Black scholes continued with Delta effect

(Last Updated On: July 21, 2015)

Black scholes continued with Delta effect

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Black Scholes formula is used widely to price financial assets basically derivatives. In our case for using options on futures. We focus on Delta fact where Delta measures change in premium with the change in futures price.

Diagrams can use to explain the Greek components without the actual formula. This is the Delta effect in the attached image. This is an important concept because once we hedging with options it is possible to adjust the number of options positions you hold to account for the effect Delta effect we have more than one. For example, if you hedge with the futures contract and you’re looking for a hedge ratio of one, then you try to match the futures position with the cash position. If you have 5000 bushels of corn, use one futures contract because it matches the optimal hedge but you may want to deviate from it. Try to get as close to one as possible. You pick one futures contract. If you choose an option edge instead of futures contract, option premium only move in tandem with with the underlying futures price if the Delta is one. If the futures price moves 10% the premium will move  10% if the delta is one. If the delta is 0.5 futures price may move 10% of the premium will only move 5%. When hedging with options you may get left behind if you have delta with a very low percent because the futures price can change the spot price changing, but the value of the option may not change. This is the premium.

If you take the reciprocal of the Delta, it is .5 then choose 2 option contracts for every futures contract. If you have 5000 bushels of corn we would have one futures contract since it is 5000 bushels and we would have 2 option contract. The Delta is also important for speculators because if you’re buying options, you’re predicting the price to rise or fall, if the prediction is correct but the premium does not change. You will be disappointed.   If you predict the price of gold will decrease this week and you buy a put option, at the end of the week the premium has hardly changed you will be disappointed. You are right about the price move but the premium did not change, the value of your option did not change. It would only happen if the Delta is very low. If you buy an option very low that was out of the money, you could see the market moving but the premium does not change you will not be able to profit from the move. This is why speculators are interested in the Delta. Hedgers  are interested in the Delta since the hedge position will be one over the Delta for every unit underlying spot.

Another way to show this is through a supply demand diagram in the attached images. This will show volume of options that are purchased by the buyers or  sold by the writers. Writers are offered  the supply curve while demand willingness to pay by the speculators. We have the volume of the futures contracts written plus we have the price of the premium on the other axis. The higher the premium holding everything else constant, the greater the number of contracts are willing to supply. The higher the premium the lower tha demand will be on behalf of speculators. Remember there is a diagram for a box on theoretical range For calls and puts premiums where you have minimum and maximum Price. Saw the maximum price for call was the futures price. This is the most irrational buyer will pay for call option is the futures price. As a result the demand curve will hit the supply curve at the futures price since no one will pay more than that. You will have an equilibrium point a point e. This equilibrium gives a premium of the market on the volume on a given day. We are now interested in the Delta which captures the change in the premium when the future Price changes.

Let’s say the future price increases, there is a maximum willingness to pay is the futures price. If the futures price rises the willingness to pay is higher so the intercept on the demand curve has to shift it up. The demand curve shifts to the right in a parallel fashion. At the same time the supply curve will change because the writers of the option are assessing the probability as to whether the option is in or out of the money. If it moves into the money, the writer starts to lose profit where the call option with a fixed strike price. As a future price increases. Given that it is a call option for the fixed strike price, as a future price increases while out of the money, the strike price will be above that the futures price for a call. As a future price moves closer to the money, the rational writers of the option well since the futures price increased, for any given premium they want to hire the normal rate. The probability of this option moving into the money is now increased. You will have a left rotation of the supply curve with the right shift in the demand curve. You’ll move from point E of the equilibrium premium P1 with E2. Higher future price will lead to a higher premium in the case will call option. The Delta will therefore be positive. The Delta will range in value from in value from 0 to 1 if positive

This diagram help so will show premium on a call option rises because there’s a shift in the shift in the demand curve and a rotation of the supply curve.

 

If the futures price falls, does the Delta become negative? This is not true for a call option since you just reversed course of the above example.   If it falls the demand curve will shift down or left, the supply curve will rotate to the right. The man curve will shift left since the futures price Will fall. The supply curve rotates right since it is out of the money, it will move further out of the money. The writers are willing to write more options or the Same out for a lower premium.   We will have a fall in the premium as the fall in the futures price. It will still be positive. With a call option for both positive going in the same direction.

 

Referred to the lambda effect diagram in The attached images

 

There’s also the volatility factor which is part of the Black scholes formula. This is associated with Lamda. You can estimate the volatility historically. Or you could use the formula to compute the implied volatility if you use the market premium into the formula. In the diagram you have the same situation with an out of the money call option, you have to supply demand curves which is exactly as in the previous example. Intercept on the demand curve hits the vertical axis at The futures price because that is the most out of buyer is willing to pay. Solid lines referred to the current situation with an equilibrium in the price P1. If There is an increase in volatility look in the example of crude oil. Volatility will rise if there is uncertainty in the marketplace. You start with the premium of P1 as volatility has increased.  If you have buyers of the call option which is out of the money, as volatility increases, volume of purchases of Q that the Demand curve will rotate to the right. This means that buyers are willing to pay more for a higher premium for the same volume. With a fixed-rate strike price no change in the future price, it will move a lot more. you have uncertainty in the marketplace, as a buyer you’re willing to pay for a higher premium because of the additional volatility as it moves into the money. The writer of the option will sell the call option for this level. If the premium is two dollars in and out of money option and if the volatility goes up, there is a greater probability that the option will move into the money they will increase the premium now. That is why the supply curve will rotate to the left while the demand curve rotates the right. There’s a new equilibrium of E2 from equilibrium eat one is a premium has increased.

As before when the volatility increases, the premium will increase as well. This is the economics of these increases.

 

The third important greek component is theta affect. Theta measures change in premium with the change in time to maturity. If you look at one strike price December February June options, you are looking at more distant months. The time of maturity increases. You observe a different theta. Look At the economics and attached diagram of a call option in the money. We have an initial equilibrium a point E as in previous examples. Supply meets demand. We Allow decrease The time of maturity. We do not change the volatility or the futures price. Supposed we start a week ago over the previous week, the future price has not changed orThe volatility has not changed. The only thing that has changed is we are closer to maturity or a decrease in time to maturity. As a result the premium will fall. We have to scrub options as a decaying asset. As the value erodes towards maturity. Reason is we have a rotation to the right in the supply curve  and they left rotation in the demand curve. For a given volume the demand curve rotate in a given Q. The option is for out of the money. If you are out of the money closer to maturity, you may have paid two dollars for the premium

But you’re not willing to pay the same premium so time is on your side but it is out of the money. An event that may push this into the money, you are willing to pay less for the option a week later. If you know it could move into the money, as the week goes by the probability of it being in the money his less. So for any given Q, you’re willing to pay a lower price on the premium.

For the writer of the option, you will have to reduce the premium. You’re willing to sell it at a lower premium because time is marching down for less time to maturity. The lower the probability of that moving into money. This is good for you as a writer as you don’t wanted to move into the money. Both the writer and the buyer willingness is lower to execute a trade at a lower price.

 

Referred to value of Delta attached image

For speculatorThe Delta will indicate the premium will change. There’s a trade off when the out of money option is really cheap since you’re not paying much for it. The Delta will be low. As a price may change the premium will not change. If you have a deep out of the money option then the Delta will be close to zero. These will have a low probability of moving into the money. Even if the price does change it will not be enough to put it into the money. As it moves into the money the Delta will range between zero to .5. When it is at the money, the Delta is a .5. When an option at the money given the efficiency,  is a 50-50 chance that It could move in the money. So if the futures price increases by $10 then the premium rise by five dollars. It is one half of the change in price. Once you moved into the money, Delta will exceed .5 and if you’re deep in the money it will be close to one. You can use the Delta as a probability that an option will stay in the money. If it stays out of the money in the Delta zero, it is a zero probability that it will stay in the money.

 

If you buy an option today and it has a certain delta of .5, does it remain constant? It will change where the gamma affect records the Delta. It will change daily but if it is in the money the change is relatively small. It gets really interesting when it is at the money where you’ll see the wide swings. You will see more variation of the money.

 

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