Derman
(My life as a Quant)
2. Option Pricing
• Built on arbitrage. Arbitrage guides prices in all
financial markets, particularly in option pricing.
• We don’t think of fruit salad when we think of
arbitrage or option prices.
• Complex problems are best understood when
distilled to their basic building blocks and option
pricing can be approached in the same way.
• An option can be valued by averaging the prices
of its underlying components.
• Arbitrage sets boundaries on the ultimate result.
3. Determinants of Option Price/Premium
1. Spot/Market Price. (S)
2. Strike Price. (K)
3. Time to expiry. (T)
4. Volatility (Standard Deviation σ)
5. Risk free Interest rate ( r )-Why risk free?
4. Option Pricing
• Figuring out how to price stock options occupied
some of the best financial minds for most of the
20th century.
• Two of the three individuals who figured out how
to value options ultimately won the Nobel Price
in 1997. (Black Scholes- Merton Miller, Myron
Scholes).
• A key unresolved issue was the appropriate rate
of return for valuing the future payoffs generated
by a stock option. Quantifying the discount rate
applicable to a stock option proved exceptionally
challenging.
5. Key insight-Riskless portfolio of Stock and Call option .
• One of the greatest insights of option pricing theory, is
the importance of the risk free rate and hedging in
option valuation.
• The capacity to combine a stock option with its
underlying stock in a portfolio that is riskless ,makes the
risk-free rate ,the appropriate return to use in option
valuation.
• Key Question-Is there a mix of stock and options ,that
provides a riskless, hedged position irrespective of
whether the stock price increases or decreases?
• Yes. Because a call option’s value is tied to that of the
underlying stock, a riskless combination/mix of both,
the option and the stock will contain offsetting
elements. What is this combination/mix/ratio?
7. Key insight-Riskless portfolio of Stock and Call option .
• Hedge ratio is the number of shares bought for a
single call option sold.
• To be riskless , the portfolio (Stock+ Option)
payoffs associated with the stock moving up (Su)
or down (Sd), at the end of the period should be
equal.
• The portfolio is not exposed to variations in the
value of the stock or the option, because an
appropriate , offsetting long/short mixture of the
two has been adopted. What is lost on one
position is offset by gains on the other.
8. Option Pricing Models
• Two major approaches.
1. Binomial Option Pricing Model (BOPM), which shows
the underlying stock prices and associated option
payoffs, over a series of discrete periods of time in
which the underlying asset’s price can either go up or
down by a specified amount. Only two potential future
stock prices , are considered at a time. It is widely used
and only needs some basic arithmetic and algebra.
2. Nobel Prize winning Black-Scholes Option Pricing Model
(BSOPM) ,which views stock prices and option payoffs
from a continuous time perspective. It portrays a
probability distribution of future stock prices. It broadens
the range of future stock prices.
9. BOPM-2 period price path
Price= 100 u= 1.1 d= 0.9091
(Expected stock prices in two periods-price path)
90.91
121
100
82.64
Time 0 Time 1 Time 2
100
110
Suppose that a stock , currently priced at 100, can either rise by a
factor 'u', which equals 1.1 or fall by a factor 'd',( d= 1/u) in each
period.The possible stock prices after 1 and 2 periods are as below:
10. BOPM-2 period price path
Time 0 Time 1 Time 2
S.U2
S.U
S S.U.D
S.D
S.D2
(Alternate stock price paths)
11. BOPM-2 period price path
100 = p X (110) + (1-p) X 90.91
Strike = 100 p= 0.476 1-p = 0.524
0
90.91
82.64
0
(Stock and Option Prices)
0
4.76
121
21
110
100 100
10.00
Time 0 Time 1 Time 2
These assumptions imply that the expected stock price stock price in
period 1 is 100.That is:
It is further assumed that the stock pays no dividends and that the
expected return on the stock equals the interest rate and the interest rate is
0% per period.
12. Suu= 43.70
Cuu= 13.70
639 Shares long
639 calls short
Su= 36.42 Debt = 2713.27
Cu= 7.29 Value= 16451.83
h0= 1.000
639 Shares long
639 calls short
Borrow= 2634.25 Sud/Sdu= 30.23
Value= 15972.65 Cud/Cud 0.23
639 Shares long
639 calls short
S0= 30.35 Debt = 2713.3
C0= 3.88 Value= 16451.8
h0= 0.639
639 Shares long 25 Shares long
1000 calls short 1000 calls short
Value= 15507.4 Lend= 15939.0
Value= 16451.8
Sd= 25.19
Cd= 0.12
h0= 0.025
25 Shares long
1000 calls short Sdd= 20.91
Lend= 15474.8 Cdd= 0.00
Value= 15972.7 25 Shares long
1000 calls short
Lend = 15939.0
Value= 16451.8
OR
Two Period Binomial Option Pricing Model Framework
Borrowing and lending is due to
rebalancing of the portfolio as per the
new hedge ratio.
13. one annual period
two six monthly periods
four quarterly periods
twelve monthly periods
The period is being chopped into
smaller and smaller intervals
14. a)The Binomial Option Pricing Formula is given below:
N
Ct=
∑
n!
Pu
j
.Pd
(n-j)
Max(0,Uj
.D(n-j)
St-K
j!(n-j)!
j=0 rn
The above formula gives the present value of a the expected payoffs at expiration
where:
Ct= Present value of call on expiration.
n= total number of periods(not years) until expiration. A feature of the binomial
model is that the stock can either go up or go down in each period.
J= number of periods that the stock goes up. If it goes up j periods it goes down (n-j)
periods.
U= magnitude of upward movement .
D= magnitude of downward probability
Pu=Risk Neutral probability of upward movement.
Pd=Risk Neutral probability of downward movement.
St= Current stock price.
K=Strike price
r=discount factor for present value
15. How do we arrive at up and down moves?
• Recall that in our example of one period and two period
binomial model we took a 20% up move and a 17% down
move. How did we arrive at this?
• Formula which relates the up and down changes to the
standard deviation of the stock returns.
1 + upside change = u = eσ√t
1 + downside change = d = 1/u.
where e=base for natural log = 2.7182, σ= annual
standard deviation of continuously compounded stock
returns, t = interval as fraction of a year.
• To find σ given u, σ = log(u)/√t
16. a)Annual standard deviation = 57.55%
b) Time t= 3 months= 0.25 years
c) e= 2.7182
First calculate for a single period of 3 months. Next calculate for
two periods of 11/2 months, each period.
Calculate the up move and the down move for the Binomial option
pricing model given the following information.
17. a)Annual standard deviation = 57.55%
b) Time t= 3 months= 0.25 years
c) e= 2.7182
Answer:
Single 3 month period
1 + upside change = u = eσ√t
1.3334 33.3%
1 + downside change = 1/u= 0.75 -25.0%
Two 1 & 1/2 periods. t= 0.125 years
1 + upside change = u = eσ√t
1.2256 22.56%
1 + downside change = 1/u= 0.8159 -18.41%
First calculate for a single period of 3 months. Next calculate for
two periods of 11/2 months, each period.
Calculate the up move and the down move for the Binomial option
pricing model given the following information.
18. a)Annual standard deviation = 57.55%
b) Time t= 3 months= 0.25 years
c) e= 2.7182
60
60
0.50%
c) Risk free interest rate=
(per period)
First calculate for a single period of 3 months. Next calculate for two
periods of 11/2 months, each period.
Also calculate the price of a two period call option given the above and
following information:
a)Current stock price=
b)Strike price=
Calculate the up move and the down move for the Binomial option
pricing model given the following information.
19. Option Pricing Models
• Two major approaches.
1. Binomial Option Pricing Model (BOPM), which shows
the underlying stock prices and associated option
payoffs, over a series of discrete periods of time in
which the underlying asset’s price can either go up or
down by a specified amount. Only two potential future
stock prices , are considered at a time. It is widely used
and only needs some basic arithmetic and algebra.
2. Nobel Prize winning Black-Scholes Option Pricing Model
(BSOPM) ,which views stock prices and option payoffs
from a continuous time perspective. It portrays a
probability distribution of future stock prices. It broadens
the range of future stock prices.
21. C = S.N(d1)-Ke-rt.N(d2)
• It may not look like much, but it is the most
dangerous equation since E=MC2.
• Just as Albert Einstein’s equation eventually led
to Hiroshima and Nagasaki, this one has had the
financial impact of a nuclear bomb.
• It contributed to stock market booms and busts,
to a succession of financial crises and to
economic slumps that lost millions of people
their livelihood.
• It is the Black Scholes formula.
22. Black Scholes Option Pricing Model
-rt
C=S.N(d1)-Ke .N(d2)
C=Call optionPrice
S=Current price ofstock
K=Strike price ofthe stock
N(d1) /N(d2)= Probabilities(Cumulative Normal DistributionFunction)
e=2.71828
t=time toexpiry inyears
r=riskfree annualisedinterest rate as adecimal
24. Assumptions of the Black-Scholes model
1. It applies only to European options.
2. There are no transaction costs or taxes.
3. There are no restrictions on short selling.
4. The stock price follows a lognormal probability
distribution.
5. The interest rate is constant.
6. The stock does not pay dividends.
7. The stock price moves continuously without
jumps.
8. Volatility is constant across all strike prices , on
the same underlying.
25. The main insight of the Black Scholes model
• Value of a derivative depends on underlying
security prices which are constantly changing,
increasing risk.
• BS found a way to neutralize this risk in options.
• Since risk is driven by the stock, the risks of the
option can be eliminated by taking opposite
position in the stock.
• There is a specific ratio between the stock and
option positions such that risks of the stock and
option exactly cancel each other (Hedge ratio-Δ).
• The arguments main issue concerns a search for
this ratio.
27. Spot(S) 47
Strike(K) 50
Time in years (t) 0.5
Volatality(σ) 40.00%
RF-Rate -( r) 10%
Find the value of a call/put option using the Black-Scholes
option pricing model,given the following information.
30. Spot(S) 220
Strike(K) 225
Time in years (t) 0.5
Volatality(σ) 12.00%
RF-Rate -( r) 8%
Find the value of a call/put option using the Black-Scholes
option pricing model,given the following information.
33. Spot(S) 45 50 55
Strike(K) 50 50 50
Time in years (t) 0.25 0.25 0.25
Volatality(σ) 50.00% 50.00% 50.00%
RF-Rate -( r) 6% 6% 6%
In each case calculate the intrinsic and time value.
Find the value of a call/put option using the Black-Scholes
option pricing model,given the following information.
37. S0 45.000 50.000 55.000
P 7.117 4.573 2.807
52.117 54.573 57.807
C 2.861 5.318 8.551
Ke-rt
49.256 49.256 49.256
52.117 54.573 57.807
Put call Parity- S0 + P = C + K/(1+ Rf)
T
38. Time to expire
• Can be given in months (divide by 12) or days
(divide by 365)
39. Efficient Market Hypothesis
• The hypothesis holds that in an ideal
market, all relevant information is already
priced into a security today.
• Yesterday’s change does not influence
today’s , nor today’s tomorrows.
• Each price change is “independent” from
the last.(coin tossing)
40. Efficient Market Hypothesis
Founded on two critical assumptions:
1. Price changes are statistically
independent.
2. Price changes are normally distributed.
41. Geometric Brownian Motion
• One assumption used in deriving the BSOPM is
that the stock price randomly wanders through
time following a price path called GBM.
• As a result of this assumed stock price behaviour,
prices over a given interval of time are
lognormally distributed and continuously
compounded returns are normally distributed.
• If the stock’s price follows GBM, the variance of
returns is proportional to time and standard
deviation is proportional to square root of time.
42. GBM
• If stock prices vary according to the square
root of time, they bear a remarkable
resemblance to molecules colliding with one
another as they move in space. (Drunkards
walk).
• An Englishman Robert Brown discovered this
in the 19th century, and it is generally known
as Brownian motion- a critical ingredient of
Einstein’s theory of the atom.
43.
44. Ito’s lemma
• Jack Treynor was searching for an answer to the question of how
rational investors take risk into consideration when they are
deciding whether an asset is cheap or expensive over several
time periods.
• The mathematics required is very intricate-something known as
Ito’s lemma.
• Developed by Kiyoshi Ito a Japanese mathematician, and
introduced by Robert Merton, an economist and mathematician
at MIT, into the theory of portfolio management.
• Merton describes Ito’s lemma as providing the ‘’differentiation
rule for the generalized stochastic calculus’’.
• In plain English it provides the key for describing how randomly
fluctuating security prices change from one short period to the
next period i.e in ‘’continuous time’’..
