Greek Letters

The Greek Letters

B. B. ChakrabartiProfessor of Finance

Indian Institute of Management, Calcutta

Option Portfolio Value and Greeks

B. B. Chakrabarti: [email protected] 2

2

2

2

2

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Delta• Delta () is the rate of change of the option price with

respect to the underlying.• If c is the price of the call option and S is the stock price,

Optionprice

A

BSlope =

Stock price

Sc

then,

Delta• Delta denotes the movement of the option position

relative to the movement of the underlying position. So, Delta is the speed of change of option value when underlying asset value changes.

• Delta is another way of expressing the probability of an option expiring in the money.

• ATM call options have a Delta of 0.5 or 50% meaning a 50% chance of expiring ITM.

• Deep ITM call will have a Delta of 1 meaning a 100% chance of expiring ITM.

• Deep OTM call will have a Delta close to zero meaning a near zero chance of expiring ITM.


DeltaDelta RangeLong call 0 to 1.00 Short call 0 to -1.00 Long put 0 to -1.00 Short put 0 to 1.00 Delta Range Rules of Thumb (Long Call)Deep-in-the-money 0.75 to 1.00Slightly-in-the-money 0.55 to 0.75At-the-money 0.45 to 0.55Slightly-out-of-the-money 0.25 to 0.45Deep-out-of-the-money 0 to 0.25Delta of long stock is 1 and -1 for short stock.



Delta of European Options• The delta for a European call on a non-

dividend paying stock = N (d 1)• The delta for a European put on a non-

dividend paying stock =N (d 1) – 1• The delta of a European call on a asst

paying dividends at rate q is N (d 1)e– qT

• The delta of a European put on this asset is e– qT [N (d 1) – 1]


Position Delta• Position Delta or Delta of a portfolio of options or

other derivatives dependent on a single asset whose price is S and the value of the portfolio is Π is

• The delta of the portfolio can be calculated from the deltas of the individual options in the portfolio.

• If a portfolio consists of a quantity wi of option i (1 < i < n), the Position Delta is

• where ∆i is the delta of ith option.

S

n

iiiw

1


Delta of a Portfolio – Example• Suppose a financial institution in the United States

has the following three positions in options on the Australian dollar:1. A long position in 100,000 call options with strike price

0.55 and an expiration date in three months. – The delta of each option is 0.533.

2. A short position in 200,000 call options with strike price 0.56 and an expiration date in five months. – The delta of each option is 0.468.

3. A short position in 50,000 put options with strike price 0.56 and an expiration date in two months. – The delta of each option is – 0.508.

• The delta of the whole portfolio is= 100,000*0.533 – 200,000*0.468 – 50,000*(– 0.508) = – 14,900

• This means that the portfolio can be made delta neutral with a long position of 14,900 Australian dollars.


Delta – Hedging Schemes• A position with a delta of zero is referred to

as being delta neutral.• Because delta changes, the investor's

position remains delta hedged (or delta neutral) only for a relatively short period of time. The hedge has to be adjusted periodically. This is known as rebalancing.

• The delta-hedging scheme with intermediate adjustment is called dynamic-hedging scheme.


Making Position Delta Neutral – Example

• A U.S. bank has sold six-month put options on £1 million with a strike price of 1.6000 and wishes to make its portfolio delta neutral.

• Suppose that the current exchange rate is 1.6200, the risk-free interest rate in the United Kingdom is 13% per annum, the risk-free interest rate in the United States is 10% per annum, and the volatility of sterling is 15%.

• In this case, So = 1.6200, K = 1.6000, r = 0.10, rf = 0.13, σ = 0.15, and T = 0.5.

• The delta of a put option on a currency ise– r

fT [N (d 1) – 1]


Making Position Delta Neutral – Example – contd.

• It can be shown that• d1 = 0.0287 and N(d1) = 0.51 15• Hence, the delta of the put option is – 0.458.

– This is the delta of a long position in one put option.– It means that when the exchange rate increases by ∆S,

the price of the put goes down by 45.8% of ∆S.– The delta of the bank's total short option position is

+458,000. • To make the position delta neutral, we must

therefore add a short sterling position of £458,000 to the option position. – This short sterling position has a delta of – 458,000 and

neutralizes the delta of the option position.


Using Futures for Delta Hedging

• The delta of a futures contract is e(r-q)T times the delta of a spot contract.

• The position required in futures for delta hedging is therefore e-(r-q)T times the position required in the corresponding spot contract.


Theta• Theta () of a derivative (or portfolio of

derivatives) is the rate of change of the value with respect to the passage of time.

• Theta stands for the option position’s sensitivity to time decay.

• Long (Short) options have negative (positive) Theta meaning that time decay is eroding the time value portion of the option value as days pass.

• Time decay thus hurts an option buyer and helps option writer’s position.

tf put or call a of θ


Theta for NDPS European Calls and Puts

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dNrKeTdNS

rT

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where

(put)

:stock paying dividend-non on put European aFor

(call)

:stock paying dividend-non on call European aFor


Theta for DPS European Calls and Puts

TdT

TqrKSd

TTqrKSd

dNrKeedNqST

edNS

dNrKeedNqST

edNS

rTqTqT

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)2/2()/ln(

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where

(put)

:q rate at dividend paying asset on put European aFor

(call)

:q rate at dividend paying asset on call European aFor


Theta – Example• Consider a four-month put option on a stock index.

The current value of the index is 305, the strike price is 300, the dividend yield is 3% per annum, the risk-free interest rate is 8% per annum, and the volatility of the index is 25% per annum.

• In this case So = 305, K = 300, q = 0.03, r = 0.08, σ = 0.25, and T = 0.3333.

• The option's theta is

= – 18.15= – 18.15/365 = -0.0497 per calendar day= – 18.15/252 = -0.0720 per trading day

)()(2

)('210

10 dNrKeedNqST

edNS rTqTqT


Gamma• Gamma () is the rate of change of delta ()

with respect to the price of the underlying asset.

• If gamma is small, delta changes slowly, and adjustments to keep a portfolio delta neutral need to be made only relatively infrequently.

• If gamma is large in absolute terms, delta is highly sensitive to the price of the underlying asset.

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Gamma• Gamma can be viewed in two ways.

a) as the acceleration of the option position relative to the underlying stock price.b) as the odds of a change in Delta.

• Gamma is effectively an early warning that Delta could be about to change.

• Both calls and puts have positive Gammas.• Deep OTM and deep ITM options have near zero

Gamma because the odds of a change of delta are very low.

• Logically Gamma tends to peak around the strike price.


Gamma as Curvature in Option Price

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CStock priceS

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Interpretation of Gamma• For a delta neutral portfolio, t + ½S 2

S

Slightly Negative Gamma

S

Slightly Positive Gamma


Interpretation of Gamma – contd.

S

Large Negative Gamma

S

Large Positive Gamma


Making a Portfolio Gamma Neutral• Making a delta-neutral portfolio gamma

neutral can be regarded as a first correction for the fact that the position in the underlying asset cannot be changed continuously when delta hedging is used. An option on the underlying asset is used for this purpose. The underlying asset cannot be used because its gamma is zero.

• Delta neutrality provides protection against relatively small stock price moves between rebalancing.

• Gamma neutrality provides protection against larger movements in the stock price between hedge rebalancing.


Making a Portfolio Gamma Neutral – Example

• Suppose that a portfolio is delta neutral and has a gamma of -3,000.

• The delta and gamma of a particular traded call option are 0.62 and 1.50, respectively.

• The portfolio can be made gamma neutral by including in the portfolio a long position of (3,000/1.5) = 2,000 in the call option.

• However, the delta of the portfolio will then change from zero to 2,000*0.62 = 1,240.

• A quantity 1,240 of the underlying asset must therefore be sold from the portfolio to keep it delta neutral.


Calculation of Gamma

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where

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where

:stock paying dividend-non on putor call European aFor


Gamma – Example• Consider a four-month put option on a stock index.

The current value of the index is 305, the strike price is 300, the dividend yield is 3% per annum, the risk-free interest rate is 8% per annum, and volatility of the index is 25% per annum.

• In this case, So = 305, K = 300, q = 0.03, r = 0.08, σ = 0.25, and T = 4/12.

• The gamma of the index option is given by

=0.00857

• Thus, an increase of 1 in the index (from 305 to 306) increases the delta of the option by approximately 0.00857.

TSedN qT

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Relationship Between Delta, Gamma, and Theta

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Relationship Between Delta, Gamma, and Theta – With Dividend

For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q

( )r q S S r12

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Vega• Vega () is the rate of change of the value of a

derivatives portfolio with respect to volatility.

• Options tend to increase in value when the underlying asset’s volatility increases.

• Volatility helps the option buyers and hurts the writers.

• Vega is positive for long options and negative for short options.

• Vega tends to be greatest for options that are close to at-the-money.


Calculation of Vega

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where

:stock paying dividend-non on putor call European aFor


Managing Delta, Gamma, & Vega

• Delta can be changed by taking a position in the underlying.

• To adjust & it is necessary to take a position in an option or other derivative.


Rho• Rho is the rate of change of the value

of a derivative with respect to the interest rate.

• For currency options there are 2 rhos corresponding to two interest rates.

r

rho

)N(-dKTe- rho(put)and )N(dKTe rho(call)

stocks, paying dividend-noon optionsEuropean For

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Rho• Rho stands for the option position’s

sensitivity to interest rates. • A positive Rho means that higher

interest rates are helping the position and a negative Rho means that higher interest rates are hurting the position.

• Rho is the least important of all the Greeks as far as stock options are concerned.

Positive or Negative Sign of Greeks

• A plus or minus sign for a Greek is based on the assumption that the underlying stock price rises, time is moving forward, volatility and interest arte are increasing.

• A Greek can be positive or negative number , depending on long or short option and whether it is a call or put.


Positive or Negative Sign of Greeks

Option Delta Gamma Vega ThetaLong call + + + -Short call - - - +

Long put - + + -Short put + - - +


Hedging Principle


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Hedging in Practice

• Traders usually ensure that their portfolios are delta-neutral at least once a day.

• Whenever the opportunity arises, they improve gamma and vega.

• As portfolio becomes larger hedging becomes less expensive.


Scenario Analysis

A scenario analysis involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilities.


Portfolio Insurance

• Portfolio insurance is done by buying a put option on the portfolio.

• One approach is to buy put options on a market index or to create the options synthetically.

• This involves initially selling enough of the portfolio (or of index futures) and the proceeds invested in a riskless asset to match the of the put option.


Portfolio Insurance – contd.

• As the value of the portfolio increases, the of the put becomes less negative and some of the original portfolio is repurchased.

• As the value of the portfolio decreases, the of the put becomes more negative and more of the portfolio must be sold.


Portfolio Insurance – Example 17.9• A portfolio is worth $90 million. For protection

against market decline, a 6-month European put option on the portfolio at strike price $87 million is required.

• r=9% pa, q=3% pa, T=0.5 yr, σ=25% pa, S&P 500 index stands at 900, K=870. The value of one index contract is $100,000. Assume that the portfolio mimics S&P 500 index closely.

• Delta of the required put option = e– qT [N (d 1) – 1] = -0.3215

• So, 32.15% of the portfolio should be sold initially and invested in a riskless asset to match the delta of the required option. If after 1 day, the value of the portfolio reduces to $88 million and the delta changes to -0.3679, a further 4.64% of the original portfolio should be sold.


Problem No. 17.14 (Hull – 7th Ed.)

• A financial institution has just sold 1,000 seven-month European call options on the Japanese yen. Suppose that the spot exchange rate is 0.80 cents per yen, the exercise price is 0.81 cents per yen, the risk-free interest rate in the United States is 8% per annum, the risk-free interest rate in Japan is 5% per annum, and the volatility of the yen is 15% per annum. Calculate the delta, gamma, vega, theta, and rho of the financial institution's position. Interpret each number.


Problem No. 17.14 (Ans.)

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Problem No. 17.14 (Ans. – contd.)

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• A bank's position in options on the dollar-euro exchange rate has a delta of 30,000 and a gamma of -80,000. Explain how these numbers can be interpreted. The exchange rate (dollars per euro) is 0.90. What position would you take to make the position delta neutral? After a short period of time, the exchange rate moves to 0.93. Estimate the new delta. What additional trade is necessary to keep the position delta neutral? Assuming the bank did set up a delta-neutral position originally, has it gained or lost money from the exchange rate movement?


Problem No. 17.22 (Ans.)• The delta indicates that when the value of the euro

exchange rate increases by $0.01, the value of the bank’s position increases by 0.01*30,000= $300.

• The gamma indicates that when the value of the euro exchange rate increases by $0.01, the delta of the portfolio decreases by 0.01*80,000= $800.

• For delta neutrality 30,000 euros should be shorted.• When the exchange rate moves up to 0.93, we expect

the delta of the portfolio to decrease by (0.93 – 0.90)*80,000 = 2,400 so that it becomes 27,600.

• To maintain delta neutrality, it is therefore necessary for the bank to unwind its short position 2,400 euros so that a net 27,600 has been shorted.


Problem No. 17.22 (Ans. – contd.)• As shown in the figure below, when a portfolio is

delta neutral and has a negative gamma, a loss is experienced when there is a large movement in the underlying asset price. We can conclude that the bank is likely to have lost money.

S

Slightly Negative Gamma

S

Large Negative Gamma



• A financial institution has the following portfolio of over-the-counter options on sterling:

• A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8.a. What position in the traded option and in sterling would make

the portfolio both gamma neutral and delta neutral?b. What position in the traded option and in sterling would make

the portfolio both vega neutral and delta neutral?

Type

Position

Delta of option

Gamma of option

Vega of option

Call -1,000 0.5 2.2 1.8Call -500 0.8 0.6 0.2Put -2,000 -0.4 1.3 0.7Call -500 0.7 1.8 1.4


Problem No. 17.25 (Ans.)• The delta of the portfolio

= – 1,000*0.50 – 500*0.8 – 2,000*(–0.40) – 500*0.70 = – 450

• The gamma of the portfolio= – 1,000*2.2 – 500*0.6 – 2,000*(1.3) – 500*1.8 = – 6,000

• The vega of the portfolio= – 1,000*1.8 – 500*0.2 – 2,000*0.7 – 500*1.4 = – 4,000


Problem No. 17.25 (Ans. – contd.)Part a:• A long position in 4,000 traded option will

give a gamma-neutral portfolio since the long position has a gamma of 4,000*1.5 = 6,000.

• The delta value of the whole portfolio (including traded options) is then= 4,000*0.6 – 450 = 1,950

• Hence in addition to the 4,000 traded options, a short position in £1,950 is necessary so that the portfolio is both gamma and delta neutral.


Problem No. 17.25 (Ans. – contd.)Part b:• A long position in 5,000 traded option will give a

vega-neutral portfolio since the long position has a vega of 5,000*0.8 = 4,000.

• The delta value of the whole portfolio (including traded options) is then= 5,000*0.6 – 450 = 2,550

• Hence in addition to the 5,000 traded options, a short position in £2,550 is necessary so that the portfolio is both vega and delta neutral.

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