LECTURE 10:
OPTION HEDGING AND
THE GREEK LETTERS
LEARNING OBJECTIVES
1)
To understand how financial institutions that sells options manage its risks.
2)
To understand key measures of options sensitivity analysis: Greek Letters represented by the Delta,
Gamma and Vega.
3)
Perform the Delta and Gamme hedging adjustments for a European option using Black-Scholes
model.
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OPTION WRITER’S RISK MANAGEMENT
 Option writers always face the risk of unlimited loss if the option they sold are exercised.
 If an option expires (unexercised), the writer retains the premium and has nothing to lose.
 If in the event an option is exercised then the writer has the obligation to pay what ever the
excess between the strike and the terminal underlying asset price.
 Therefore, the option writers will hedge (or protect ) their loss through various avenues.
 Here we will explore certain simple hedge techniques and move on to the sophisticated
Greek letters hedge techniques.
 Lets begin with Covered and Uncovered options and its underlying portfolios, the stop loss
strategies and slowly move on to the Delta hedging, the Gamma hedging and the Vega
hedging.
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THE HEDGE REQUIREMENT
 Assuming that a bank sold 100,000 Call options on a non-dividend paying shares of QFC for $300,000 ($3 per option).
 These options have the data: S0 = 49, K = 50, r = 5%, volatility = 20% p.a., T = 20 weeks,
 The Black-Scholes value of the option is $240,000 (so the “spread” for making market is $60,000).
 If the bank does nothing (i.e. taking a naked position), it makes $300,000 gain if the options expires unexercised, or
potentially face huge loss if share price goes up, say if ST = $60, the net loss will be (60 – 50) x 100,000 – 300,000 =
$700,000 loss
 How does the bank hedge its risk so that it can secures a profit of approximately $60,000?
 This is the requirement to hedge its risk to assure the bank can make $60,000 profit against the dynamic movements of
the underlying asset price due to the changes of its factors affecting the price of the options.
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STRATEGY 1: TAKING COVERED POSITION
1. Buy 100,000 shares of QFC as soon as the call option been sold, that is to buy shares at S0=$49:
effectively a covered call strategy.
2. If the options are exercised, total profit is (50 – 49)*100,000 + 300,000 = 400K (less financing cost for
the share purchase)

If option expires worthless because ST < K, lets say ST = $40, the loss on the shares is (49 – 40) x
100,000 = $900,000, making net loss = 900K - 300K= 600K loss

This is not a good hedge avenue.
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STRATEGY 2: A STOP-LOSS STRATEGY
1.
When the bank has sold 100,000 CALL Options on the shares of QFC, at a strike price of $50, it has to manage its loss in the
event the ST of this shares at expiry of the options is higher than K =$50.
2.
The bank can adopt the covered call strategy i.e. to buy the underlying shares to cover the possible loss if the options expire
with ST > K.
3.
Buy 100,000 underlying shares as soon as share price surpasses K=$50, and sell the shares as soon as share price falls below
$50 again.
4.
This is known as continuous series of buying and selling of the underlying shares until the options expire.
5.
To see the outcome of this strategy, consider 2 scenarios:
i.
The option starts its life, out-of-the-money, i.e. S0< K and
ii.
The option starts its life, in-the-money, S0 > K
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PROBLEMS WITH THE STOP-LOSS STRATEGY
 Therefore, the cost of writing and hedging the option is zero if the option starts out-of-the-money and its costs S0 – K
if the option starts in-the money
 That is, the cost of hedging is max(0; S0 – K), which is the intrinsic value of the option.
 Since option price = intrinsic value + time value, this reasoning implies this stop-loss strategy to hedge will always
lock in profit = time value of the option (plus market making spread).
Two problems:
1) the calculation ignores the time value of the cash flows and
2) also ignores the fact that such strategy often buys at price slightly higher than K and sell at price slightly lower than
K, plus transaction costs.
 Actual cost may be much larger than intrinsic value.
 Simulation shows this procedure does not produce good hedge outcome.
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DELTA AND DELTA HEDGING
ON A COMBINATION OF OPTION AND ITS UNDERLYING ASSET
REVIEW THE CONCEPT OF OPTION DELTA
 Delta of an option, denoted  , is the change in the option price as the result of a $1 change in the underlying
stock price.
 Delta can also be the equivalent number of units of the underlying asset i.e. stocks of a company
 Diagrammatically,
Call Option price
Slope =
B
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A
Stock price
USING THE OPTION DELTA TO APPROXIMATE CHANGES IN
THE OPTION PRICE

The delta has a simple interpretation, i.e. if the underlying stock price changes by a small amount dS, the delta predicts a
change in it’s option price of dS
𝒅𝑪 ≈ ∆𝑪 𝒙 𝒅𝑺
𝒅𝑷 ≈ ∆𝑷 𝒙 𝒅𝑺

Example: Suppose a call option is trading at c = $8.45, delta of this option is 0.7. Suppose the underlying price increases by
$0.4. The delta estimates a change in the call price of (0.70) x (+0.4) = +$0.28

So the new call price should be about 8.45 + 0.28 = $8.73

Note that this is an approximation only, which works extremely well for very small d, but it is progressively becomes less
accurate as d increases
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DELTA HEDGING
 Delta hedging means undertaking a position in the underlying asset to achieve a zero delta for the
overall position.(i.e. to neutralize the delta to zero).
 Note that delta of the underlying asset itself is 1.
 A position with a delta of zero is referred to as being delta neutral.
 Delta neutral positions/portfolios do not react much to small changes in the underlying’s price.
 However, because delta changes over time, the position remains delta hedged (or delta neutral) for
only a short period of time. The hedge has to be adjusted periodically. Periodical rebalancing the
overhanging delta is known as dynamic hedging.
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EXAMPLE 1
 Suppose a bank just sold call options on 2,000 shares, where  = 0.6 and S0=$100.
 =0.6 means a $1 increase in the stock price will lead to approximately $0.6 increase in the call option
price per share.
 Total delta on the short call would be 0.6 x - 2000 = -1200.
 Delta hedging means offsetting this delta of -1200 with something that has a delta of 1200: that is
buying 1200 shares (to neutralize the delta position to zero) .
 Assume stock price increases by $1 to $101. The short call position will lose approximately
$1x1200=$1200. The long position in 1200 shares gains $1200: overall impact is approximately zero:
this is because the overall position has been made delta neutral.
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EXAMPLE 1 (CONT’D)
 But since  will keep changing as St change by time, (the delta ratio is not static, it will change by
time) the hedger needs to recalculate  and readjust their stock position accordingly: this is called
dynamic hedging.
 Assuming the bank has sold 2000 call options on a stock price of $100 each and the delta was 0.6
 Therefore, the delta of a short position is 0.6 x -2000 = -1200
 This delta can be neutralised by buying additional 1,200 shares.
 Next day, suppose stock price increases to $110, and as the result, delta changes to 0.65. Total delta
on the short call changes to 0.65 x -2000 = -1,300.
 To rebalance the hedge, the trader needs to top up another 100 shares to maintain delta neutrality.
 This is call dynamic hedging: the hedging needs to be continually rebalanced to adjust to new delta
in order to maintain delta neutral.
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COMPUTING OPTION DELTA USING BLACKSCHOLES
 Mathematically, delta is the partial derivative of the option pricing function with respect to
S.
∆𝐶 =
𝜕𝐶
𝜕𝑆
∆𝑃 =
𝜕𝑃
𝜕𝑆
 Different pricing models produce different delta (and other Greek letters – examined
shortly)
 Black-Scholes model allows delta (and other Greek letters) to be computed analytically.
 It can be shown that, based on the Black-Scholes model,
∆ 𝑪 = 𝑵 𝒅𝟏 ,
∆𝑃 = 𝑁 −𝑑1 = 1 − 𝑁 𝑑1 or N(- d1)
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COMPUTING AND DELTA HEDGING OF EUROPEAN STOCK
OPTION
For a European Call option on a non-dividend-paying stock, delta is given by: ∆(call) = N(d1)
Where, d1 is defined as = ln(S0/K) + (r + σ2/2)T / σ √ T and N(x) is the cumulative distribution function for a
standard normal distribution.
The above formula gives the delta of a long position in one call option. The delta of a short position in one call
option is – N(d1).
Using delta hedging for a short position in a European call option involves maintaining a long position in the
delta number of shares for each option sold.
Similarly, using delta hedging for a long position in a European call option involves maintaining a short position
of N(d1) shares for each option purchased.
For a European put option on a non-dividend-paying stock, delta is given by: ∆(put) = 1- N(d1) or N(-d1)
Delta is negative, which means that a long position in a put option should be hedged with a long position in the
underlying stock, and a short position in a put option should be hedged with a short position in the underlying
stock.
In simple form; a negative delta can be neutralized through buying the delta number of underlying stock and a
positive delta can be neutralized through selling the delta number of underlying stocks
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EXAMPLE 2: DYNAMIC HEDGING
 Let’s go back to the original example where we need to hedge call options on 100,000 shares where S0=49,
K=50, r=5%, T=20 weeks, sigma=20%. Using BSM model, d1=0.0542 and N(d1)=0.522, and c=$240,000 (so
that since the trader sold the call for $300,000, they earned a spread of $60,000)
 Perform a delta hedge that is rebalancing the delta on weekly basis.
 During week 0, the Delta is -100,000 x N(d1)= -52,200. To neutralize this delta option seller need to buy 52,200
shares by borrowing $49 x 52,200 = $2,557,800 at an interest cost of 0.05 x $2,557,800 =$2,500 in the first week
alone.
 During week 1, the call option price is revised to be $48.12 and the delta drops to 0.458, where the new Delta is -
45,800 ( a drop of 6,400 shares that can be sold now). This will realize $308,000 in cash and cumulative
borrowing is reduced to $2,252,300 during week 1.
 This process of rebalancing on weekly basis as the option price change due to the change in the underlying stock
price and the change in the delta is called Dynamic delta hedging.
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DELTA OF A PORTFOLIO
IS SIMPLY THE SUM OF DELTAS OF INDIVIDUAL POSITIONS
DELTA OF A PORTFOLIO
 Is simply the sum of deltas of individual positions
 Suppose a financial institution has the following three positions on a stock
1) A long position in 100,000 call options with K=55, T=3 mths. Delta of each option is 0.533 (100,000 x 0.533 =
53,300)
2) A short position in 200,000 call options with K=56, T=5mths, delta = 0.468
( - 200,000 x 0.468 = - 93,600)
3) A short position in 50,000 put options with K=56, T=2 mths and delta= -0.508 ( - 50,000 x – 0.508 = 25,400)
Total Portfolio’s delta = 53,300 + (- 93,600) + 25,400 = - 14,900
** Portfolio can be made delta neutral by buying 14,900 shares **
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GAMMA AND GAMMA HEDGING
GAMMA
 The gamma, , of a portfolio of options on an underlying asset is the rate of change
of the portfolio’s delta for a change in the underlying asset’s price
Γ𝐶 =
𝜕Δ𝐶
𝜕𝑆
=
𝜕2 𝐶
𝜕𝑆 2
Γ𝑃 =
𝜕Δ𝑃
𝜕𝑆
=
𝜕2 𝑃
𝜕𝑆 2
 Small gamma means delta changes slowly and delta hedging requires infrequent
rebalancing.
 High gamma means delta changes quickly such that delta hedging got to be
rebalanced frequently (hence more costly) to minimise errors.
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CALCULATION OF GAMMA
Using the Black-Scholes model, for both European calls and puts, gamma is
calculated as
N (d1 )

, where
S T
1  x2 / 2
N ( x) 
e
2
 Call option with S0=49, K= 50, r=5%, sigma=20% and T=20 weeks, has a gamma of 0.066
 When stock price changes by say $1, the delta of the option changes by 0.066x1=0.066.
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USING THE OPTION GAMMA TO APPROXIMATE CHANGES IN THE OPTION
PRICE
Recall that if the underlying price changes by a small amount dS, the delta predicts a
change in the option price of  dS
𝑑𝐶 ≈ ∆𝐶 𝑥 𝑑𝑆
𝑑𝑃 ≈ ∆𝑃 𝑥 𝑑𝑆
A more accurate estimate is obtained by augmenting this with the gamma as follows
1
𝑑𝐶 ≈ ∆𝐶 𝑥 𝑑𝑆 + 2 Γ𝐶 (𝑑𝑆)2
1
𝑑𝑃 ≈ ∆𝑃 𝑥 𝑑𝑆 + 2 Γ𝑃 (𝑑𝑆)2
Thus, the gamma picks up the curvature error that is not captured by the delta
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GAMMA HEDGING
 Gamma hedging is seeking gamma neutrality so that the portfolio is insensitive to large changes in the
underlying price.
 Since gamma on the underlying asset is always zero, it is impossible to trade in the underlying asset to
make portfolio gamma neutral.
 Therefore, to achieve gamma neutrality we need to trade other options within the portfolio.
 But bringing in new options to the portfolio will alter its delta position: we need to also trade in the
underlying asset to make the portfolio delta neutral again.
 Once you neutralize the Gamma, you have to neutralize the delta again that would have changed due
to additional options traded within the portfolio to zero the gamma.
 Therefore, you have make gamma neutral first before you can delta neutral the portfolio.
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MAKING A PORTFOLIO GAMMA AND DELTA NEUTRAL –
EXAMPLE 3
 A trader’s portfolio is delta neutral and has a gamma of -3000.
 Assume a traded option has a delta of 0.62 and gamma of 1.5.
 To make portfolio gamma neutral, the trader needs to buy 2000 units of that
option. ( 3,000 / 1.5 = 2,000)
 Since the resulting portfolio (gamma neutral) now ends up with a altered delta of
0.62 x 2000=1240, the trader can sell 1240 units of the underlying asset to
maintain delta neutrality
 Therefore, you need to gamma neutral first and then eventually delta neutral again.
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OTHER GREEK LETTERS
ON COMBINATION OF OPTIONS AND ITS UNDERLYING ASSETS
THE VEGA
 So far we’ve assumed that volatility of the underlying stock is constant, which we
know is counterfactual.
 This means the value of a option can change because of movements in volatility as
well as because of changes in the underlying asset price.
 Come the concept of Vega
 Vega of a portfolio of options is the rate of change of the value of the portfolio with
respect to the change in the volatility of the underlying asset
 𝑉𝐶 =
𝜕𝐶
𝜕𝜎
𝑉𝑃 =
𝜕𝑃
𝜕𝜎
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OTHER GREEK LETTERS
 Theta, , is defined as the rate of change in option value with respect to time passage
 Theta is usually negative for an option because as time passes, all else remaining the
same, the option tends to be less valuable.
 Option writers normally do not theta hedge because there is no uncertainty in time
passage
 Rho,  , of a portfolio is the rate of change in option value with respect to the change
in interest rate.
 Since interest rate is typically stable, traders do not hedge this effect.
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LECTURE SUMMARY
 To manage its risk of loss due to the sold options being exercised, the option seller must make the
option sensitivities, the Delta, Gamma and Vega to be neutral.
 Delta of a stock option can keep changing due to the change in the underlying stock price,
therefore, the delta need to be neutralized on a periodical basis.
 The Gamma captures small errors of delta changes in order to maintain absolute neutrality of the
delta.
 The Vega can also induce delta variations due to the changes in the volatility of the underlying
stock price changes.
 Therefore we need to neutralized the Gamma and Vega first and then finally neutralize the Delta.
 Only Delta can be neutralized by trading the underlying stocks. The Gamma and Vega can only be
neutralized by trading other options within the option portfolio.
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