Market making and delta hedging Flashcards
what is sort of the market maker assumption regarding black scholes?
That market makers are profit maximizers who want to hedge their positions. Market makers can hedge by taking a stock position in a way that offset the delta.
Because delta change, market makers must continuously change their position to align with their risk exposure.
what do market makers buy?
They buy based on demand. They stand ready to buy and sell based on demand.
They will buy low price, and sell high price. They create the bid-ask spread.
what is the job of market makers
Supply immediacy, making it possible for traders to trade whenever they want to
how can market makers control their risk
delta hedging
is delta hedging referring to buying shares or options?
both satisfy the requirement. The key is that the position has a net delta of 0. At that point, the position is not sensitive to movement in the market.
is a delta hedged position a “zero value position”?
no, not in general. Thisis because the price of the option position and the shares position are generally not equal. Because of this, the market maker must have capital to hold the position.
what can we say about the return on a hedged portfolio?
since the portfolio is not zero value, we have tied up capital in it. Therefore, we expect a return. However, since we have no risk invovled, as the position is delta hedged, we expect the risk free rate.
As a result, we expect the risk free return from the delta hedged portfolio.
what happens to the greeks if we sell option instead of buying?
Signs flip
if we do marking-to-market, what do we do?
We consider the question “what happens if we liquidate now”. What is the gain/loss?
if nothing on an option change overnight, what happens in regards to a market maker?
If the market maker wrote the option, he will make money on this. This is because of theta. Why? Because now he can buy back the option cheaper than he sold it for. Thus, he has the ability to exit the position at a gain.
elaborate on using delta in regards to a price movement in terms of accuracy
It is an approximation. Delta is point accurate. Not interval accurate.
If we use the delta when a stock price is at $40, and we consider some price increase, then we know that delta will become larger and larger, evenutally approaching 1. As a result, using the delta from the initial stock price will UNDERSTATE the new option price/option price change.
Similar for a price decrease, but now it will OVERSTATE it.
why does delta change, and as a result, either overstate or understate
because of the effect that GAMMA has
What delta do we ideally use to compute option price change as a result of stock price change?
average delta
elaborate on the process of delta hedigng a written call
we write a call, get the premium, and purchase shares so that delta is 0. Since delta is additive or whatever, we can simply buy ∆*100 amount of shares.
The differnece in cost between the shares and the premium is the amount we need to borrow (if cost is positive). otherwise, we earn interest on it. if the cost of the shares is larger than the premium we received on the call, we need to borrow, which gives us overnight cost equal to the interest.
The next day we can check what we have earned by marking-to-market. We consider the new call price and the new share price. Together with the interest, we use this to compute the profit/loss.
Then we need to rebalance. If the share price increased, we need to buy more shares. This means that we need to borrow more funds.
If share price decreased, we sell some shares. We buy at market price, so this doesnt affect marking-to-market.
elaborate on interpreting the profit calculation of the hedged position
We can conider the results as the extend to which teh position require cash infusions in order to maintain a delta neutral hedge.
we first need to fund the position. typically, the shares cost more than the premium we receive. This difference is borrowed, and payed interest on.
the next day, the share price increase. As a result, the option price increase.
we need to buy more shares to remain delta neutral. but, because we also own shares, we have made money on these shares.
gain on shares + gain on option + interest = profit
if profit is positive, say 0.5, we are better of.
Then we rebalance, and this is done by buying more shares. After delta neutral buying, the portfolio has increased on value by the amount we spent. but our portfolio has also increased as a result of the new value of the shares (and the effect of the new value of call). As a result, we will see that the amount we need to borrow is smaller than the amount our portfolio has increased by. This represent the profit. or loss, if that is the case.
elaborate deeply on the market maker perspective on writing a call option and being delta hedged.
if we write an option, say a call, we need to hedge by being delta neutral. This is done by buying shares equal to the delta etc. The value of this position is given as its cost. The cost is the share price times shares we bought less the premium received (value of option).
This represent the borrowed amount as well, which we need to pay interest on, alternatively consider it as opportunity cost. In either case, we treat it as interest we need to pay.
The next day, we will likely see changes in the stock price. This will affect our portfolio.
First of all, due to the effect of delta, the value of the option has increased. Therefore, we have seen a certain incresae in our portfolio value due to increased share price (since we own shares) but we also see a decrease as a result of the higher valued call option. If delta never changed, these would perfectly offset each other. however, because of the gamma effect, delta will change continusously, which creates a net result that is positive or negative (most likely). We also have theta, which drags the option price a little but down. We ignore the other effects, like volatility and interest etc, by considering them fixed.
Because the gamma force change delta, we will no longer be delta neutral. therefore, we need to look at the change in delta, and buy the shares we need. Since delta has the property that the delta of a portfolio is equal to the sum of individual deltas, we can simply buy more shares and add them to our already existing portfolio to match the new delta.
but the question is what this will cost us. we need to borrow this amount. Specifically, our portfolio has increased value in some parts, and decreased in others. When we then consider the value of our stock position less the cost of the new value of the option/call position, and then pay the interest we owed overnight, we will likely see that the amount by which our portfolio value has increased is not the same as the amount of cash we need to borrow in order to become delta neutral. This difference represent profit.
if the process was differnet, we could have lost some money.
so if we only delta rebalance our position every 10 point price increase or decresae, we will likely lose significant amount of profit. if the stock goes up, we dont have as much as we should of it, and if it goes down, we own too much of it. So the only way for us to make bank in such a scenario is for theta, vega, rho or psi to work in our favor. if we consider rho and psi as fixed, we have theta and vega. What we wish, is for vega to be large negative and theta to be large negative. Meaning, we want the volatility of the stock to flatten as much as possible, and the time decay to be significant. As a result, it is possible for the delta hedger to actually make significant profit even with shitty rebalancing. It just sort of contradicts the purpose of hedging, at least to some degree. We are still somewhat hedged, just not perfectly.
what happens if we are able t ocontiuously rebalance our portfolio?
We eliminate delta exposure, and as a result, also gamma. However, we still have the other greeks to consider.
If we write a call, theta works in our favor. We also want volatility to drop etc.
many things still affect our position, being delta hedged is just a first order hedge.
key to understand regarding delta hedging
we are considering the effect of changes in stock price vs interest we owe and the effect that the various greeks have on the option.
what behavior in regards to the stock price will fuck us the most if we write a call?
Big movements. Our position takes a hit if the stock price moves with great magnitude because of the gamma effect. Gamma change the delta, and if it change delta a lot, we become severely non-neutral in our position.
The effect is that t
what is gamma squeeze
I suppose it refer to what happens when you push a stock through its entire delta band. Deep OTM to Deep ITM. if this happens quickly, it is a struggle to become delta hedged. you lose a lot, quickly. It generates a pressure to buy more shares, which further makes the options ITM.
is delta an accurate predictor of the change in option price?
No, because delta change with the stock price
Use average delta a perfect measure?
No, if we want perfect we need to integrate (sum) the entire price change.
What can we do to make a better-than-delta-alone heuristic for option price changes?
We want to use average delta, but doing this is difficult since we dont necessarily have the future delta.
therefore, we can turn to using gamma to estimate the average delta.
future delta = price move x gamma + current delta
gamma is the rate of change in delta that comes from a single point move in the price. Therefore, we can find the future delta by using gamma.
This allows us to find a measure of the average delta, which we can then use as a proxy for the change in option price that follow from a stock price movement.
when using gamma to predict average delta, and use average delta to predict option price change, what do we call it?
delta-gamma-approximation
