Forwards and Options

Question 1

Put call parity

Answer 1

P-C = Z*(K-F)

Question 2

How cna you sue linear interpolation to find option prices

Answer 2

If they are European and put call parity holds then P-C for options where strikes are equally spaced will have equal step size between them

Question 3

What is Z graphically

Answer 3

Plot P-C against strike - z is slope, should be the same if put call parity holds

Question 4

What type of function are options

Answer 4

Convex functions of strike. Convex means. If the line segment between any two distinct points on the graph of the function lies above the graph.
Has increasing first derivative

Question 5

Why doesnt Put call parity work in some cases

Answer 5

Arbitrage opportunity or dealing wiht american option. Put call parity only works if one exercise date

Question 6

How to compute an arbitrage trade

Answer 6

Find forward structures C-P costs and payoffs then to get arbitrage was to hedge exposure to future share prices so you subtract forwards to get a risk free payoff
Subtracting forwards gives risk free bonds.
Arbitrage would be borrow at lower rate and invest at higher.

Other way to show : Payoff for all ranges of K an show its never negative sometimes positive.

Question 7

How to show valid probability density function

Answer 7

Non negative, and integrates to 1

Question 8

How to calculate forward value from risk neutral density

Answer 8

Mean of the risk neutral density

Question 9

How to calculate call price from risk neutral density

Answer 9

Integrate 1-cdf is the undiscounted call price so we must then multiply by Z

Question 10

PV get and PV pay of call otpion

Answer 10

ZF = PV get
ZK = PV Pay

Question 11

How to find ZCB price from call option price

Answer 11

Differentiate and let strike go to zero then -Ce’0 = ZCB

Question 12

What is risk neutral cdf in terms of call option prices

Answer 12

Fk=1-Ce’k/Ce’0

Question 13

Relationship of cdf to pdf

Answer 13

Pdf = differentiated cdf

Question 14

How to calculate forward value of an index from Call option price

Answer 14

Ce0 is an entitlement to the share at time 10. So forward value is this divided by ZCB

OR
Calculate mean of risk neutral density

Question 15

Limit of BS as st dev goes to 0 or infinity

Answer 15

As goes to 0: max(PVget-PVPay,0)
As goes to infinity: PVGet

Question 16

How to show something is strictly increasing

Answer 16

Show gradient in relation to variable of interest : if gradient has to be positive must be increasing formula

Question 17

How to show something is concave/convex

Answer 17

Differentiate twice. If second derivative is positive then means convex

Question 18

What does convexity mean for american call options

Answer 18

Call option struck at the average of two strikes is less valuable than the average of two call options one at each strike. That is the definition of convexity.

Question 19

What should I check for when assessing a call option formula to be used for all strikes and extrpolated

Answer 19

Call options should be decreasing functions of strike. decreasing to zero - does this happen at the boundary.

Question 20

How to tell without maths if Put or call option prices permit arbitrage

Answer 20

The call prices do not decrease for large k and the put prices are not increasing for small k

Question 21

Compare put option to insurance

Answer 21

Buying a put option is like buying insurance against adverse effect of market falling. Insurance buyers are prepared to pay a premium, pay more than the expected claims to reduce downside risk. The same goes for options. The option buyer will pay a premium above the option payoff to reduce downside risk, espeically for extreme downside risk.

Question 22

Whats the minimum and maximum value of a trade involving options

Answer 22

Being forced into exercise the option or if we were forced not to exercise the option so has value 0.
Max value would be options cannot be worth more than outright entitlement to the underlying