Financial Options

Options

components of option price

m.v of option = time premium + intrinsic value
intrinsic value: the difference of m.v of underlying and strike price of call option

Minimum value of call
American call : Ca(S0, T, X) >= max(0, S0 - X)
European call : Ce(S0, T, X) >= max(0, S0 - X(1+r)^-T)

Minimum value of put
American put : max(0, X - S0 )
European put : max(0, X(1+r)^-T - S0 )

Ca(S0, T, X) > Ce(S0, T, X) 

but prior to expiration, S0 - X(1+r)^-T > S0 - X

early exercise?

- for call , if dividend > time premium, then you exercise the call option

- for put , if interest rat is large enough

Maximum value of put

American put : X

European put: X(1+r)^-T

Put call parity

S0 = C(S0, T, X) - P(S0, T, X) + X/(1 + r )

ATM european call or ATM european put, which one has higher price?
S = c - p + pv(x)
c - p = S - pv(x)   ;   S=X for ATM option
c - p =  S - pv(S)  ;  S - pv(S) > 0
So : c > p

Put call forward parity
S0 = C(S0, T, X) - P(S0, T, X) + X/(1 + r )
F = S0(1+r)
S0 = F/(1+r)
So : F/(1+r)  = C(S0, T, X) - P(S0, T, X) + X/(1 + r )
  P(S0, T, X) = C(S0, T, X) + (X-F)/(1+r)

Binomial Model
- discrete time model
- if infinite samples, converges to BSM model
- if interval time getting smaller, converges to BSM model
p = (1+r-d) / (u -d)
c = [ pCu + (1-p)Cd ] / (1+r)
if stock price up by 20%, and down by 10%, then: u = 1.2, d = 0.9
S⁺ = Su; S^- = Sd, find Cu, Cd, then find c

Black Scholes Model
- continuous time model
- assume rf and vol are constant
- no taxes and transaction fee
- assume options are european
- assume stock price is normally distributed
- S, T, X, rf, vol -> find c

calculate implied vol
- work backwards to find it
- C, S, X, T, rf ->BSM model -> find implied vol

volatility smile
- shows implied vol is not consistent
- implied vol depends on exercise price
- violates the constant vol assumption of BSM

Implied vol > forecast vol
- option overvalued
- sell option

selling options, the trade is short volatility. if actual vol is lower than what he priced it at, he makes money

Interest rate Cap
- series of interest rate call options

Interest rate Floor
- series of interest rate put options