The critical factor when trading in options, is determining a fair price for the option.
Computes the value of a call option
The value can change only at the end of the period (t+1) and the possible maximal and minimal values are currently (t) known. The binomial model for option pricing
The Binomial Option Pricing Formula
Explanations of the parameters
C = Call option
= Call value at t+1, when stock price goes to max.
= Call value at t+1, when stock price goes to min.
= Riskless interest rate
u = Multiplicative upward movement in stock price (S)
d = Multiplicative downward movement in stock price (S)
Features of the binomial model
The formula doesn’t depend on investors’ attitudes towards risk
Investors can agree on the relationship between C, S and r even if they have different expectations about the upward or downward movement of C
The only random variable on which the call value depends, is the stock price itself
p can take values between 0 and 1 (0 < p < 1). If investors were risk neutral then p=q.
A Short Example
K = 15 mu (striking price)
S = 20 mu
= 25 mu
= 15 mu = 10 monetary units
= 0 mu
r = 1,06 ( 6 %)
u = 1,25
d = 0,75
The Black & Scholes formula (A continuous time formula)
“It is possible to create a risk free portfolio” by owning 1 stock and writing h call options on it.”
The most frequently used option pricing formula
Originally a heat transfer equation in physics
Assumptions behind the formula: 1. The stock price follows a continuous Wiener- process and the future stock prices are lognormally distributed.
2. There exist no transaction costs or taxes.
3. No dividends are paid during the lifetime of the option.
4. The capital market is perfect: there exist no arbitrage-possibilities.
5. The composition of the portfolio can be continuously adjusted.
6. The risk free interest rate is constant during the lifetime of the option.
Black & Scholes Option Pricing Formula
Copeland & Weston. 1988. p 276. where
Parameters
C = The price of the call option
S = Stock price
N = The standard normaldistribution
= The continuous risk free rate of return [=ln(1+ )]
t = The time to expiration
(if 63 days, then t=63/365)
K = Striking price
= The variance of the stock return
N(d1) = the inverse hedge ratio
e.g. for each stock that is owned, 1/N(d ) options has to be written for the portfolio to be risk free.
= the discounted value of the striking price.
N(d2) = the probability for the option to be “in the money” on due date (e.g. the option will be exercised).
The price of an option is dependent on the following parameters:
P C
Current stock price (S)
Time to expiration (t)
Striking price (K)
Stock volatility
Interest rates
(Cash dividends)
References
Cox & Rubinstein. 1985. p 37.
The Put-Call Parity:
There is a connection between the price of a call and a put option
If the prices differ from this equation, there exist arbitrage opportunities
References
Cox John C.&Rubinstein Mark: OPTIONS MARKETS. (USA 1985). Pp 41-42.
An intuitive example
1 + 19 = MAX(19-20,0) + 20 = 20 Buy a share: S = 19 mu
Buy a put option: P = 1 mu (striking price K = 20 mu)
Buy a call option with the same striking price and maturity: C = 1,50 mu
Deposit 18,50 mu at the risk free rate r = 10,95%
Outcome 9 months later on expiration date
out of the money in the money Input vaules: r=10,95%;t=0,75; K=18,50;C=1,50;P=1;S=19 Discounted to t(0)
Prolonged to t Put + Share = Call + Deposition
3 + 17 = 0 + 20
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