Bounding the Strike Sensitivity of a Call Using No Dominance
Summary
The document derives bounds on the derivative of a call price with respect to strike. It compares calls with nearby strikes and uses the fact that, at maturity, the difference between their payoffs is bounded by zero and the negative of the strike gap. Taking conditional expectations translates those payoff bounds into price bounds, with the bond price supplying the present value of the strike difference.
Dividing by the strike gap bounds the corresponding difference quotient between the negative bond price and zero. As the strikes converge, the quotient approaches the call price’s strike derivative, giving the stated result, including one-sided derivatives where needed. The argument relies on the payoff inequalities and the pricing relation used to take conditional expectations. The document’s initial attempted portfolio reasoning is incomplete; the accepted solution supplies the payoff comparison and limiting step that establish the bound.
Key ideas
- The difference between call payoffs at two strikes lies between the negative strike gap and zero.
- Conditional expectation converts these payoff bounds into bounds on call price differences.
- Dividing by the strike gap bounds the call price’s strike difference quotient.
- Taking the limit gives a lower bound of negative bond value and an upper bound of zero for the strike derivative.
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Full text
# Options and bond related to convexity
# Options and bond related to convexity
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.
## Answer by Gordon (score 1, accepted)
https://quant.stackexchange.com/a/23083
Note that, for $K_1 < K < K_2$, \begin{align*} -(K_2-K_1) \le (S_T-K_2)^+ - (S_T-K_1)^+ \le 0. \end{align*} Taking the conditional expectation with respect to information set $\mathcal{F}_t$, \begin{align*} -(K_2-K_1)B_t(T) \le C(T, K_2, S, t) - C(T, K_1, S, t) \le 0. \end{align*} That is, \begin{align*} -B_t(T) \le \frac{C(T, K_2, S, t) - C(T, K_1, S, t)}{K_2-K_1} \le 0. \end{align*} Since \begin{align*} \lim_{K_2-K_1\rightarrow 0}\frac{C(T, K_2, S, t) - C(T, K_1, S, t)}{K_2-K_1} = \frac{\partial C(T, K, S, t)}{\partial K}. \end{align*} The result follows immediately.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.