MMS Derivatives Lec 3

7/30/2019 MMS Derivatives Lec 3

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Option Pricing Models

By Vaibhav KabraM.F.S.M, F.R.M.


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Binomial Model


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Binomial Model

A stock is currently at $100.

We are interested in valuing an European call option to buy thestock at the strike price of $102.

This option will have one of the 2 values at the end of 3months i.e. At the end of 3 months, the stock will either be at$105 or $ 95.

If the stock price rises to $ 105, the value of the call optionwill be $ 3, if the stock price goes down to $95, the value ofthe call option will be zero.


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Binomial Model

We calculate the value of that makes the portfolio risk less

The portfolio is risk less if the value of is chosen so that the finalvalue of the portfolio is same for both alternatives

If the stock price moves up from $100 to $105, the value of shares is105 and the value of the option is $3

Therefore the total value of the portfolio is (105 3)

If the stock price moves down from $100 to $95, the value of sharesis $95 and the value of the option is $0

Therefore the total value of the portfolio is (95 )


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The portfolio is risk less if the value of is chosen so that the final value ofthe portfolio is same for both alternatives

Therefore 105 3 = 95 105 -95 =3

=3/10=0.3

A risk less portfolio is therefore

Long : 0.3 Shares Short : 1 call option

If the stock price moves up to $105 , the value of the portfolio is (105 *0.3)3 = 28.5 If the stock price moves down to $ 95 , the value of portfolio is 95 * 0.3 =

28.5


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Binomial Model Risk less portfolios must, in the absence of arbitrage opportunities,

earn the risk free rate of interest

If the risk free rate is 10% per annum, the value of the portfoliotoday must be present value of 28.5

28.5 * e-0.10*(3/12) = 27.79

The value of the portfolio today is100 * c

=100 * 0.3c

Hence, 100 * 0.3c = 27.79c = 3027.79

c = 2.21

In the absence of arbitrage opportunities, the current value of the

call option must be $2.21.


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Binomial Model

In the absence of arbitrage opportunities, the current value of thecall option must be $2.21.

If the value of the option were more than $ 2.21, the portfoliowould cost less than 27.79 to set up and would earn more than the

risk free rate

If the value of the option were less than $ 2.21, shorting theportfolio would provide a way of borrowing money at less than the

risk free rate


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Binomial Model-

A Generalization

Consider the stock Price is S0 An option on the stock whose price is f

We suppose than the option lasts for time T

The stock price can either move up from S0to S0u . (u >1)

The stock price can either move down from S0to S0d . (d


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Binomial Model-

A Generalization

The portfolio consists of a long position in shares and a shortposition in one call option We calculate the value of that will make the portfolio risk less If there is an up movement in the stock price, the value of the

portfolio at the end of the life of the option is (S0u fu)

If there is a down movement in the stock price, the value of theportfolio at the end of the life of the option is (S0d fd) For a risk less portfolio both should be equal.

(S0u fu) = (S0d fd)

(S0 uS0 d) = (fufd) = (fufd) / (S0 uS0 d)

As per the equation, is the ratio of the change in the optionprice to the change in the stock price.


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Binomial Model-

A Generalization

The portfolio is risk less and must earn the risk free interest rate If the risk free interest rate is denoted by r , the present value of theportfolio is (S0u fu) e-rT

The cost of setting the portfolio is (S0- f)

(S0 - f) = (S0u fu) e-rT

f = S0 S0u e-rT + fu e-rT

f = S0(1 u e-rT) + fu e-rT

Substituting for We get,f = e-rT [p fu + (1-p) fd]

p = (erTd) / (u - d)*** Please refer your note books for the entire proof


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Risk Neutral Valuation

p is the probability of an upward movement 1-p is the probability of downward movement The expected payoff from the option is [p fu + (1-p) fd] The expected payoff from the option today is obtained by discounting it

by risk free rate

f = e-rT [p fu + (1-p) fd] The expected stock price at time T , E(ST)

E(ST) = p S0 u + (1-p) S0 d

= p S0 u + S0 d - p S0 d

= p S0 (ud) + S0 d= S0 (ud) ( erTd) / (ud) + S0d= S0 e

rTS0d + S0dE(ST) = S0 e

rT

Hence the stock price grows on average at the risk free rate. Setting theprobability of up movement equal to p, is therefore equivalent toassuming that the return on the stock equal to the risk free rate


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Risk Neutral Valuation

-Formulae to Remember

Term Symbol Formula

Size of up move u e T

Size of down move d e - T = 1/eT = 1/u

Probability of up movePu (e

rTd) / (u - d)

Probability of down

move

Pd 1 - Pu


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Concept Checkers

The stock price of Shah Inc. is $100. The annual S.D. is 15%.Continuously Compounded Risk Free rate is 15% p.a.

Compute the value of 6 month European call option with a

strike price of $ 100 using a one period Binomial Model.

*** Refer your notebooks for solution


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Two step Binomial Model


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Concept Checker

The stock price of Shah Inc. is $100. The annual S.D. is 15%.Continuously Compounded Risk Free rate is 15% p.a.Compute the value of I year European CALL option with astrike price of $ 100 using a two period Binomial Model.



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Concept Checker

The stock price of Shah Inc. is $100. The annual S.D. is 15%.Continuously Compounded Risk Free rate is 15% p.a.Compute the value of I year EuropeanPUT option with astrike price of $ 100 using a two period Binomial Model.



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American Options

Valuing American Options using Binomial Model

The Binomial Model is also well suited for handling American Styleoptions

At any point in the Binomial tree, we can see whether the calculatedvalue of the option is exceeded by its value if exercised early

If the payoff from early exercise (the intrinsic value of the option )is greater than the options value (the present value of the expected

payoff at the end of the second period) then it optimal to exerciseearly .

If early exercise is optimal, we replace the calculated value with theexercise value


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Concept Checker

The stock price of Shah Inc. is $100. The annual S.D. is 15%.Continuously Compounded Risk Free rate is 15% p.a.Compute the value of I year American CALL option with astrike price of $ 88 using a one period Binomial Model.



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Impact of Dividends

When dividends are to be considered then there is a minor change inthe formula of probability up move.

The formula for Pu becomes

Pu = (e(r-q)Td) / (ud)

where q is the dividend yield

Up and down movement factors remains the same.


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Black- Scholes - Merton Model


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In the binomial model, we divided an options life into a given number ofperiods.

Suppose we are pricing a one year option. If we use only one binomialperiod, it will give us only two prices for the underlying, and we areunlikely to get a very good result.

If we use two binomial periods, we will have three prices for theunderlying at expiration. This result would probably be better but still notvery good.

But as we shorten the length of intervals and increase the number ofperiods, there will be more branches to consider and hence the result shouldbecome more accurate.

Increasing the number of periods by considering arbitrarily small lengthof intervals, we are moving from discrete time to continuous time.


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Black- Scholes - Merton Model for option valuation is acontinuous time model.

The Binomial model converges to this continuous timemodel as we make the time periods arbitrarily small.


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Assumptions

1. The Underlying Price follows a Geometric Lognormal

Diffusion Process

The underlying price follows a lognormal probability distribution as

it evolves through time. A lognormal probability distribution is one in which the log return is

normally distributed.

For e.g., if a stock moves from 100 to 110, the return is 10 % but thelog return is ln(1.10) = 0.0953 or 9.53%.

Log returns are often called continuously compounded returns. The lognormal distribution is skewed, reaching further out to the

right and truncated on the left side, reflecting the limitation that an

asset cannot be worth less than zero.


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Assumptions1. The Underlying Price follows a Geometric Lognormal Diffusion Process

ln (ST / S0) ~ N [(2/2)T , T)]

ln ST ~ N [ ln S0 + (2/2)T , T)]

where:

ST = stock price at time T

S0 = stock price at time 0

= expected return on the stock per year = volatility of the stock price per yearN [m , s] = normal distribution with mean = m and standard

deviation = s


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Assumptions

1. The Underlying Price follows a Geometric LognormalDiffusion Process

ln (ST / S0) ~ N [(2

/2)T , T)] Dividing the mean and standard deviation by T, results in

a continuously compounded annual return of a stockprice. The continuously compounded annual returns are

normally distributed with aMean of (2/2)Standard deviation of /T


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Assumptions

Expected Value:

Using the properties of a lognormal distribution, we can

show that the expected value of ST, E(ST) isE(ST) = S0 e

T


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Assumptions

Expected Annual Return =

Mean Return = (2/2)

The difference between the Expected Annual Return onthe stock, and the mean return (2/2), is closely relatedto the difference between the arithmetic mean and the

geometric mean.

The mean return will always be slightly less than theexpected return, just as the geometric return will always be

slightly less than the arithmetic return.

Using a Geometric Return produces a more accuraterepresentation of portfolio returns.


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Assumptions

2. The Risk Free Rate is constant and known

The Black- Scholes - Merton Model does not allow

interest rates to be random.

Generally, we assume that the risk free rate is constant.


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Assumptions

3. The volatility of the underlying asset is constant andknown

The volatility of the underlying asset, specified in the form ofstandard deviation of the log return, is assumed to be known at

all times and does not change over the life of the option.

In reality, volatility is definitely not known and must beestimated and obtained from some other source.

In addition, volatility is generally not constant. Obviously thestock market is more volatile at some times than at others.

Nonetheless, this assumption is very critical for this model.


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Assumptions

4. There are no Taxes or Transaction Costs

Taxes and Transaction Costs greatlycomplicate our models and keep us from

seeing the essential financial principles

involved in the models.


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Assumptions

5. There are no Cash flows on the underlying

The basic form of Black- Scholes - Merton Model makesthe assumption that the underlying asset pays nocoupons, dividends or and other cash flows.

However, this assumption can be relaxed and will bediscussed in the later sections.


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Assumptions

6. The Options are European

The Black- Scholes - Merton Model does not priceAmerican Options.

Users of the model must keep this in mind, or they maybadly misprice these options.

For pricing American options, the best approach is thebinomial model with a large number of time periods.


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Formulae

The BlackScholesMerton formulae for the prices of calland put options are

c = S0

N(d1)X e rcT N(d

2)

p = X ercT[1 - N(d2)] - S0 [1 - N(d1)]

whered1 = ln(S0 / X) + [r

c + 2/2)]T / ( T)

d2 = d1 - T


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Formulae

= the annualized standard deviation of thecontinuously compounded return on the stock

rc = the continuously compounded risk free rate of

return T = time to maturity

S0 = Stock or Asset Price

X = Strike or Exercise Price N ( ) = Cumulative Normal Probability


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Formulae

Note:

If you are given any one of the prices, then the other oneneed not be substituted in the BSM Formula. The otheroption price can be calculated by using the put-call parity

equation. For e.g. if the call price is known, then the put value can be

found by simply substituting the values in the put-call parityequation

p = cS + X e rt

Similarly, if the put price is known, then the call value canbe found by simply substituting the values in the put-callparity equation

c = p + S - X e -rt


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Concept Checker

S0 = $100

X = $95

T = 3 months

rc

= 4%= 15%

Calculate the call and put option value using the BlackScholesMerton Model



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with Dividends

Since the BlackScholesMerton Model is in continuoustime, in practice S0 e

qT is substituted for S0 in the BSMformula, where q is equal to the continuously compounded rateof dividend payment.

The asset price is discounted by a greater amount to accountfor the greater amount of cash flows.

Cash flows will increase put values and decrease call values


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Concept Checker

S0 = $100

X = $95

T = 3 months

rc = 4%

= 15%q = 3%

Calculate the call and put option value using the BlackScholesMerton Model



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Historical Volatility

Volatility is one of the important variable in the BSM modeland is unobservable.

The steps in computing historical volatility for use as aninput in the BSM continuous time options pricing model are:

1. Convert a time series of N prices to returnsRi = PiPi-1 / Pi-1 , i= 1 to N

2. Convert the returns to continuously compounded returns

Ric = ln (1+Ri), i= 1 to N

3. Calculate the Variance and Standard Deviation of thecontinuously compounded returns

2 = (Ric - Ri)2 / (N-1) and = 2


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Historical Volatility

Calculation

R1 = P1P0 / P0 = 15%R2 = P2P1 / P1 = 20%R3 = P3P2 / P2 = 14%R4 = P4P3 / P3 = 13%R5 = P5P4 / P4 = 18%

R = 16%

2 = (R1-R)2 + (R2R)2 + (R3R)2 + (R4R)2 + (R5R)2 / (N-1)

2 = (1 + 16 + 4 + 9 + 4) / 4 ; 2 = 8.5 and = 2.92


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Implied Volatility

The one parameter in the BSM model that can not be directlyobserved is the volatility of the stock price. In the previous slide,we observed how volatility is obtained from Historical Prices.Historical data can serve as a basis for what volatility might begoing forward, but is not always representative of the currentmarket.

In practice, traders usually work withImplied Volatilities.

These are the volatilities implied by option prices observed in themarket.

To illustrate how implied volatility is calculated, suppose that thevalue of a European Call on a non dividend paying stock is 1.875when S0 = 21, K = 20, r = 0.1, and T = 0.25. The implied volatility isthe value of that , when substituted in the BSM equation gives c =

1.875.


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Implied Volatility

Unfortunately it is not possible to invert equation so that is expressed as afunction of S0, K, r, T and c. i.e. Volatility enters in the equation in acomplex way, and there is no closed form solution for the volatility thatwill satisfy the equation. Thus, by setting the BSM price equal to themarket price, we can work backwards to infer the volatility. Hence aniterative search procedure can be used to find the implied volatility.

For e.g. we can start by trying = 0.20. This gives the value c = 1.76,which is too low. Option value and volatility are positively correlated i.e. cis an increasing function of. Hence a higher value of is required here

We can try a value of 0.30 for . This gives the value of call = 2.10, whichis too high and means that should lie between 0.2 and 0.3. The value of0.25 for also proves to be high. Hence should lie between 0.2 and 0.25.

The correct value for the call price is obtained at = 0.235 or 23.5%


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Option Greeks


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Naked and Covered Call

A naked position occurs when one party sells a call option without owningthe underlying asset.

A covered position occurs when the party selling a call option owns theunderlying asset

A firm sells 1,000 call options S

0= $100 and X = $105

Option premium received = $10 per option A naked position would generate $10,000 in revenue, if the stock price

remains below $105

If at expiration, S0 = 106, then loss on the written call = $1,000 and profitreduces to $9,000

If at expiration, S0 = 110, then loss on the written call = $5,000 and profitreduces to $5,000 If at expiration, S0 = 125, then loss on the written call = $20,000 and loss =

$10,000


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Naked and Covered Call

Therefore, the maximum potential gain is capped at the level of thepremium received, whereas the potential loss from a naked written positionis unlimited.

With a Covered Call, the firm already owns 1000 shares of the underlyingstock, so if the stock price rises above $105 and the option is exercised, thefirm will sell shares that it already owns. For e.g. if the stock is bought at$105

If at expiration, S0 = 106, then loss on the written call = $1,000 but there isa gain of $1000 on the underlying stock. Therefore the revenue of $10,000on the option premium is saved.

If at expiration, S0 = 125, then loss on the written call = $20,000 but thereis a gain of $20,000 on the underlying stock. Therefore the revenue of$10,000 on the option premium is saved.

However, if the stock price reduces to $50, then there is a loss of 55,000 onthe stock and a net loss of 45,000


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Stop Loss Strategy

Stop Loss Strategies with Call options are designed to limit thelosses associated with short option positions, especially the callwriters.

Hold naked Call Position when the option is out of the money

Hold Covered Call position (buy the underlying) when the call is inthe money

Sell the asset as soon as the option goes out of the money ( the stockprice is below the strike price)

This approach is very simplistic, however, the transaction costs arevery high and there is great price uncertainty as well.


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Option Delta

Delta ( )gives the relationship between option price and theunderlying price.

Delta ( )of an option is the ratio of the change in the optionprice(c or p) to the change in the underlying price (S).

Delta = Change in Option Price / Change in the underlying price

= (c /S) . for Call Options = (p /S) . for Put Options

Delta is the slope of the call option pricing function at the currentstock price


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Option Delta

A Delta of 0.8 means that the option price will change by $0.8 forevery $1 change in the underlying price

To completely hedge a long stock or a short call position, aninvestor must purchase the number of shares of stock equal to deltatimes the number of options sold.

For e.g. If an investor is short 1000 call options, then he needs tobuy 800 shares of the underlying stock. Now if the underlying pricerises by $1, the value of sold options decreases $800 but there iscorresponding increase in the stock price by $800.


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Option Delta

Call Deltas range from 0 to 1. i.e. the delta will increase towards 1as the underlying price moves up and will decrease towards 0 as theunderlying price moves down.

Call Deltas range from 0 to -1. i.e. the delta will decrease towards -1as the underlying price moves down and will increase towards 0 as

the underlying price moves up.

Delta of an at the money call is +0.5 whereas Delta of an at themoney put is -0.5


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Option DeltaDeltas Formula

European call non dividend paying

stock

N ( d1)

European call dividend paying stock e q T *N( d1)

European put non dividend payingstock

N ( d1)1

European put dividend paying stock e q T *[ N ( d1)1]

Forward contract 1

Forward contract (dividend payingstock)

e q T

Futures contract e rt

Futures contract ( dividend paying

stock)

e (rq )t


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Dynamic aspect of

Delta hedging

Delta is constantly changing which means that delta hedging is a dynamicprocess. In fact Delta hedging is also referred to as dynamic hedging.

In theory, the delta is changing continuously and the hedge should beadjusted continuously to maintain a delta neutral position, i.e. the investor

will need to either purchase or sell the underlying asset depending on thenew delta.

However the continuous adjustment is not possible in reality.

When the hedge is not adjusted continuously, we are admitting thepossibility of much larger moves in the price of the underlying.


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Portfolio Delta (P)

The delta of a portfolio of options on a single underlying asset can becalculated as the weighted average delta of each option position in theportfolio:

Portfolio Delta = P = wii

where wi = Portfolio weight of each option position i = Delta of each option position

Therefore, portfolio delta represents the expected change of the overalloption portfolio value given a small change in the price of the underlyingasset.


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Option Vega ()

Vega () is option prices sensitivity to volatility changes in the underlyingstock

Vega () is positive for both, calls and puts, meaning that if the volatilityincreases, both call and put prices increase.

Vega () of 0.5 means that if volatility increases by $1, options price willincrease by $0.5

Vega () is maximum when Options are At- theMoney

Deep In The Money and Deep Out of The Money options have very littlesensitivity to changes in volatility thus small Vega() i.e. Vega() is closeto zero.


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Option Rho ()

Rho () is option prices sensitivity to the changes in the Risk freerate

Equity Options are less sensitive to changes in interest rates as theyare to changes in other variables like volatility and stock prices.

Rho () is a much important risk factor for fixed income derivatives.

In The Money calls and puts are more sensitive to interest ratechanges than Out of The Money options

Increase in rates cause larger increases for in the money calls versusout of the money calls

Increase in rates cause larger decreases for in the money puts versusout of the money puts


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Option Theta ()

Theta, () , measures the options sensitivity to a decrease in time toexpiration. It is the rate of change of the option price with respect to thepassage of time with all else remaining the same.

Theta () is sometimes referred to as the time decay of the portfolio.

Theta () is usually negative for an option. This is because, the option

price decreases as time moves forward all else remaining constant.

Both call and put values decrease as the time to expiration decreases. Theta() is most pronounced when option is atthemoney especially nearer tothe expiration.

For some cases, European put options can increase in value as the time toexpiration decreases, i.e. it has a positive Theta (). This occurs when theput is deep in the money, the volatility is low, the interest rate is high andthe time to expiration is low.

Most of the time, option prices are higher the longer the time to expiration.


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Option Gamma ()

Gamma is the rate of change of delta with respect to the price of theunderlying asset. It is the second order partial derivative withrespect to the asset price

= 2c /S2

2c = second partial derivative of call price S2 = second partial derivative of stock price

Gamma is Maximum when Option is AttheMoney

When option is DeepintheMoney or OutofMoney there islittle effect on gamma


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Option Gamma ()

Large Gamma implies that Delta is changing rapidly. Small GammaImplies that Delta is changing slowly

Since Gamma represents the curvature component of the call price functionnot accounted for by delta, it can be used to minimize the hedging errorassociated with the linear relationship to represent the curvature of the callprice function

Delta hedging will help for small changes in stock price but for largechanges in stock price the portfolio has to be gamma hedged

Thus we should not only create a delta neutral position, but also we need to

create a gamma neutral position

Underlying assets have linear payoffs and thus cant be used for gammahedging , we use options (non linear pay offs) to create gamma neutralpositions


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Option Gamma () If a delta-neutral portfolio has a gamma of (P) and a traded optionhas a gamma of (T). To make the portfolio and gamma neutral the

position in the traded option necessary is

- (P /T)


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Option Gamma () - Gamma Hedging:

An existing short position is delta neutral but has a Gamma of -4000(negative because we are short the options). Also, there exists atraded option with a delta of 0.7 and a gamma of 1.60. Create aGamma Neutral Position.

We need to buy options as we are currently short options , in orderto gamma hedge our position

Options to buy = ( 4000) / 1.60 = 2500 Thus buy 2500 options to be gamma neutral But this changes our delta neutral position ( We added 2500 more

options) thus we will need to sell (delta * options added) of

underlying stock to maintain original delta hedge position Thus sell (2500 * 0.7) 1750 underlying stock to maintain original

delta hedge position

In this way a Gamma Neutral position is created.


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Relationship Among Delta, Theta and

Gamma

r = + r S + (1/2) 2

S2

where

r = risk-free rate

= Price of the Option = options theta = options delta= options Gamma2 = underlying stocks variance

For delta neutral position the above equation reduces to

r = + (1/2) 2 S2


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Thank You !

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