Exotic Options: Introduction, Path dependent options, Discontinuous payoff options

Question 1

What are different types of exotic options?

Answer 1

1. Path Dependent
   * Asian options
   * Barrier options
2. Discontinuous payoff
   * Digital options
   * (Digital barrier options)
3. Multiasset (rainbow)
   * Basket options
   * Best options/Worst options
   * (Quanto options)

Question 2

What are Asian options?

Answer 2

The payoff of an Asian option depends on an average price of the underlying assets.

Question 3

What are barrier options?

Answer 3

Barrier options payoff is similar to a European option but is conditioned on whether or not a given price barrier has been touched during the life of the option.

Question 4

What is the formula for the payoff of an asian call?

Answer 4

Payoff_{Asian call} = Max[Σ ⁿ_i=1(P_i/n) - K; 0]
Payoff_{Asian put} = Max[K - Σ ⁿ_i=1(P_i/n); 0]

Where i=1, 2, … n are the fixing dates and P₁, P₂…P_n the corresponding prices of the underlying asset

Question 5

How does the cost of an Asian option compare with a European option?

Answer 5

Asian options are less costly than European options, as their payoff is less dispersed.

Question 6

How does the frequency of fixings influence the value of an asian option?

Answer 6

An asian option with quarterly fixings will be worth more than corresponding Asian call with monthly fixings, as more fixings reducees the dispersion of the payof and hence the value of the call.

Question 7

What are Asian options used for?

Answer 7

Asian options are often used for corporate hedging. They cost less than European options and are better suited to hedge an exposure which is continuous over time.

Question 8

What are the greeks of an Asian option?

Answer 8

• Asian options have a lower Delta than European options (which get even lower as time passes and number of fixings increases)
• They also have a lower Gamma than European options, even when ATM and close to maturity

Question 9

What are the types of Barrier options?

Answer 9

• A knock-in option pays at maturity only if:
  a. The option is ITM at maturity
  b. The barrier has been touched
• A knock-out option pays at maturity only if:
  a. The option is ITM at maturity
  b. The barrier has NOT
  been touched

Question 10

What is the relationship between a knock-in option and a European option?

Answer 10

If a knock-in option touches the barrier for the first time it just becomes a European option. In fact one can combine a knock-in and a knock-out option to create a European call.

Question 11

What is the formula for the knock-in knock-out parity?

Answer 11

KI call_K,H,t + KO call_K,H,t = European call_K,t

As long as K,H, t, and underlying are the same

Question 12

What are the relationships of Greeks that can be derived from the knock-in knock-out parity?

Answer 12

Delta_{KI call K,H,t} + Delta_{KO call K,H,t} = Delta_{European call K,t}
Gamma_{KI call K,H,t} + Gamma_{KO call K,H,t} = Gamma_{European call K,t}
Vega_{KI call K,H,t} + Vega_{KO call K,H,t} = Vega_{European call K,t}
Theta_{KI call K,H,t} + Theta_{KO call K,H,t} = Theta_{European call K,t}

Question 13

What are the more detailed barrier option types depending on the position of the barrier?

Answer 13

1. down-and-out
2. up-and-out
3. down-and-in
4. up-and-in

Question 14

What is a digital option?

Answer 14

A digital option has a discontinuous payoff, depending on whether the final price of the underlying is greater or smaller than the strike price.

Digital call payoff = (Q if S_final >= K; 0 otherwise)

Question 15

What is the formula for the value of a digital option?

Answer 15

C (call)_digital = e^-rt * Q * N(d₂)

P (put)_digital = e^-rt * Q * N(-d₂)

Question 16

What are the greeks for a digital call?

Answer 16

• Delta is clearly positive
• Low but positive Gamma and Vega below strike
• Low but negative Gamma and Vega above strike price
  (Caution, in the last few days close to the strike pirce, very high Gammas and very unstable Delta)