Relevant definition:
Assumption 2.1 (No dominance). If the payoff $P$ of a financial instrument is nonnegative, then the price $p$ of the financial instrument is nonnegative.
Notation:
$T$ - the maturity
$K_1$,$K_2$ - Strike prices
$S$ - stock price
$t$ - current time
$B_t(T)$ - price of bond
It is well known that a convex function has right and left derivatives at all points. From the above exercise it follows that (exercise) $\delta_{K_{\pm}}C(T,K,S,t)$ exists. Use no dominance to show $$-B_t(T) \leq \delta_{K_{\pm}}C(T,K,S,t)\leq 0$$. (hint: Consider a portfolio made of a long position in a call with strike $K_2$ and short positions in a call option with strike $K_1$ and $K_2 - K_1$ bonds.
Attempted solution: Suppose we have a portfolio made up of a long position in a call with strike $K_2$ and a short position in a call with strike $K_1$ and $K_2 - K_1$ bonds. The price of the long call is defined by $$C(T,K_2,S,t)$$ the price of the short position in the call is defined by $$-C(T,K_1,S,t)$$ and the price of the $K_2 - K_1$ bonds is defines as $$B_t(T)$$ The portfolio value is thus $$P_v = C(T,K_2,S,t) - C(T,K_1,S,t) + (K_2 - K_1)B_t(T)$$ (Taking the hint provided by the user: barrycarter) Suppose the return on $P_v$ is less than the risk-free rate, then one would have to be compensated to take this position. Therefore by the no dominance assumption the price of this position would be negative which is a contradiction. Does the result follow from exercise 2.2?
I am not sure where to go from here, any suggestions is greatly appreciated.
