Deriving the Strike Sensitivity of a Call Price
Summary
The document derives how a call option’s price changes as its strike changes. Starting from the risk-neutral expected payoff, it writes the option value as a discounted integral over terminal underlying prices above the strike. Differentiating that expression with respect to strike gives the negative discounted risk-neutral probability that the underlying finishes above the strike.
It clarifies that this strike derivative is often called dual delta; ordinary delta refers to sensitivity to the spot price. The derivation does not require the Black–Scholes model, because it follows from the payoff and risk-neutral pricing representation. The result is stated for a call price with discounting and a risk-neutral distribution; the exercise’s undiscounted expression omits that discount factor. The answer points toward the related connection between option prices and the risk-neutral distribution, without developing that broader result.
Key ideas
- A call’s derivative with respect to strike is distinct from its usual spot delta.
- The strike derivative equals the negative discounted risk-neutral probability of finishing above the strike.
- The result follows by differentiating the risk-neutral payoff integral.
- This relationship does not depend on Black–Scholes assumptions.
- Discounting must be included when the option price is expressed at the current date.
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Full text
# Call option Delta
# Call option Delta
I have an exercise where I need to show that the prices of call options $ C(t,K)=E((S_t-K)^+),t \in [0,T]$ with Strike $K$ for fixed $t$: $$\frac{\partial ^+C(t,K)}{\partial K}=-P(S_t>K).$$ We havent discussed Black Scholes model yet. I guess this will be the introduction exercises for the BS formulas. With: $$\frac{\partial ^+}{\partial K}=\lim_{h↓0}\frac{C(t,K+h)-C(t,K)}{h}$$ I get: $\frac{\partial ^+C(t,K)}{\partial K}=\lim_{h↓0}\frac{C(t,K+h)-C(t,K)}{h}=\lim_{h↓0}\frac{E((S_t-(K+h))^+)-E((S_t-K)^+)}{h}=\lim_{h↓0}\frac{P(S_t>K+h)(E(S_t|S_t>K+h)-(K+h))-P(S_t>K)(E(S_t|S_t>K)-K)}{h}...$
From there I dont know how to proceed further. Using L'Hospital b/c we have $"\frac{0}{0}"$ or left term could be 0. Please help.
## Answer by LocalVolatility (score 4, accepted)
https://quant.stackexchange.com/a/37638
First note that delta is the derivative w.r.t. to the spot and not the strike. The latter is often called "dual delta". Also, you don't need any knowledge of Black-Scholes as this is a model-independent result.
The result follows from the general expression of the call price
\begin{equation} C_0 = e^{-r T} \mathbb{E}_\mathbb{Q} \left[ \left( S_T - K \right)^+ \right] = e^{-r T} \int_K^\infty (x - K) \mathrm{d}F(x), \end{equation}
where $F$ is the risk-neutral distribution function of $S_T$. Differentiating w.r.t. $K$ yields
\begin{equation} \frac{\partial C_0}{\partial K} = -e^{-r T} \int_K^\infty \mathrm{d}F(x) = -e^{-r T} \mathbb{Q} \left\{ S_T > K \right\}. \end{equation}
This is probably one of the most common questions here; search for "Breeden-Litzenberger" for related answers.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.