Question

6. Use arbitrage arguments to prove the following bounds on the price C(So, K,T) of a European call with strike K and maturity T (assuming the underlying pays no dividend) (a) The call price is no greater than the stock price: C(So, K,T) (b) For otherwise identical calls with strikes Ki < K2, So (c) For othwerwise identical calls with maturities Ti < T2

Answer 1

Put call parity-

Call – Put = Stock – Strike

For parameters- S₀, K, T

1.

C < S

This can be proved by logical deduction without using mathematics. Since the question mentions ‘arbitrage arguments’, it can be assumed that the mathematical derivation is not required.

If call price is equal to or greater than the spot price, there is no point of holding the call at a price greater than the spot price. The trader would rather buy the underlying stock and realize similar (if C = S), or better (if C > S) payoff

2.

For K₁ < K₂

0 <= C(K₁) – C(K₂) <= K₂ – K₁

The price of call option is dependent on the volatility of price of the underlying stock. A call option with lower strike price has a greater probability of maturing in the money as compared to otherwise identical call option with higher strike price. Hence, the price of former call option is greater than the latter. If C(K₁) – C(K₂) > K₂ – K₁ then the trader could execute a short call spread strategy by buying an OTM call at K₂ and selling an OTM call at K₁. This could produce a risk free profit equal to the difference: [C(K₁) – C(K₂)] – [K₂ – K₁]

3.

For T₁ < T₂

C(T₁) <= C(T₂)

The price of the call option is dependent on the time to maturity. For otherwise identical options, the lesser the time to maturity, the lower the probability of option maturing in the money. If the relationship did not hold true, then the trader could execute long calendar spread of time spread strategy with call options and realize a risk free profit. He will sell a call with earlier expiration and buy a call with later expiration. As the time to maturity approached for the earlier option, they will close the position and realize profit.