Topic 3: Risk Management of Option Positions

Question 1

Delta Hedging (5)

Answer 1

1. Compute option delta
2. Take offsetting position in shares
3. Position is hedged, but IS NOT A ZERO VALUE POSITION. The cost of the shares required to hedge is not the same as the cost of the options
4. Therefore, market maker must invest capital to maintain a delta hedged position
5. Key to derivatives: a hedged position must earn the risk free rate.

Question 2

Calculate interest earned on delta hedged position

Answer 2

Interest per share per day = (exp(rf/365) -1) x (value invested or borrowed)

Question 3

Issues with delta

Answer 3

1. Delta changes as stock price changes
2. Delta will understate when prices are rising and overstate when prices are falling
3. GAMMA measures the change in delta when the stock price changes. Use together with delta to better approximate price changes

Question 4

Calculating a delta hedge

Answer 4

1. Calculate the delta, multiply by number of shares and the So.
2. Calculate the option price, multiply by number of shares
3. Calculate the interest at the risk free rate of $ borrowed or lent
4. Add the strategy together for overnight profit
5. Ensure T rolls down by appropriate days when calculating
6. At Day 1; delta may have moved, may need to buy additional shares to maintain hedge

Question 5

Greeks
S: name the 3 greeks acting directly on it
Volatility: name the greek

Answer 5

S: delta, gamma and elasticity
Vol: Vega

Question 6

delta

• define
• sign for calls
• sign for puts

Answer 6

1. delta = change in option value / change in underlying asset
2. For call options delta is positive
3. For put options delta is negative
   = slope of payoff diagram

Question 7

Gamma (4)

Answer 7

1. Gamma = change in delta for change in stock price, = curvature of the option’s payoff diagram
2. Always positive for purchased call or put.
3. Gamma and vega have the same sign.
4. Where gamma is always positive = convex

Question 8

Theta

Answer 8

1. Theta = change in option price for a drop in time to maturity by one day
2. Options are generally less valuable as the time to expiration declines
3. At the Money: time decay is most rapid, otherwise more steady
4. On chart - shows as a collapse of the curved line onto the kinked line. A vertical shift of the entire option diagram
5. Puts (European) on non dividend paying stock: theta can be positive or negative

Question 9

vega (no greek symbol)

Answer 9

1. Vega = change in option price when there is an increase in volatility of one percentage point
2. increase in volatility leads to an increase in the price of a put or call
3. Vega measures sensitivity of option price to volatility
4. For both puts and calls vega is positive

Question 10

Rho

Answer 10

1. Rho = change in value of option when interest rate r changes
2. rho is positive for European call options and negative for European put options
3. Rho has the opposite sign to psi
4. put entitles the owner to receive cash; the value of this is lower when r is higher
5. As time to expiration increases, rho increases
6. As call option becomes more ITM, rho increases

Question 11

Psi

Answer 11

1. psi = change in value when the div (convenience) yield changes
2. psi is negative for European call and positive for European puts, opposite to rho
3. Call entitles the holder to receive stock but without receiving dividends paid. When PV of stock is higher, div yield is lower
4. WIth a put - may deliver a share that may have a lower PV due to the div yld
5. Absolute value of psi increases with (T-t). HIgher div yield has little effect on a short maturity.

Question 12

Net cashflow =

Answer 12

Net cashflow = change in borrowing capacity
minus cash used to purchase additional shares
minus interest

Question 13

How do gamma, theta and interest impact the market maker

Answer 13

Theta is negative - ie time decay benefits the market maker

Interest and gamma work against the market maker

Question 14

Delta (3)

• define
• sign for call, sign for put
• delta is what in the payoff diagram
• ITM options

Answer 14

1. Delta is how much an option’s value (price, premium) changes when the underlying changes a little upward
2. For bought call options, delta is positive (sold = negative). For bought put options, delta is negative (sold = positive).
3. Delta is the slope of the option’s payoff diagram
4. ITM options are more sensitive to stock price than Otm OPTIONS. Deep ITM - are likely to be exercised and therefore delta approaches 1; behaves like leveraged position
   Small changes only.

Question 15

Gamma (4)

1. Define
2. Sign for call, sign for put
3. gamma is what in the payoff diagram
4. consider vega

Answer 15

1. Define: Gamma: change in delta for a change in stock price (underlying)
2. Sign for call, sign for put: BOTH are positive for a PURCHASED put and PURCHASED call
3. gamma is what in the payoff diagram: curvature
4. consider vega: sign is the same, ie gamma and vega are positive for purchased call and purchased put

Option’s value pre expiry is a curve, not straight line, therefore the slope is changing as S changes
Focus on what happens if asset price moves by a large amount.

Question 16

Theta (5)

1. Define
2. Time decay At the Money vs otherwise:
3. Theta is what in the payoff diagram
4. As time to expiry decreases, :
5. Theta on European div paying stock: sign
6. Bought European calls:

Answer 16

THETA

1. Define: change in option price for a drop in time to maturity by 1 day
2. At the money: time decay is more rapid, otherwise more steady
3. Theta shows a collapse in the entire curve onto the kinked line. A vertical shift downwards for the entire diagram
4. As time to expiry decreases, options are generally less valuable
5. European puts on non dividend paying stock: theta can be positive or negative. If deep ITM, cannot exercise European option, therefore the option effectively becomes a T Bill.
6. Bought European calls: theta is always negative

Question 17

Vega
1. Define
2. Sign for call, sign for put
3. increase in volatility:
Note: there is no Greek symbol for vega

Answer 17

Vega

1. Define: Change in price when there is an increase in volatility of one percentage [point. Measures sensitivity of an option to price volatility
2. Sign for call, sign for put: positive
3. increase in volatility leads to an increase in the price of a call or put.

Question 18

Rho

1. Define
2. Sign for call, sign for put
3. As time to expiry increases:
4. consider psi
5. As call option becomes more in the money

Answer 18

Rho

1. Define: change in the value of an option when interest rate r changes
2. Sign is positive for European calls and Negative for European puts. A put entitles the owner to receive cash; the value of this is lower when r is higher
3. As time to expiry increases: rho increases
4. has the opposite sign to psi
5. As call option becomes more in the money, rho increases

Question 19

Impact of larger market movements:

Answer 19

Focus on Greek sensitivity is micro.
If the market gaps or volatility spikes, the sensitivity can be misleading. (Delta-gamma approximation to the value change will be inaccurate.
Need to do full revaluation of the entire portfolio.

Question 20

How large is large (asset movement)

Answer 20

Think of market price movements as multiples or fractions of the underlying’s price/value volatility, sigma. Eg, 3 sigma movement.

Question 21

Psi

1. Define
2. Sign for call, sign for put
3. As time to expiry increases:

Answer 21

Psi

1. Define: Psi measures change in price when there is a change in dividend rate of 1%
2. Sign for call, sign for put: negative for stock call, negative for put. (call entitles holder to receive stock but not the divs. therefore when PV of the stock is higher, div yld is lower.) Put: deliver a share that may have a lower PV due to div yld.)
3. As time to expiry increases: psi increases

Question 22

Elasticity

1. Define
2. Leverage, K
3. Sign
4. Replicate call:

Answer 22

1. Define: Option elasticity = risk of the option relative to the stock in percentage terms
   ie, if stock changes by 1%, how much does the option change by (in % terms)
2. Leverage: pay a small amount for the option, Leveraged position is always riskier than the actual position, therefore it increases as K increases and decreases when K decreases
3. Sign: positive for bought calls and negative for bought puts
4. Replicate call: leveraged position in the stock

Question 23

Greeks - are they constant?

Answer 23

Greeks are not constant;
Vary with other parameters and market movements
In particular most are sensitive to moneyness (where S is relative to K) and to time to expiry

Question 24

For a CALL describe errors for delta, delta-gamma and delta-gamma-theta

Answer 24

CALL:

1. Delta: At higher S; delta UNDERESTIMATES. At lower S; delta UNDERESTIMATES (due to convexity)
2. Delta-Gamma: At higher S delta-gamma OVERESTIMATES. At lower S, delta-gamma UNDERESTIMATES
3. Delta-Gamma-Theta: At higher S delta-gamma-theta OVERESTIMATES. At lower S, delta-gamma-theta UNDERESTIMATES. Both are a small improvement on D-G though

Question 25

For a PUT describe errors for delta, delta-gamma and delta-gamma-theta

Answer 25

CALL: 1. Delta: At higher S; delta UNDERESTIMATES. At lower S; delta UNDERESTIMATES (due to convexity) 2. Delta-Gamma: At higher S delta-gamma OVERESTIMATES. At lower S, delta-gamma UNDERESTIMATES 3. Delta-Gamma-Theta: At higher S delta-gamma-theta OVERESTIMATES. At lower S, delta-gamma-theta UNDERESTIMATES. Both are a small improvement on D-G though

Question 26

An option is at the money if::

Answer 26

Option is ATM if underlying price = strike price. | ATM is Ao = Ke^-rT

Question 27

Approximate delta of option:

Answer 27

- Delta ranges from -1 to 0; and 0 to +1. - delta is linked to the risk neutral probability of exercise. - ATM option, statistically speaking probability of exercise is around 50%. Therefore for a bought put delta would be around -50%

Question 28

what happens to delta as option moves OTM

Answer 28

Delta will fall in absolute terms. In the case of a put it will approach -1

Question 29

what happens to delta if volatility is increased

Answer 29

If vol is increased there is a greater likelihood of exercise. Therefore delta will increase in absolute terms. ie put delta will approach -1

Question 30

What is the sign of gamma for long and for short positions

Answer 30

Gamma is positive for long positions | Gamma is negative for short positions

Question 31

HOw does gamma change over time for a long put option?

Answer 31

- Gamma increases as maturity approaches - SQRT (T) is in the denominator - Delta becomes more sensitive to changes in price of underlying - market maker will have to adjust a delta hedge more frequently - long option will make more money if market gaps and position is delta hedged by market maker

Question 32

Is vega positive or negative

Answer 32

All options have positive vega, as volatility increases so does the price of the option

Question 33

How does vega change over time assuming no change in underlying

Answer 33

Vega reduces over time vega is a function of SQRT (T) options become less sensitive to change in volatility as T shrinks

Question 34

Explain differences between vega and gamma

Answer 34

Vega: change in option value for a change in volatility Gamma: Impact on delta of a change in price. Gamma relates to actual market movements whereas vega relates to implied or expected vol.

Question 35

At expiry, what will delta of long put be if ITM

Answer 35

If ITM, a long put will have delta = -1 | At expiry, delta will either be zero or -1. Probability of exercise is certain at expiry.

Question 36

gamma and term to maturity - decrease term to maturity, gamma = ? - relate to the equation - frequency of hedge for market maker - if market gaps?

Answer 36

- gamma increases as term to maturity decreases - sqrt (t) is in denominator - delta becomes more sensitive, market maker will need to adjust hedge more frequently - option will make money if market gaps

Question 37

Hedge bought calls against negative market movement: | Should you gamma hedge

Answer 37

- hedge by buying puts. This will result in a straddle if executed at the same strike price - straddle - want to profit from an increase in volatility, in which case don't hedge gamma as GAMMA tends to eliminate VEGA

Question 38

Vega: - implied volatilities and time - time and value of option

Answer 38

Vega: - longer term options are more sensitive to a change in vol AND - volatility is more variable for shorter dated options AS SUCH: volatility risk is potentially more important (if not more important) for ST options.

Question 39

what greek option works in your favour in the case of gapping prices

Answer 39

Gamma. Convexity. Note difference if a position was merely delta hedged

Question 40

1. direction of psi for call option if underlying S increases 2. direction of psi as maturity increases

Answer 40

Psi moves down as underlying S increases for psi | Psi moves down with maturity

Question 41

Direction of rho for call option if underlying increases | Direction of rho for increase in maturity

Answer 41

Rho increases for a call option if underlying increases | Rho increases with maturity

Question 42

``` Theta For puts, when is theta highest When puts are ATM... Time value for a European put ITM calls vs OTM calls ITM puts vs OTM puts ```

Answer 42

Theta 1. For puts theta is highest when they are deeply ITM 2. SOme but not all puts have high theta when they're ATM 3. Time value for a European put can be negative 4. ITM calls tend to have more time value than OTM calls 5. ITM puts tend to have more time value than OTM puts

Question 43

TIme value for non dividend paying European call is always

Answer 43

positive

Question 44

Vega on long dated calls is (higher?/lower?) than vega on short dated calls

Answer 44

Vega is higher on long dated calls vs short dated calls

Question 45

Gamma of short dated calls tends to be higher when... Vega of calls tends to be higher when... Theta on calls tends to be largest (in absolute terms) when....

Answer 45

Gamma on short dated calls tends to be higher when they're ATM or around the money. Likewise for vega Theta tends to be largest on calls when they're ATM. ALso, short dated calls have more difficult to manage theta than longer calls.

Question 46

Longer dated calls that are OTM: - gamma - delta

Answer 46

Longer dated OTM calls: - gamma is higher - delta is lower

Question 47

Compare dominance of vega vs gamma for short / long dated options

Answer 47

FOr short dated options gamma is dominant over vega | FOr long dated options vega is dominant over gamma

Question 48

Volatility smile and smirk 1. Chart 2. B-S 3. Explain

Answer 48

Volatility smile or smirk 1. Chart implied vol on vertical axis vs K on horizontal axis 2. Black Scholes assumes constant vol 3. Smirk has to do with the distribution of the underlying. Equity prices go up stairs and down ladders. Have a negative skew (greater probability of negative return) Low strike put suddnetnly becomes ITM and stock price plunges. Put call parity means similar vol is assumed for puts and calls; therefore the market maker charges more for low strike options. Vol varies according to the term of the option. Market participants believe vol should be higher for lower strikes (ie ITM puts and OTM calls)