portfolios and risk

Question 1

short selling:

Answer 1

when you buy basically an ‘I owe you’ contract with the owner of the shares, sell them, wait for the price to drop, then buy them back and return them to the original owner (it is not very legal cause it can cause Problems)

Question 2

portfolio:

Answer 2

the range of investments held by an individual, your position towards each asset is positive (long) if you want the investment to go up and negative if you want it to go down (short, like if you’re tryna short sell)

Question 3

straddle:

Answer 3

buying a call and put option with the same strike price and expiry time, value is Π(S,t)=C(S,t;E)+P(S,t;E)

Question 4

payoff of a straddle:

Answer 4

E-S if S<E (you exercise the put) or S-E if S>=E (you exercise the call) - it’s good if the stock price changes a lot in either direction, but that means it’s expensive to set up

Question 5

short straddle:

Answer 5

when you sell the call and put options, everything is negative basically, it’s worth it cause you charge people to set it up with you so you are hoping the share price doesn’t change much so you make a profit still because of the initial cost

Question 6

investors expected profit from a portfolio:

Answer 6

funky E[Π(S,T)]-Π0e^(rT)

Question 7

investors expected return from a portfolio:

Answer 7

funky E[∆Π]/Π0=(funky E[Π(S,T)]-Π0)/Π0

Question 8

bull spread:

Answer 8

buy a call then sell a call with a slightly higher exercise price, betting on upward movement of share price still, Π(S,t)=C(S,t;E1)-C(S,t;E2) where E2>E1, this is considered a hedge because if the stock only goes up slightly then this is much cheaper than just buying a call option, even though you can’t gain as much if it goes up a lot

Question 9

payoff of a bull spread:

Answer 9

0 if S<E1 (neither call is exercised), S-E1 if E1<S<E2 (you exercise the first call), or E2-E1 if S>=E2 (you exercise the first call, the buyer exercises the second)

Question 10

bull market:

Answer 10

confidence in the market is high, stocks and shares are growing in value

Question 11

bear market:

Answer 11

stocks and shares are losing value so confidence lower, people are betting on more reduction

Question 12

bear spread:

Answer 12

buying and selling a put option, Π(S,t)=P(S,t;E1)-P(S,t;E2) but E1>E2

Question 13

bond:

Answer 13

a contract that pays a known amount F, the face value, at a known time T, the maturity (or redemption) date, we assume they are risk-free (that the seller will not go bankrupt) for maths purposes basically
banks are essentially selling bonds

Question 14

coupon:

Answer 14

payments from the seller to the buyer of a bond, what makes bonds like. actually valuable since otherwise why buy them

Question 15

zero-coupon bond:

Answer 15

what is says on the tin, bonds with no coupons

Question 16

return on a risk free bond:

Answer 16

dB/B=r dt

Question 17

arbitrage opportunity:

Answer 17

risk-free profit essentially, always yields a positive profit with 0 initial investment (mathematically)
Π(S,t)>0 for all S in [0,∞) at t=T

Question 18

no arbitrage principle:

Answer 18

arbitrage opportunities don’t exist lmao, you can never make risk free profit on the stock market, mathematically this is a portfolio with a non-negative payout must be at least equally valuable as holding nothing
ΠT>=0 => Πt>=0, ∀t<=T (Πt=Π(S,t))

Question 19

comparing contracts:

Answer 19

if XT>=YT, then Xt>=Yt for all t<=T

Question 20

equivalent contracts:

Answer 20

if XT=YT, then Xt=Yt for all t<=T

Question 21

return on risk-free contracts:

Answer 21

dΠ/Π=r dt

Question 22

practically why does all this no-arbitrage stuff work:

Answer 22

if two risk-free products A and B are being sold, paying the holder £10 at T, but A costs £5 and B costs £7, then demand for A will be higher so A will increase in price and demand for B will be lower so B will decrease in price until they meet in the middle and become equal

Question 23

put-call parity relationship:

Answer 23

St+Pt-Ct=Ee^(-r(T-t)), can rearrange this to find a call option price from a put option price and etc.