From James’s Page

Actuary: Parity

MFE topic list

Put-call parity formula for stock options

Variables:
{$ T $} = expiration time
{$ K $} = strike price
{$ S_0 $} = stock price at time 0
{$ r $} = risk-free interest rate
Suppose the stock has a continuous dividend rate of {$ \delta $}. The put-call parity formula is {$$ C(K,T)-P(K,T)=S_0e^{-\delta \, T} - Ke^{-rT}, $$} where {$C$} is the call option price, and {$P$} is the put option price.

Justification: The key is that buying a call and selling a put amount to a synthetic forward. The cash flow for buying a call and selling a put at time 0 is {$ -C+P $}. There is an additional cash flow at time {$ T $} of +{$ K $}. The cash flow at time {$ T $} for a buying a forward is -forward price = {$ - S_0e^{(r-\delta)T} $}. Thus the cash flows at time 0 are {$$ -C+P +PV(K) = PV (- S_0e^{(r-\delta)T}) $$} and the formula follows.

General put-call parity formula

Now the underlying asset can be stock with different dividend schemes, foreign currency, futures, or bonds.
Variables:
{$ F_{0,T} $} = prepaid forward price of the underlying asset
{$ K $} = strike price in dollars per share of asset
The formula is {$$ C(K,T)-P(K,T)=F_{0,T}^P - Ke^{-rT}. $$}

Properties

Price ranges

• A call cannot cost more than the stock price {$ S $}.
• A put cannot cost more than the strike price {$ K $}.
• Both {$C$} and {$P$} should be positive; this also puts lower bounds on both

• American options are always going to be more expensive because they have the potential to early exercise.

We thus have the following string of inequalities: {$$ \max[0, F_{0,T}^P - Ke^{-rT}] \leq C_{\text{Eur}} \leq C_{\text{Amer}} \leq S; $$} {$$ \max[0, Ke^{-rT} - F_{0,T}^P ] \leq P_{\text{Eur}} \leq P_{\text{Amer}} \leq K; $$}

Early Exercise

Price vs. {$t$}

Price vs. {$K$}

{$C$} as a function of {$K$} is

1. decreasing;
2. has slope {$\geq -1 $};
3. concave up.

{$P$} as a function of {$K$} is

1. decreasing;
2. has slope {$\leq 1 $};
3. concave up.

If any of the properties are violated, we can have arbitrage. For the first two conditions, this is done by creating a spread, i.e., buy the option that's mispriced to be lower than it should, and sell the option mispriced to be higher. When the third (convexity) condition is violated, we can create an asymmetrical butterfly spread with "{$\lambda$}".