Chapter 16 - Market making and delta hedging

Question 1

what do market makers do

Answer 1

they stand ready to sell to buyers, and buy from sellers

Question 2

recall bid and ask

Answer 2

bid is what the market maker pays for buying the shit.
the ask is what hte market maker reuqire to sell the shit.

Naturally, the ask is larger than the bit, and this markup must cover the cost of doing business.

Question 3

important aspect of market making

Answer 3

there is no speculation. There is no personal preference. It is strictly expected demand + markup

Question 4

must market makers hold assets to sell?

Answer 4

No they can short

Question 5

most important aspect of market making, in order ot make it viable

Answer 5

Controlling the risk. Without hedign the positions, they are fucked. Uncotnrollable risk

Question 6

One way for MM to control risk

Answer 6

Delta hedging

Question 7

briefly discuss delta hedign

Answer 7

compute option delta and take an offsetting position in the shares.

it also reuqire investment of capital to balance the equation.

Question 8

what is the “key” idea in derivatives markets in regards to delta hedging?

Answer 8

we have tied up capital, and should earn return on it.

If properly hedged, we earn risk free rate.

Question 9

what do we mean by marking-to-market?

Answer 9

if we liquidate now, what do we get in profits.

marking-to-market is a daily re-set thing I believe

Question 10

say a market maker sell a call option, and the stokc price doesnt move. What is hte marking-to-market outcome the next day?

Answer 10

this is a profit for the market maker because of time decay. there is less time value left, so he can buy back the option for less than the original price.

Question 11

nice thing about delta

Answer 11

delta represent the sensitivity of hte option, which means that it measures the exposure of the position. Meaning, the delta is crucial for understanding the exposure of the market maker.

Question 12

if we consider the delta of an option. What happens if the stock price incresae+

Answer 12

because of the delta, the option value increase. however, as the option move more ITM the delta increase as well. Delta ocnverge towards 1.

THe crucial thing here is that if we use delta with large “gap” (not continuous), the delta will understate the risk exposure.

At the same time, if the stock drops instead, and the delta decrease, the delta will overstate the exposure of negative movements.

Question 13

change in delta is measured by

Answer 13

Gamma

Question 14

What is the ultimate change in option price in regards to delta over a stock price movement?

Answer 14

The average delta.

Question 15

what can we say that mark-to-market profits measure?

Answer 15

Net cash infusions required to maintain the hedged position

Question 16

in a regular “sell call, buy shares” delta hedging, what are the sources of cash infusions?

Answer 16

borrowing.

purchase/sale of shares of the asset.

Interest.

Question 17

how can we calculate net cash flow fro mthe hedged position?

Answer 17

Question 18

elaborate on the “borrowing” as used in this context. what is meant by “borrowing capacity”, and why is it equal to the portfolio value

Answer 18

the idea is that if “i” want to establish a hedge, and the position value is worth X, and I tell the bank this, then the bank can agree to lend this amount to me, since the value of the position serve as collateral if they want me to repay. Of course, theoretical example, not necessarily practical

Question 19

what can we say about hte marking-to-market overnight profits of a delta hedged portfolio+

Answer 19

the profit we can pocket, or the cash we need to pay, in order to keep the position delta hedged.

Question 20

what do we call a posiitoon (delta hedged posiiton) that doesnt require that we take cash out or supply cash?

Answer 20

self financingn

Question 21

elaborate on what determine the pattern of gains and losses of a delta hedged daily marking-to-market portfolio

Answer 21

Broadly speaking, there are 3 effects that controbute:
1) Gamma
2) Theta
3) Cost of carrying the position

GAMMA
for the larger moves, the market maker is negative while on the smaller moves, the market maker makes profit.
This can be explained like this: If the stock price changes ,the position becomes unhedged. If we short the option, a stock price increase makes us lose money because while we earn the upside from the stock price movement, since the gamma force the delta to increase, the larger the up-move in the stock price is, the further out-of-hedged our position becomes. Meaning, we would actually require more shares to balance it. In other words, the call lose money faster than the stock earns money.
The same sort of thing happens if the stock price decrease. If the stock price decrease, the delta decrease, and the call makes money more slowly than the stocks lose money. We have too many shares, and we therefore carry too much of the downward asset return momentum.

THETA
Each day that pass by, the option lose value as as result of less time value. If we originally sold the option, this is beneficial for us.

Cost of carrying position
This is also called interest cost. In order to hedge, the market maker must purchase shares.

Question 22

consider market makers who write options. What do they not want?

Answer 22

big moves

Question 23

when is the position self-financing?

Answer 23

when it moves every day by exactly one standard deviation.

As a reuslt, the binomial model produce approximately self-financing results, meaning htat if the stock move according to the binomial model, the delta hedged portfolio would be ish self-financing.

Question 24

what is financial forensics

Answer 24

in this context, looking at a price movement, seeing the effects, and tring to explain why the effects occured

Question 25

why is delta alone a bad predictor of optio nvalue changes

Answer 25

because delta change with stoc kprice movement. We need the gamma, and we need the theta as well.

Question 26

when is the inaccuracy of using only delta as a predictor largest?

Answer 26

When the delta is most sensitive. This is when the gamma is at its largest.

Question 27

there are two ways to compute an estimate for option price that is based around average delta. elaborate on them

Answer 27

One use only the average delta, but this require that we have the delta at the intiial and terminal stock price. if we have these, we compute average delta, and take the option price at the first time, and add the average delta multiplied by the unit change in stock price we are interested in, and the result is an approximation of the option price that result from the stock price change. the other variant is to use gamma to approximate the average delta. Since gmama is the rate of change in delta, we can multiply gamma by delta, and multiply this by the unit change in stock price, and the outocme is the expected future delta. Then we can use this future delta to compute the average delta (like before) and then compute hte estimate for the next option price like before, using the average delta. So the difference is that one method assume we have both values that we need to compute hte average delta, while the other doesnt, and instead approximate it based on the gamma. the second method is called **delta-gamma-approximation**.

Question 28

what is the algebraic relationship between delta change and gamma?

Answer 28

epsilon is the stock price change. h is the time interval. new delta is equal to old delta + the gamma multuplied by the stock price change.

Question 29

how can we use this to help us compute option price changes?

Answer 29

if gamma is constant, we can compute average delta. We add ∆(S_{t+h}) to both sides, and then divide both sides on 2. We know have RHS expression for average delta. Then we use the following relationship we have earlier: new_option_price = old_option_price + epsilon [average delta] but we insert the expression for average delta. The result is at the picture.

Question 30

using the formula we derived in the last question, what do we need to know regarding the stock price change direction in regards to gamma?

Answer 30

nothing. epsilon is squared, so the effect that the stock price has on the gamma term is only dependent on the magnitude. This makes perfect sense because if the stock price move up, we know that delta will increase. Using constant delta wil ltherefore understate the future option price. Therefore, we need to add correction term. If the stock price mvoe down, the delta decrease. As a result, using the initial delta alone will predict too negatively. therefore we need a positive correction term. it is worth noting that the epislon matters in regards to the initial delta. Meaning that if the stock price movement is negative, the delta affects negatively on the option price.

Question 31

is delta-gamma-approximation perfect?

Answer 31

no, because gamma is not constant. we could make it better by including more and more terms that represent the sensitivity of the gamma, and the sensitivtiy of the sensitivity of the gamma and so on.. but this is not worht it. besides, the formula we dervied for delta-gamma approximatoin is basically a taylor series approximation for the change in the call price.

Question 32

what is theta

Answer 32

time decay, the loss in value holding stock price fixed

Question 33

what happens with our expression for new call option price if we add theta

Answer 33

Question 34

how do we find the algebraic representation of hte profit of the market maker?

Answer 34

This is simply the net value the market maker receive. As the formula show, he lose money from gamma effect on larger price moves. Pay attention to the hidden signs. Theta on a call is negative, and since market maker sell the call, it becomes positive (double negative). as a result, the market maker benefits from high theta. finally, interest is a net cost. The key is to remember that theta is negative.

Question 35

relate sigma to the market maker's profit equation

Answer 35

The idea is that the market maker makes zero profit if the movment of the stock price is 1 standard deviation. So, if we instead of having epsilon directly in the equation, we take substitute it by standard deviation. If annual volatility is sigma, then the interval h appropriate volatility is sigma x sqrt(h). Then we multiply it by the initial stock price to arrive at what one stnadard deviation price move look like. we set this to be epsilon. epsilon = sigma sqrt(h) S_t Then, since epislon is purely in the function squared, we square this shit. e^2 = sigma^2 h S_t^2 We enter this into the equation instead of epsilon squared.

Question 36

we get this result by taking the market maker's profit and dividing on h, and re-arranging the terms. Relate this to BS

Answer 36

The greeks represent partial derivatives. delta is the partial derivative of the stock price. Gamma is the double partial derivative. theta is also partial derivative, but with regards to time and not the stock price.