Explain why margin accounts are only required when clients write options but not when they buy options?
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Chapter10-BinomialOptionPricing-BasicConcepts-Students1.pptxChapter 10 & 11 Binomial Option Pricing: Basic Concepts
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1
A Simple Binomial Model
A stock price is currently $20
In 3 months it will be either $22 or $18
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Stock Price = $18
Stock Price = $22
Stock price = $20
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A Call Option
A 3-month call option on the stock has a strike price of $21
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Stock Price = $18
Option Price = $0
Stock Price = $22
Option Price = $1
Stock price = $20
Option Price=?
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Setting Up a Riskless Portfolio
For a portfolio that is long Δ shares and a short 1 call option values are
Portfolio is riskless when 22Δ – 1 = 18Δ
Δ = 0.25
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22Δ – 1
18Δ
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Valuing the Portfolio
The riskless portfolio is:
long 0.25 shares
short 1 call option
The value of the portfolio in 3 months is
22 × 0.25 – 1 = $4.50
Assume risk free is 12% per annum with continuous compounding
The value of the portfolio today is
4.5e–0.12×0.25 = $4.3670
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Valuing the Option
The portfolio that is worth $4.367
long 0.25 shares
short 1 option
The value of the shares is $5.000 = 0.25 × 20
The value of the option is therefore $0.633 = ( 5.000 – C = 4.367 )
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Creating a Synthetic Call
Buy ¼ share of the stock today by borrowing $4.367 and paying $0.633
In 3 months
It’s payout is the same as the options which means they must cost the same
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22 x ¼ – 4.50 = 1
18 x ¼ - 4.50 = 0
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Arbitrage
Let’s say the option is priced at $0.70. How is it possible to arbitrage it?
What if the price is $0.60?
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Generalization
A derivative lasts for time T and is dependent on a stock
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S0u
ƒu
S0d
ƒd
S0
ƒ
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Generalization
Value of a portfolio that is long Δ shares and short 1 derivative:
The portfolio is riskless when S0uΔ – ƒu = S0dΔ – ƒd or
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S0uD – ƒu
S0dD – ƒd
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Generalization
Value of the portfolio at time T is S0uΔ – ƒu
Value of the portfolio today is (S0uΔ – ƒu)e–rT
Another expression for the portfolio value today is S0Δ – f
Hence
ƒ = S0Δ – (S0u Δ – ƒu )e–rT
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Generalization
Substituting for Δ we obtain
ƒ = [ pƒu + (1 – p)ƒd ]e–rT
where
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Proof
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p as a Probability
It is natural to interpret p and 1-p as probabilities of up and down movements
The value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rate
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S0u
ƒu
S0d
ƒd
S0
ƒ
p
(1 – p )
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Risk-Neutral Valuation
When the probability of an up and down movements are p and 1-p the expected stock price at time T is S0erT
This shows that the stock price earns the risk-free rate
Binomial trees illustrate the general result that to value a derivative we can assume that the expected return on the underlying asset is the risk-free rate and discount at the risk-free rate
Risk-neutral means investors do not increase the expected return they require from an investment to compensate for increased risk
2 simplifying features of risk-neutral world are
Expected return on any asset is the risk-free rate
The discount rate used on the expected payoff of an option is the risk-free rate
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Original Example Revisited
p is the probability that gives a return on the stock equal to the risk-free rate:
20e 0.12 ×0.25 = 22p + 18(1 – p ) so that
p = 0.6523
Alternatively:
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S0u = 22
ƒu = 1
S0d = 18
ƒd = 0
S0=20
ƒ
p
(1 – p )
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Option Value Using Risk-Neutral Valuation
The value of the option is:
e–0.12×0.25 (0.6523×1 + 0.3477×0)
= $0.633
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S0u = 22
ƒu = 1
S0d = 18
ƒd = 0
S0=20
ƒ
0.6523
0.3477
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Irrelevance of Stock’s Expected Return
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When we are valuing an option in terms of the price of the underlying asset, the probability of up and down movements in the real world are irrelevant
This is an example of a more general result stating that the expected return on the underlying asset in the real world is irrelevant
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A Two-Step Example
K = 21, r = 12%
Each time step is 3 months
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24.2
19.8
16.2
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Valuing a Call Option
Value at node B
= e–0.12×0.25(0.6523×3.2 + 0.3477×0) = 2.0257
Value at node A
= e–0.12×0.25(0.6523×2.0257 + 0.3477×0) = 1.2823
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16.2
0.0
20
1.2823
22
18
24.2
3.2
19.8
0.0
2.0257
0.0
A
B
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A Put Option Example
K = 52, time step =1yr
r = 5%, u =1.2, d = 0.8 p = 0.6282
21
50
4.1923
60
40
72
0
48
4
32
20
1.4147
9.4636
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When the Put Option is American
22
50
5.0894
60
40
72
0
48
4
32
20
1.4147
12.0
C
The American feature increases the value at node C from 9.4636 to 12.0000.
This increases the value of the option from 4.1923 to 5.0894.
An American option is similar to a European option but now at each node you need to consider early exercise
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Delta
Delta (Δ) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock
The value of Δ varies from node to node
The Δ of a call is positive and negative for a put
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Choosing u and d
One way of matching the volatility is to set
where s is the volatility and Dt is the length of the time step
Thus the standard deviation is
This is the approach used by Cox, Ross, and Rubinstein
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Choosing u and d MacDonald’s Way – Forward Tree
One way of matching the volatility is to set
where s is the volatility, r is risk-free rate, q is dividend yield, and Dt is the length of the time step
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Girsanov’s Theorem
Volatility is the same in the real world and the risk-neutral world
We can therefore measure volatility in the real world and use it to build a tree for the an asset in the risk-neutral world
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Assets Other than Non-Dividend Paying Stocks
For options on stock indices, currencies and futures the basic procedure for constructing the tree is the same except for the calculation of p
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The Probability of an Up Move
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