期权应用教程(四)

8
© True Partner Education Ltd 2012
The Gamma / Theta ratio
The ratio of Gamma to Theta reflects the relative cost of Gamma Gamma/Theta Ratio = Theta In words: “how many units of Gamma do I get for HKD 1 payment in Theta” It can be applied both on individual options, as well as on a portfolio level. The Gamma / Theta Ratio is driven by: • Volatility (i.e. higher volatility will make Gamma’s more valuable) • Share Price (i.e. higher share price will make a unit of Gamma more valuable) Gamma
Normal Distribution
One Standard Deviation
Mean
* Over and over again, we learn the hard way not to make this assumption! More later…
3
© True Partner Education Ltd 2012
© True Partner Education Ltd 2012
The concept of volatility
When Premium Long, we crave for movement When Premium Short, we fear it How do we express it? Annualized Volatility: 67% of observations fall within 1 standard deviation of the mean. If we assume share prices to be normally distributed*, we would have 67% of the possible share prices one year from now to be within one standard deviation of the future.
b)Long 0.1 Gamma in Standard Chartered -
10
© True Partner Education Ltd 2012
Normalizing Delta & Gamma: Example
When a portfolio consists of options on different underlying instruments, it is difficult to compare the Greeks as the share prices differ The solution is to ‘normalize’ Greeks to a standard instrument (for example a HKD 100 stock) Suppose we have the following options position that we want to ‘normalize’: Underlying: 200 HKD Stock - Portfolio Delta:50 - Portfolio Gamma: 2 Underlying: 100 HKD Stock - Portfolio Delta:? - Portfolio Gamma: ?
9
© True Partner Education Ltd 2012
The Gamma / Theta ratio
Back to our previous example: a)Long 0.1 Gamma in the Hang Seng Index Last index value is 20,000 Volatility 16 Expected Theta: HKD 2,000 Gamma / Theta ratio of 0.00005 (0.00005 Gamma for HKD 1 in Theta) Last share price is HKD 200 Volatility 16 Expected Theta: HKD 0.20 Gamma / Theta ratio of 0.5 (0.5 Gamma for HKD 1 in Theta)
- If portfolio value is to change 100 HKD for the 1 HKD change in share price, delta has to be 100/1 = 100
12
© True Partner Education Ltd 2012
Normalizing Delta & Gamma: Example
6
© True Partner Education Ltd 2012
Are all Gamma’s created equal?
Assume the following choice: a)Long 0.1 Gamma in the Hang Seng Index Last index value is 20,000 Volatility 16 Expected Theta ? Last share price is HKD 200 Volatility 16 Expected Theta ?
1
© True Partner Education Ltd 2012
C U H K
FINA6250: Fundamentals of Derivatives Trading Strategies
Lecture 4
Vega Volatility Final Aspects Assignment 1
2
Delta:
- Portfolio Delta:100 - Portfolio Gamma: ?
For a 200 HKD stock, 1% change in share price is 2 HKD - With delta 50, portfolio value changes 50 x 2 = 100 HKD For a 100 HKD stock, 1% change in share price is 1 HKD
Annualized Volatility
Daily Movement = SQRT (# of trading days) Example: •Share price of HKD 100 •Volatility of 32% Expected daily movement is HKD 2
x Share Price
4
© True Partner Education Ltd 2012
Volatility, Gamma and Theta
Assume the following situation: Stock Price: Gamma: Volatility: HKD 20 -/- 1,000 24
a)What is the expected daily movement? b)How much should I receive in theta? c)If we move HKD 0.40, do I earn or lose money if I hedge at the close? d)Same as c), but now I delta hedge every 10 cents?
11
© True Partner Education Ltd 2012
Normalizing Delta & Gamma: Example
Underlying: 200 HKD Stock Underlying: 100 HKD Stock
- Portfolio Delta:50 - Portfolio Gamma: 2
MSc in Finance, Term 3 2014 - 2015
C
U
H
K
Fundamentals of Derivatives Trading
Applying Options Theory in the Marketplace
Tobias Hekster -Senior Strategist, True Partner Fund -Adjunct Associate Professor, CUHK -Adjunct Professor, National Taiwan University
b)Long 源自文库.1 Gamma in Standard Chartered
7
© True Partner Education Ltd 2012
Are all Gamma’s created equal?
Assume the following choice: a)Long 0.1 Gamma in the Hang Seng Index Last index value is 20,000 Volatility 16 Expected Theta: 0.5 x 0.1 x 200 ^ 2 = HKD 2,000 Last share price is HKD 200 Volatility 16 Expected Theta: 0.5 x 0.1 x 2 ^ 2 = HKD 0.20
5
© True Partner Education Ltd 2012
Volatility, Gamma and Theta
Assume the following situation: Stock Price: Gamma: Volatility: HKD 20 -/- 1,000 24%
a)Expected daily move is 24% / 16 x HKD 20 = HKD 0.30 b)Gamma Revenue = 0.5 x – 1000 x 0.30 ^ 2 = - 45 c)Gamma Revenue = 0.5 x – 1000 x 0.40 ^ 2 = - 80 Therefore my loss from movement exceeds the theta received. a)Gamma Revenue = 0.5 x – 1000 x 0.10 ^ 2 = -5 (per increment) Over 4 increments I lose 20 from movement. Therefore, I have received more theta then I lost in the movement.
The concept of volatility
From Annualized volatility to Daily volatility: Assume there are 256 trading days in one year (the square root of 256 conveniently equals 16)
b)Long 0.1 Gamma in Standard Chartered
Thus a difference of a factor 100 in the share price has a magnified effect of 10,000x on the revenue of Gamma. (this is just a reformulation of the Gamma value formula: 0.5 x Gamma x Move ^2