Lecture 5: Calculate Black-Scholes option prices

Question 1

What is Black-Scholes equation for European call option?

Answer 1

c = So*N(d1) - Ke^(-rT) * N(d2)

Question 2

What is Black-Scholes equation for European put option?

Answer 2

p = Ke^(-rT) N(-d2) - So*N(-d1)

Question 3

d1 = ?

Answer 3

d1 = [ln(So/K) + (r+ 0.5sigma^2)T] / sigma* T^0.5

Question 4

d2 =?

Answer 4

d2 = d1 - sigma * T^0.5

Question 5

If a stock were to pay dividend during the life of a option, call price ?, put price ?.

Answer 5

If a stock were to pay dividend during the life of a option, call price FALLS, put price RISES.

Question 6

If underlying asset pays dividend, calculate ?? then ? it from So and use ?? to calculate call/put B-S price.

Answer 6

If underlying asset pays dividend, calculate present value of dividend = De ^(-rt) then subtract it from So and use So’ = So - De^(-rt) to calculate call/put B-S price.

Question 7

Stock Index (e.g. s&P500, FTSE) Options:
d1 =  [ln(So/K) + (r ? + 0.5sigma^2)T] / sigma* T^0.5

c = S0 ?? *N(d1) - Ke^(-rT) * N(d2)

Answer 7

Stock Index (e.g. s&P500, FTSE) Options:
d1 =  [ln(So/K) + (r - q + 0.5sigma^2)T] / sigma* T^0.5
d2 = d1 - sigma * T^0.5
c = S0 e^(-qT)*N(d1) - Ke^(-rT) * N(d2)

Question 8

? position:

• writer sell option without taking a position in t’ underlying asset.
• risk is ? when writing a call but ? to a fall in asset price to ? when writing a put.

Answer 8

Naked position:

• writer sell option without taking a position in t’ underlying asset.
• risk is unlimited when writing a call but limited to a fall in asset price to 0 when writing a put.

Question 9

? position:

• combine ? with ?.
  e. g. ? a call and ?? to cover potential option exercise
• if stock price falls significantly, loss in value can be ? ?? writer received.

Answer 9

Covered position:

• combine option with u.a.
  e. g. write a call and hold stock to cover potential option exercise
• if stock price falls significantly, loss in value can be&raquo_space; option premium writer received.

Question 10

?-? strategy:
- ? u.a. each time option goes in the money
& then ? it whenever option goes ? t’ money
- significant ??, esp. when options are close to money or at t’ money.

Answer 10

Stop-Loss strategy:
- buy u.a. each time option goes in the money
& then sell it whenever option goes out t’ money
- significant transaction costs, esp. when options are close to money or at t’ money.

Question 11

??:

• most common hedging strategy
• option writer takes a position in u.a. depending on magnitude of option ?.
• writer will sell options if an attractive price based on writer’s estimate of volatility is identified & then hold ?shares.
  e. g. If short (write) n calls, writers can ?? this position by buying (??) shares.

Answer 11

Delta Hedge:

• most common strategy
• option writer takes a position in u.a. depending on magnitude of option Delta.
• writer will sell options if an attractive price based on writer’s estimate of volatility is identified & then hold Delta shares.
  e. g. If short (write) n calls, writers can Delta hedge this position by buying (delta*n) shares.

Question 12

Delta of a portfolio of options = ?? of individual Delta of options in t’ portf.

Answer 12

Delta of a portfolio of options = weighted average of individual Delta of options in t’ portf.