Options with pricing ( 9-10) Flashcards
What is an option?
An option gives its owner the right, but not the obligation, to buy or sell a given
quantity of a specified asset at some particular future date for a pre-determined price, known as the strike price or exercise price
An individual who has purchased an option contract is said ….
an individual who has sold an option contract is said to ….
An individual who has purchased an option contract is said to be long the option
and an individual who has sold an option contract is said to have written the
option.
What does K and St mean ?
K = strike price St = underlying value of asset.
What is the call payoff at maturity and explain and example with strike price 100 and values of underlying asset 80 and 120?
E.g. if stike price is 100 and the underlying value of asset = 80, then would make a lose and not exercise option.
If strike price was 120 and underlying value of asset = 100 then you would make a profit and exercise option
What is the put value at Maturity ( assuming we are longing it) ?
If the strike price is 100 and underlying value of asset is 80 then you will sell because you make a profit( you are selling something for 100 worth 80) .
If the strike price is 100 and the underlying value of asset is 120 you will not long the put you will keep and do nothing.
How does the payoff diagram look like when longing 1 call option based on one unit? What does this diagram not factor? Why is gradient 1.
This diagram doesn’t factor that when you buy the call option at t=0 you have a cash outflow.
If i assume 1 call option is based on one unit of the underlying, the y axis = the difference between value of underlying and strike price and the same of x axis.
What is the payoff diagram for a person who shorts ( sells) 1 call option ( the person who has written the call option) ? What does this diagram ignore?
The person who sells gets a cash inflow at time c, if we forget about time value of money, we would shift diagram up by c.
What is the payoff diagram for longing 1 put option?
What is the payoff diagram of shorting a put option?
What do Profit diagrams show?
Profit diagrams (also known as net payoff diagrams) include the initial cost of establishing the option position i.e. the option premium.
What is the profit diagram for longing and shorting a call option?
What is the profit diagrams for longing and shorting put options?
What is the straddle option strategy?
And when is it typically used?
The straddle involves going long in a put option with strike price K and going long in a call option on the same underlying with the same exercise date and strike price K.
An investor may use the straddle strategy if they believe that a stock’s price will move significantly but is unsure as to which direction it will move.
What is the reverse straddle option strategy?
And when is it typically used?
The reverse straddle involves going short in a put option with strike price K and going short in a call option on the same underlying with the same exercise date
and strike price K.
An investor may use the reverse straddle strategy if they believe that the stock will have little volatility and the stock price at time T will be close to the exercise price.
What is the bull spread using calls and why is it used?
The strategy for the bull spread using calls involves going long a call option with strike price K1 and going short a call option on the same underlying and exercise date with a higher strike price K2.
An investor may use the bull spread strategy if they have a mildly bullish view but who wants a portfolio that’s protected against extreme price moves.
(when it gets to k2 the gradient of long +1 and the gradient of put = -1 then the gradient flattens to 0.
What is the bear spread using puts option strategy?
And when is it typically used?
The strategy for the bear spread using puts involves going short a put option with strike price K1 and going long a put option on the same underlying and exercise date with a higher strike price K2.
An investor may use the bear spread strategy if they have a mildly bearish view about the price of the underlying but wants a position that’s not too sensitive to extreme market movements.
If you want to get the upside of owning a stock, whilst still mitigating the downside in case the stock price goes down, we can buy a stock and put option, so that when a stock goes below a certain price the put option starts to have value and mitigate the downside. What can we do to replicate the payoff?
Buy a call option on the same stock S at time = 0 where the exercise price is the same
And invest the present value of the stock price e.g. into a bond or something.
What is the put call parity without dividends( THE UNDERLYING STOCK DOESNT PAY DIVIDENDS) And when can it be used?? AND WHAT DOES THIS ONLY APPLY TOO?
The put and call options both relate to the same stock and both have the same exercise price and exercise date.
It only applies to European options ( only excerise at maturity date)
What can we do with the put call parity and what cant we do yet?
We can rearrange the formula to find the price of call or put option conditional that we know either value of one of the put or call, but we cant yet find the original price of either the call or put ( we will see later.)
So what are the specififc terms in the put-call parity?
We are going to look at the put call parity a bit more, 1) So it essentially states Long call 2) Invest K/(1+r)^T risk free rate 3) Long stock 4) Long put Show how the net payoffs are the same
Show the first part of put call parity on a payoff diagram?
Long call and invest K / ( 1 + r) ^T
Here the net payoff will be the same with other side of Put call parity.
Show cash flows at t=0 for the put call parity
g
Since the put call parity shows that same payoff the price of each strategy,
under no arbitrage arguments should be the same, if they are not when what can we generate?
Aribtritage profits.
Lets say the put is more expensive and we rearrange the put call parity, what is our aribtrage strategy?
be caregul the stock stock is minus, so we need to do the reverse ( short stock)
Work out the put call parity, is there mispricing?
How can we take advantage of the mispricing? ( HINT WHAT WILL BE CASHFLOW TODAY AND AT MATURITY)
What is the formula for the Put-Call Parity with dividends?
Addition you invest in the pv of dividends at t = 0
So for the binominal model in a single period ( t0 to t1) the underlying price can move from its current level S0 to one of only 2 new levels. it can move up and down u(S) or d(S) where u>d show this on a tree diagram where S0 = 100 and u = 1.25 and d = 0.20. ? In our course can you make arbitrage profit with the up and down factors?
in our course no, because we will assume that the up and down factors are given, so the Up factor has > risk free rate> down factor, you cant make aribtrage profit.
We are going to have a replicating portfolio that replicates the call option to start with and do the same for a put option, but first What is the payoff diagram at maturity of owning 1 unit of the stock? what about if i own 2 units of the stock and stock price is 100 . ( e.g i buy 1 unit of stock for 100?
So if stock price is 100 and you own 1 unit then payoff is 100. If the stock price is 200 and you own 1 unit of the stock then payoff is 200. if i own 2 units of the stock and stock price is 200, then payoff at maturity is 200. hence gradient is 2
So what determines the gradient of the stock payoff?
The gradient of the stock payoff line depends on the number of stocks held.
If the payoff of a bond at maturity is positive what does it mean?
You invested PV(B) at risk free rate
If the payoff of a bond at maturity is negative what does it mean?
Lets say we want to replicate payoff of call option, now from previous couple flashcards, the payoff of a call option can have an upside or downside, ( value of call in downside = 0 and value of call in upside St-K), as there are 2 outcomes, we only need to replicate 2 payoffs ( the up and down state. ) Now what is the replicating portfolio that goes through the 2 points?
The payoff of the green line will replicate the payoff of the call option, as stock price can have 2 options
How can we replicate the green line( hence call option) ?
We can buy delta shares in the underlying asset and borrow money ( buy government bond at risk free rate
We can buy delta shares in the underlying asset and borrow money ( buy government bond at risk free rate to replicate call option we said show this on the diagram ( HINT EXPLAIN WHAT DELTA IS AND WHAT ITS MEANT TO BE THE SAME AS)
1) If at time 0 i buy delta shares, the gradient of this will be delta, we need delta to be that of the green line, but that green line doesn’t by itself replicate the green line, it needs to be shifted downwards.
2) Hence i need a cash outflow at maturity date of -B ( borrowing PV(B)
3) if you sum of the 2 red lines the end result is the green line.
HENCE REPLICATING PORTFOLIO IS SAME PAYOFF AS CALL OPTION.
So this is the replicating portfolio of the call option, so how can we solve for the amount of shares ( NOTICE HERE OBVIOUSLY B IS NEGATIVE )
We have 2 equations and 2 unknowns ( delta and B) so we can solve simultaneously.
We can use algebra to solve for the number of shares but just tell me the formula?
Whats another way of getting number of shares
Its just the gradient Cu-Cd/Su- Sd ( gradient of green line)
What is the formula value of B( B is amount you need to repay at T from borrowing the PV(B) at t=0 )?
( ITs also the intercept of y intercept)
So by working working out B and number of shares what is the value of the call (
I am choosing to work with value so this B is negative and s0( delta) is positive)
We are going to use the fact that use the equation of a line to solve for b and Delta which will help us solve for the call price for the replicaitng portfolio formula.
What is the European call price today, t=0, if the strike price is £20, the exercise date is in 3 months and the risk-free rate for 3 months is 1.25%. probability of upstate is 20% and probability of downstate is 10%
1) Work out the number of shares
2) Create a linear straight line equation
3) Plug 2 points to find y intercept.
4) Write out the the replicating portfolio
5) Then use the call price formula to find price.
( HINT BE CAREFUL WITH VALUE)
If we are just going to use formula what do we have to do first? ( HINT DRAW SUTTING)
Draw tree diagram, the value of call in upstate = 4
the value of call in downstate = 0.
Using the formulas directly solve this
We now want a replicate portfolio for put option which is what?
1) shorting delta (Δ) shares in the underlying
and
2) investing an amount of money at t=0 at the risk-free
rate. T
What is the price for a put formula?
Use the inittutive equation of the line formula.
Strike price = 90 not 100
USE FORMULA HERE
Be careful of signs ( here b value is positive )
Now we are going to look at binominal option pricing with the risk neutral method, what is the formula( HINT formula(S))? What is the trick in exam questions?
We are essentially calculating expected payoffs of the call option is up and down state/ then discount by risk free rate
Lets say the question gives information on an option you want to price and it says probability of up movement is 0.5 and down is 0.5, but it doesn’t tell you what type of probabilities they are. We don’t use these in option pricing we have to calculate risk neutral probabilities to price call option
Stock A is trading at £50 per share. The stock price will either go up or go down
by 20% in each of the next two years. The annual risk-free interest rate is 10%.
We will calculate the price of a two-year European call option with the strike
price K = £45.
Whats the first thing we do ?
Draw tree diagram ( Fill in red first then blue)
We are going to apply the method to 3 trees.
q isn’t always constant be careful
As the up and down factors don’t change we don’t need to change q but make usually they do so be aware. But firstly work the q and 1-q
Using risk neutral method. ( To note you will have to do this numerous times if the up and down factors change
Looking at the up node work out value of the call in the up node using risk neutral method at t = 1
Looking at the down node work out value of the call in the up node using risk neutral method at t =1
Now work out value of the call at t = 0 using the risk natural method. ( NOTE THE UP AND DOWN FACTORS AGAIN ARE THE SAME)
So in all the examples we have assumed European call option. what would we do if we are looking at American call option ( can exercise before maturity date)?
You need to check at cheap point whether its optimal to exercise or not.
