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Fin 288
The Greek Letters
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Fin 288Pricing Options
Both the Binomial Tree Approach and the Black Scholes approach produce the same option value as the number of steps in the Binomial tree becomes large. For this section we will concentrate on the “Theoretical value” of the option – the black scholes solution.
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Fin 288Black Scholes
Value of Call Option = SN(d1)-Xe-
rtN(d2) S = Current value of underlying asset
X = Exercise pricet = life until expiration of optionr = riskless rate2 = varianceN(d ) = the cumulative normal
distribution (the probability that a variable with a standard normal distribution will be less than d)
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Fin 288Black Scholes (Intuition)
Value of Call Option
SN(d1) - Xe-rt N(d2)The expected PV of cost Risk NeutralValue of S of investment Probability
ofif S > X S > X
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Fin 288Black Scholes
Value of Call Option = SN(d1)-Xe-rtN(d2)
Where:
t
trXS
d
)2()ln(2
1
tdd 12
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Fin 288Time Value of an Option
The time value of an option is the difference in the theoretical price of the option and the intrinsic value.It represents the the possibility that the value of the option will increase over the time it is owned.
Hull Chapter 15 Fundementals of Futures and Options Markets
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Fin 288An Example1:
Assume that a financial institution has sold a European Call Option on a non dividend paying stock.S= $49, X=$50, r = 0.05, =0.20, t = 20 weeks =0.3846 years. Call option value = 2.372Assume that the institution has sold the option for $3 a share or .628 more than its theoretical value.How can it hedge its risk?
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Fin 288Naked vs. Covered position
The firm can do nothing and hold only the option (a naked position). It would then be forced to buy the shares if the owner of the option exercises it in 20 weeks. The profit diagram would look like the normal short call. The firm can buy the stock today and have a covered call. This introduces a downside risk, if the value of the stock decreases the firm looses due to the decline in the value of the share.
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Fin 288Profit Diagram Covered Call
Profit
Short Call
Long Spot
Covered Call
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Fin 288
Hedging with a Stop-Loss Strategy.
One possible solution is to develop a dynamic buying strategy for the share. For example the firm could buy shares whenever the stock price is greater than the exercise price, It could then sell the shares if the stock price drops below the exercise price. It would then be hedged when the option will be exercised and unhedged when it will not be exercised.
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Fin 288Stop Loss Costs
The problem is that there are substantial transaction costs associated with the strategy.Also there is uncertainty about the actual cost of the share. Therefore you are not buying and selling each time at the exercise price.
A better approach is to use the delta of the option
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Fin 288Delta of an option
The delta of the option shows how the theoretical price of the option will change with a small change in the underlying asset.
bond underlying of pricein change
option call of pricein changedelta
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Fin 288Time Value of the Option
Plotting the value of the option compared to the profit and or payoff provides a starting point to explaining delta.Using the option above the following prices were obtained and graphed on the next slide.
Stock Call Stock Call Stock Call42 0.173 50 2.962 58
9.21146 1.107 54 5.732 62
13.025
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Fin 288Time Value of Option
-4
1
6
11
16
21
34 39 44 49 54 59 64 69
Spot Price
$
Call Value Profit Intrinsic Value
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Fin 288Call Option Value
Call Option ValueX=50, r=.05, =.2, t=20 weeks
-4
1
6
11
16
21
34 39 44 49 54 59 64 69
Spot Price
$
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Fin 288Delta Graphically
-9
-4
1
6
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16
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34 44 54 64
Spot Price
$
Delta is equal to the slope of the line tangent to the graph of the Options price
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Fin 288Delta of an option
Intuitively a higher stock price should lead to a higher call price. The relationship between changes in the call price and the stock price is expressed by a single variable, delta. The delta is the change in the call price for a very small change it the price of the underlying asset.
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Fin 288Calculating Delta
Delta can be found from the call price equation as:
Using delta hedging for a short position in a European call option would require keeping a long position of N(d1) shares at any given time. (and vice versa).
)( 1dNS
c
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Fin 288Delta explanation
Delta will be between 0 and 1.
A 1 cent change in the price of the underlying asset leads to a change of delta cents in the price of the option.
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Fin 288Delta and the stock price
For deep out-of-the money call options the delta will be close to zero. A small change in the stock price has little impact on the value of the optionFor deep in the money options delta will be close to 1. A small change in the stock price will have an almost one to one change in the option price.
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Fin 288Delta vs. Share Price
Delta of Call Option
0
0.1
0.2
0.3
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0.5
0.6
0.7
0.8
0.9
1
35 40 45 50 55 60 65 70
Spot Price
De
lta
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Fin 288
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
35 40 45 50 55 60 65 70
Stock Price
Delt
a
3 months
1 year
3 years
Delta and Time to MaturityX=50 r=0.05 =0.2
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Fin 288
Delta X=35 r=0.05 =.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time
Delt
a
out-of-the money S=30 at-the-money S=35 in-the-money S=40
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Fin 288Example of Delta Hedging
Assume that we had sold the option in our example for 100,000 shares of stock.Using the information from before: S= $49, X=$50, r = 0.05, =0.20, t = 20 weeks =0.3846 years. Call option value = 2.3715Given 100,000 shares the value of the option is $237,150Assuming a share price of 49, the delta of the option is .5828594
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Fin 288The hedged position
The bank has a portfolio of delta shares for each share it has written an option on. This implies it owns 100,000(.5828594) = 58,286 shares.If the share price increases by $1 the value of the shares will increase by $58,286However the value of the option will decline.
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Fin 288Option value
The value of the option at a price of $50 is 2.926.Therefore the value of the option will decrease by (2.926 – 2.3715)100,000 = $55,450
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Fin 288Total position
Gain on Spot position = $58,286 Loss on Option = - $55,450 Net change in portfolio = $2,836
They do not perfectly offset due to the size of the price change and rounding errors. The total value of the portfolio would change from $3,093,161.06 to $3,095,997.00
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Fin 288Dynamic Hedging
Since the value of delta changes at each stock price the amount of shares would need to be adjusted to keep the portfolio value hedged. The larger the price change the less successful the hedge.
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Fin 288Delta of a portfolio
The delta of a portfolio of options is simply the weighted average of the individual deltas. Where the weight corresponds to the quantity of the option.It is therefore possible to adjust the delta of a portfolio quickly by adjusting one or more of the option positions.
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Fin 288Delta of a put option
A long position in a put option should be hedged with a long position in the stock, (delta will be negative).Delta for the put is given by N(d1) – 1
Similar to call options, for deep in the money puts (Asset price is less than exercise price) the value of delta will be close to -1. For delta out of the money puts the delta will be close to zero.
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Fin 288Delta Hedging
The delta neutral portfolio removes much but not all of the risk associated with the position.Looking at the value of the portfolio for a small range of prices changes provides a good indication of the ability of the hedge to remove the risk associated with a change in the stock price. The change hedge is not perfect because the value of the option is not a linear function with relation to changes in the stock price. Consider the previous portfolio.
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Fin 288
Value of Delta Neutral Portfolio(1 Short Call + Delta Shares) x 100,000
shares
2500000
2600000
2700000
2800000
2900000
3000000
3100000
3200000
3300000
3400000
3500000
40 45 50 55 60Stock Price
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Fin 288Gamma
Gamma measures the curvature of the theoretical call option price line.
Asset underlying of pricein Change
detla in the changegamma
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Fin 288Gamma of an Option
The change in delta for a small change in the stock price is called the options gamma:
Call gamma TS
e
TS
dN d
2
)( 2/1
21
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Fin 288Gamma Graphically
-9
-4
1
6
11
16
21
34 44 54 64
Spot Price
$
Delta is equal to the slope of the line tangent to the graph of the Options price
Gamma measures the amount of curvature In the call price relationship, The reason the portfolio Is not perfectly hedged is because delta provides only a linear estimate of the call price change. The hedge error is from the difference between the estimate from delta and the actual relationship
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Fin 288Gamma
If gamma is small it implies that delta changes slowly which implies the cost to adjust the portfolio will be small. If gamma is large it implies that delta changes quickly and the cost to keep a portfolio delta neutral will be large.
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Fin 288Gamma Con’t
Gamma is the adjustment for the fact that the call option price dos not have a linear relationship with the spot price. Delta provides a linear approximation of the change in the value of the call option that is less accurate the large the change in the stock price.The impact of gamma is easy to see in our earlier example.The impact of gamma will be the largest when the stock price is close to the exercise price
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Fin 288Gamma
The gamma of a non dividend paying stock option will always be positive (the larger the change in the stock price the larger the change in the value)
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Fin 288Gamma and Stock Price
The impact of gamma will be the largest when the stock price is close to the exercise price.For deep in the money or deep out of the money call options gamma will be relatively small.
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Fin 288
Gamma and Time to Maturity
Gamma will be highest for at the money options close to maturity.Gamma will be low for both in the money and out of the money options that are close to maturity.
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Fin 288
Gamma vs. Stock PriceX=50, r =0.05, t
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
30 35 40 45 50 55 60 65 70
Stock Price
Ga
mm
a
GAMMA
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Fin 288
Gamma vs Time to MaturityX=35, r=0.05, =.2
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time
Gam
ma
out-of-the money S=30 at-the-money S=35 in-the-money S=40
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Fin 288Other Measures
The sensitivity of the value of the option to a change in the expiration of the option is measured by theta
expiration toin time change
option of price in the changetheta
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Fin 288Theta
Theta is generally negative for an option since as the time to maturity decreases the value of the option becomes less valuable. (Keeping everything else constant, as time passes the value of the option decreases).
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Fin 288
ThetaX=35, r=0.05, t=.5, =.2
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0 10 20 30 40 50 60 70
Stock Price
Th
eta
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Fin 288
Theta vs Time X=35, =.2, r=0.05
-9.0
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0 0.5 1 1.5 2 2.5 3
Time
Th
eta
out-of-the money S=30 at-the-money S=35 in-the-money S=40
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Fin 288
Relationship between Delta, Theta and Gamma
From the derivation of the Black-Scholes Formula it can be shown that:
(We will show this soon)In a Delta Neutral portfolio delta =0 and the portfolio value remains relatively constant. This implies that if Theta is negative, Gamma needs to be of similar size and positive and vice versa. Therefore Theta is often considered as a proxy for Gamma.
)())((2
1)()( 2 aluePortfolioVrGammaSdeltarStheta
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Fin 288Vega (or Kappa)
The rate of change of the option value with respect to the volatility of the underlying asset is given by the Vega (also sometimes called kappa)The Black Scholes Model assumes that volatility is constant, so in theory this seems to be inconsistent with the model. However variations of the Black Scholes do allow for stochastic volatility and their estimates of Vega are very close to those form the Black Scholes model so it serves as an approximation.
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Fin 288Vega
Vega will be highest for options that are at the money. As the option moves into or out of the money the impact of a volatility change is decreased.
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Fin 288Rho
The final measure is the change in the value of the option with respect to the change in the interest rate. As we have discussed the interest rate has the smallest impact on the value of the option. Therefore this is not used often in trading.
This example is from MacDonald Derivatives Markets
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Fin 288
Market Making and Delta Hedging*
Market Maker – Individual who is ready to both sell and buy a given asset.
Bid price – Price market maker is willing to pay when buying the assetAsk price – Price market maker is willing to accept to sell the asset
A market maker can end up with an arbitrary position as a result of fullfilling orders – this represents a risk that needs to be hedged. In the options market this is done with delta hedging.
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Fin 288Market Maker
Assume that the market maker receives an order for a call option.The market maker can:
Leave the position unhedgedBuy shares of the stock (a covered call) so tht if the option is exercised the firm will be able to provide the stockUse delta to hedge the risk
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Fin 288An Example
Assume that a firm is writing the following option on 100 shares of stock.S= $40X=$40=.30r=.08t=91/365
This implies a call price of $2.7804 per shareAnd a Delta of .5824
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Fin 288No Price Change
If the price of the stock does not change the market maker realizes a profit of approximateloy 1.7 cents.
This is due to the time value of the option
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Fin 288
The unhedged position with a price increase
Assume that the stock price increases to $40.75At the new stock price the new value of the call is $3.2352This implies a loss of $2.7804-3.2352 =.4548
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Fin 288
Profit / Loss after holding one day
-25
-20
-15
-10
-5
0
5
0 10 20 30 40 50 60 70
Spot Price
Pro
fit
short call 1 day marked to market
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Fin 288Price increase revisited
Inhte case of the price increase to 40.75, the position decreased by $0.4548If the market maker had hedged using the delta of .5824, the value of the shares would have increased by
$0.75(.5824) = $.4368The value of the change in the option is understated by approximately $0.018 due to the price increase. (similarly a decrease of the stock price would result in an overstatement of the change in the option price)
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Fin 288Delta Hedging for two days
Assume that the market maker uses delta and buys 58.24 shares to offset the option written on 100 shares of stock. That represents a net investment of 58.24($40) - $278.04 = $2051.56Assume that the market maker borrowed the money and the interest charge for one day is then2051.56e0.08/365-1 =$.45
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Fin 288Day 1
Assume the stock price increases to $40.50
Gain on 58.24 shares $29.12
Gain on written call (new price =3.0621)
-$28.17
Interest charge $0.45
Overnight Profit $0.50
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Fin 288Rebalancing
The new delta is .6142This implies the need to buy .6142-.5824(100) =3.18 new shares of stock at $40.50This has a total cost of $128.79
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Fin 288Day 2
Assume the stock now falls to 39.25 there is a gain on the options and a loss on the shares
Gain on 61.24 shares -$76.78
Gain on written call (new price =2.3282)
$73.39
Interest charge-e0.08/365(2181.30)
$0.48
Overnight Profit -$3.87
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Fin 288Sources of cash flow
Borrowing - limited by the market value of the securities in the portfolio.Purchase or sale of sharesInterest
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Fin 288
Delta hedging for several days
Gain on loss on a daily basis depends upon 3 thingsGamma – If there is a large change in the stock price the market maker becomes unhedged (dealt does not represent the actual change well).Theta If there is no change in the price of the share there is a gain from time vlaueInterest Cost – there is a net carrying cost to purchasing the stock
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Fin 288A self financing portfolio
The size of the stock change has a large impact on the delta neutral outcome.Assume that the share price increases or decreases by exactly one standard deviation each day.
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Fin 288Profit / Loss
0 1 2 3 4
Stock 40 40.642 40.018 39.403 38.797
Call 278.04 315 275.57 239.29 236.76
Option Delta
.5824 .6232 .5827 .5408 .4980
Investment
2051.58 2217.66 2056.08 1891.60 1894.27
Interest -.45 -.49 -.45 -.41
Cap Gain .43 .51 .46 .42
Daily Profit -.02 .02 .01 .01
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Fin 288Intuition
If the stock price moves by one standard deviation each day in the binomial tree model, it would be approximately self financing! Next: Relating the market makers profit or loss to the relationship between gamma, theta, and delta.
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Fin 288Adding Gamma
From the example before we know that :S= $40 c=2.7804 S=$40.75 c=3.2352 =.6142Assuming we want to estimate the option price at $40.75 one way to do this would be to use the delta, but this creates error since the sensitivity of the option changes as the price increases. c40.75 = c40+.75(40.75) = 2.7804+.5824 = 3.362
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Fin 288
Correcting for change in delta
Another approach would be to average the two deltas. If the stock price change is small, the average of the two deltas should be an approximation of the actual price change.
(.5824+.6142)/2 = .5983
The new call price would be C40.75= 2.7804+.75(.5983) = 3.229 Which is
closer to the actual price of c=3.2352 than using delta alone
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Fin 288Second Approximation
However if we are going to calculate Delta at the new price we might as well calculate the new price directly. Another approximation would be to use gamma of the option at $40.An approximation of at 40.75 could be found by: 40.75=
Then the new share price could be calculated by substituting at 40.745 in the average delta equation above.
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Fin 288
Delta Gamma Approximation
average=
40.75=
average=
C40.75= c40+.75(average)
= c40+.75()
=c40+.75()
=c40+.75()
=c40+.75
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Fin 288
Delat Gamma Approximation
=c40+.75
Given a gamma =.0652
2.7804+.75(.5983)+(1/2)(.0652)(.752)=3.2355
Compared to the actual price of c=3.2352
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Fin 288Generalization
Replacing the values for the share price with St for the initial price and St+h for the price after a small change of St+h – St
Gamma can be approximated by the change in delta per dollar of stock price change or
St=(S+h-S
RearrangingS+h =St+S
If gamma is constant (assuming a small change in price makes this assumption realistic) this
approximation of delta will be exact.
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Fin 288
Computing the option price change
If gamma is constant, we can use the average delta to calculate a delta to use to approximate the new price
S+h =St+S
average=S+h+S)/2
average=(St+S+S)/2 = S+(1/2)St
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Fin 288
The approximate share price
CSt+h=CSt+average=CSt+S+(1/2)St)
=CSt+St+(1/2)(St
Regardless of the direction of the price change the gamma correction adds back to the delta correction by itself. This makes sense given the graph of the value of the option. There is still an error since gamma changes as the stock price changes (the assumption of no change in gamma is close for small changes in stock price)
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Fin 288Adding Theta
Theta attempts to account for the change in time as the option moves toward expiration.For a given period of h the change from the passing of time will be h. Theta is a yearly number, so if we have a 1 day change it implies a 365 change in the value of the option. Adding this to our earlier equation provides a new call price of
=CSt+St+(1/2)(St+h
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Fin 288The Market Makers Profit
THE value of the Market Makers position is equal to being long delta shares of stock and short one call or:
St-cSt
Assume that over time period h the stock price changes to St+h. The chang in the value of the portfolio will be given by
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Fin 288Change in Portfolio value
Assume that over time period h the stock price changes to St+h. The change in the value of the portfolio will be given by:
(St+h-St) - (CSt+h-Cs) - rh(St-CSt)Change in Stock Value
Change in Value of Option
Interest Expense
=CSt+e DSt+(1/2)(e2)GSt+qh
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Fin 288Market Makers Profit
CSt+h=CSt+St+(1/2)(St+h
Substitute(St+h-St)-(CSt+h-Cs)-rh(St-CSt)
After rearranging the market makers profit becomes
-[(1/2)(St+h -rh(St-CSt)]
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Fin 288
Impact of the change in the stock price
[(1/2)(St+h -rh(St-CSt)]
It is the magnitude, not the direction, of the change in the tock price (shown by ) in the profit equation that is important.Previously we showed that a one standard deviation change in the stock price produced a delta neutral portfolio (where there was no impact on the value of the portfolio)
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Fin 288
[(1/2)(St+h -rh(St-CSt)]
If is measured annually then a one-standard deviation move over a period of length h is equal to
Therefore a squared move of one standard deviation is
hS
hS 222
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Fin 288Market Makers Profit
substitute -[(1/2)(St+h -rh(St-CSt)]
Market Makers = -[(1/2S2St + -r(St-CSt)]h
Profit
hS 222
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Fin 288Relation to Black Scholes
Black and Scholes argued that the market maker should earn a risk free return if hedged with the option. This implies that the profit for each time period will be zero.
Market Makers = -[(1/2S2St + -r(St-CSt)]h = 0
Profit Dividing by h and rearranging produces:
(1/2S2St + - rSt=rCSt
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Fin 288Black Scholes Equation
The equation on the last slide is known as the Black Scholes Partial Differential Equation and is a fundamental component of valuing financial assets, both riky and risk free.The equation holds for both American and European options (both calls and puts) assuming that volatility and the interest rate are constant and that the stock moves one standard deviation over small intervals of time and that there are no dividends paid on the stock.
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Fin 288Limits on the Equation
While the equation holds in general, it fails if an American option is so deep in-the-money that it is optimal to early exercise the option. Think about an American Option that is deep in the money with an option price equal to X-S. Then =-1, =0, and =0
(1/2S2St + - rSt=rCSt
(1/2S2 + 0 - rSt=r(X-St)0 =rX
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Fin 288Intuition
If you are the owner of the option and have delta hedged it – you loose interest ont eh strike price you could receive if it is exercised.If you have written the option and it is ot exercised, you are earning a risk free arbitrage profit of rX.In other words when it is deep-in-the money it should not earn the risk free rate – and the equation should not hold.
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Fin 288
One standard deivation change
There is not reason to expect that the price will only change by one standard deviation.It shold change by one standard deviation or less approximately 68% of the time. If it is less than one standard deviation there is a light profit to the market maker.The other 32% of the time the market maker has a loss (the large the price change the larger the loss).However, the mean return on the position works out to being close to 0!
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Fin 288Re-hedging
There is a cost to keeping the hedge in force.Often in practice the market maker will Re-Hedge infrequently.The cost to this is that there are less observations or chances to rehedge resulting in a mean return of 0
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Fin 288Delta Hedging in practice
How can the risk of extreme price moves (resulting in a large loss on the market makers portfolio) be eliminated?There are many possible strategies that combine other options to hedge the risk of the large change in the stock price.
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Fin 288Strategy 1
Buying out of the money options. An out of the money put or call (or both) could be bought so that they profit when there is a large decline or increase in the stock price. This offsets the loss in the portfolio. The downside to this is that there is a cost – However, if the options are far out of the money they will be cheap.This will not work in aggregate since there still has to be a market maker willing to write the out of the money options as well.
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Fin 288
Strategy 2 Static Option Replication
By setting the bid and ask prices to help hedge.For example if you could buy a put with the same strike price of the written call (and also buy the underlying stock) you would be perfectly hedged. By adjusting the bid price so that anyone selling the call is willing to accept it, the market maker could hedge, but at a cost of buying both the put and stock.
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Fin 288Strategy 3 Gamma Neutral
You can buy or sell options with a gamma that offsets the gamma of the original position.Assume you have sold the following call option
X=$40, t=.25, r=.08, Buying the following call option with the same gamma will hedge your positionX=$45 t= .33, r=.08,
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Fin 288Ratio of gammas
To be gamma hedged you will need more than 1 of the X=$45 option for each of the X = $40The ratio of their gammas provides the proper hedge ratioGX=40,t=.25 /GX=40,t=.25
=0.0651063/.052438=1.2408The resulting portfolio is shown in the next slide
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Fin 288
X=40 X=45 Combined 1.2408 X=45 for each X=40
Price 2.7847 1.3584 -1.0993
Delta .5825 .3285 -.1749
Gamma .0651 .0524 0.00000
Vega .0781 .0831 0.0250
Theta -.0173 -.129 0.0013
Rho .0513 .0389 -0.0031
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Fin 288Overnight Profit
Delta and gamma Neutral
Delta Neutral
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Fin 288Why not Gamma Hedge?
The profits from writing the intital all will go to buyng the other call elimnating the market makers profitIn aggregate it is not possible for all market makers to gamma hedge. Most end users buy puts and calls resulting in them having a positive gamma – implying market makers are selling puts and calls – resulting in a negative gamma.
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Fin 288Miscellaneous Topics
Implied VolatilityVolatility Smiles and Skews
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Fin 288Implied Volatility
The one input in Black Scholes that cannot be observed is volatilityImplied volatility is calculated as the volatility that would provide the observed option price when used in the Black Scholes equationThe calculation needs to be done based upon an iterative process, since the volatility cannot be calculated directly.
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Fin 288Foreign Currency Options
For foreign currency options the implied volatility is lowest for options at the money. As an option moves significantly out of the money or in the money the implied volatility increases.This creates a “smile” when graphing implied volatility
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Fin 288Implied Volatility
Strike Price
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Fin 288Implied distribution
Given the volatility smile it is possible to calculate the risk neutral probability distribution for an asset as a future timeGenerally this distribution will be narrower than the assumed log normal distribution with the same mean and standard deviation and the implied distribution will have heavier tails (high probability of a larger rise or fall in the stock price.
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Fin 288Why not Log normal?
The log normal distribution assume that Volatility of the asset price is constantThe price changes smoothly with no jumps
Neither of these conditions hold for exchange rates.The impact of this will be greater for longer maturity options
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Fin 288Volatility Skew
For equity options the volatity decrease as the strike price increases.This implies that a lower volatility is used to price a deep out of the money call (or in the money put) option as opposed to a deep in the money option call (or out of the money put).The implied distribution is thus skewed to the left with a fatter tail to the left and a skinnier tail to the right. It is also narrower overall.
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Fin 288Why Equities have a Skew
The skew was not apparent until after the stock market crash in October 1987
Crashphobia?A decline in stock price decreases the market value of the firms equity, thus increasing its leverage. This might cause an increase in the volaitlity of the stock price.
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Fin 288Greeks and Smiles
The greeks need to be adjusted if there is a smile or skew.Sticky strike Rule – The greeks are correct if implied volatility was used in their calculation (assumes that implied volatility stays constant the next day)Better models are shown in chapter 24 (but not covered in the class).
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Fin 288Large Anticipated Jumps
The final possibility is a bimodal distribution where large anticipated jumps in price are expected.In this case he implied volatility can take on the shape of a frown.
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Fin 288
Stocks paying a known dividend yield
In theory, the payment of a dividend at rate q will lower the growth rate of the stock compared to if it paid no dividend. The current value of the stock will decline by the rate q. If the stock does not pay a dividend it will grow to Sqert by the end of period t. Or is would grow to S at time t starting at S0e-qt today.The distribution of the prices of these two stocks should be the same.
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Fin 288Black Scholes
Given that the probability distributions are the same we can replace S in the black scholes equation by S0e-rt This makes the call price equal to
Se-qtN(d1) -Xe-rtN(d2)
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Fin 288
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Fin 288Black Scholes Equation
The differential equation can also be adjusted for the dividend growth rate.
(1/2S2St + - (r-q)St=rCSt
Since the total return needs to be r in the risk neutral world this implies that the growth rate of the stock must be r-q.
(1/2)(s2S2)GSt + q - rDSt=rCSt
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Fin 288Currency Options
Similarly, foreign currency options need to account for the rate of interest paid in the foregin economy. IN this case S is replaces by S0e-rf(t) where rf is the foreign risk free rate of interest. Again the equation for d1 and d2 would be adjusted in a fashion similar to the dividined yield.
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Fin 288Futures Options
For the futures option the futures price is effective at a future point in time. Therefore the PV of the futures price will replace S in the Black Scholes equation and F0 replaces S in the d1 equation
call price = Fe-rtN(d1) -Xe-rtN(d2)
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Fin 288
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