Week 10 - Option pricing theory

Question 1

Binomial option pricing using the REPLICATING PORTFOLIO - 3 formulas & the strategy at t=0 for call/long options

Answer 1

Formulas are same for call/put. Just change C to P.
1. Δ = (Cu-Cd)/(Su-Sd) = (Cu-Cd)/S0(u-d)
- Gradient of payoff line
- No. of SHARES BOUGHT at t=0

2. B = [Cd(u) - Cu(d)] / u-d
   - Cash OUTFLOW at maturity date
3. C = S0(Δ) + B/(1+rrf)
   - Price of call at t=0

Rep. portfolio strategy at t=0
1. for call option
- Long stock (buy Δ shares)
- Borrow PV(B) at risk-free rate
2. for put option
- Short stock (sell Δ shares)
- Invest PV(B) at risk-free rate

Question 2

Binomial option pricing using the RISK-NEUTRAL METHOD - 2 formulas & the necessary condition

Answer 2

1. q = (1+rrf-d) / u-d
   - Risk-neutral probability of an UP-MOVE
   - 0<q<1
2. C = [qCu + (1-q)Cd] / 1+rrf
   - Call value/price
   - Cu & Cd are expected payoffs

d<(1+r)<u MUST HOLD!

For American option, rmb to compare ST vs K to see if +ve payoff, then check if should exercise BEFORE maturity date or WAIT

Question 3

Black-Scholes option pricing

Answer 3

Makes an alternative assumption of CONTINUOUS TIME and price of underlying evolves on a continuous basis
- as opposed to the discrete time setting of binomial model

Question 4

Effect on call & put prices if…
1. current stock price increases
2. strike price increases
3. volatility increases
4. risk-free rate increases
5. the stock pays DIVIDENDS / amount of future dividends increases?

6. time to expiration (for American call & put)?

Answer 4

1. If current stock price increases,
   - call price increases, put price decreases
2. If strike price increases,
   - call price decreases, put price increases
3. If volatility increases,
   - both call and put prices increase due to the flexibility of exercising, no obligation involved
4. If risk-free rate increases,
   - call price increases, put price decreases
5. If stock pays dividends,
   - call price decreases & put price increases
6. If time to expiration increases,
   - American call & put prices increase {probably due to greater flexibility of exercising}