Dividend-Adjusted Black-Scholes Pricing for a Call-Spread Butterfly
Summary
The document analyzes a three-strike call payoff formed by buying calls at the outer strikes and selling twice the call at the midpoint. It applies a quarterly dividend-yield adjustment by scaling the underlying with a discrete dividend factor, then expresses the option value as a weighted combination of Black-Scholes call prices at adjusted strikes. The question also asks for price and standard Greeks, including theta without differentiating with respect to the underlying.
A proposed solution incorrectly reports a zero value and omits the negative coefficient in its delta combination. The accepted response corrects the price to 0.3905 and gives delta as the same signed combination of call deltas, scaled for dividends; it suggests deriving the other Greeks by combining the corresponding call Greeks. The treatment is incomplete: it does not show the full numerical Greek calculations or resolve the requested theta method. It also uses a simplified discrete dividend approximation and contains notation inconsistencies, so its formulas should be checked before reuse.
Key ideas
- A call butterfly payoff can be represented as a signed combination of three calls with equally spaced strikes.
- The response adjusts for quarterly dividends by scaling the underlying and shifting the effective strikes.
- The accepted answer reports a positive option value and corrects delta to include the negative midpoint-call term.
- Other Greeks can be assembled as the same signed combination of the component call Greeks.
- The document leaves several requested calculations unfinished and includes notation that merits verification.
Tags
Full text
# Pricing of Black-Scholes with dividend
# Pricing of Black-Scholes with dividend
Consider the payoff $g(S_T)$ shown in the figure below. Consider Black-Scholes model for the price of a risky asset with $T = 1$, $r = .04$, and $\sigma = .02$ and dividends are paid quarterly with dividend yield $10\%$. Take $S_0 = 10$, $K_1 = 9$, and $K_2 = 11$. Find the Black-Scholes price, $\Delta$, $\Gamma$, $\rho$, and $\mathcal{V}$ of this option at time $t = 0$. Find $\Theta$ at time $t = 0$ without taking derivatives with respect to $S$.
Solution: The payoff is, $$g(S_t) = (S_t - K_1)_{+} - 2(S_t - \frac{(K_1 + K_2)}{2})_{+} + (S_t - K_2)_{+}$$ The Black-Scholes formula with dividend gives \begin{align*} V(t = 0,S) &= e^{-r\tau}\hat{\mathbb{E}}[g(\tilde{d}S_T)]\\ &= \tilde{d}\left(BS_{call}(\frac{K_1}{\tilde{d}}) - 2BS_{call}(\frac{K_1+K_2}{2\tilde{d}}) + BS_{call}(\frac{K_2}{\tilde{d}})\right) \end{align*} where $$\tilde{d} = \left(1 - \frac{d}{4} \right)^{4} = .9037$$ So, $$V(t = 0,S) = e^{-r\tau}\hat{\mathbb{E}}[g(\tilde{d}S_T)] = (.9037)((0) - 2(0) + (0)) \approx 0 $$ For the Greeks we have $$\Delta = \partial_S V(t = 0,S) = \tilde{d}\left[\Phi(d_1(\frac{K_1}{\tilde{d}})) + \Phi(d_1(\frac{K_1+K_2}{2\tilde{d}})) + N(d_1(\frac{K_2}{\tilde{d}})) \right] \approx 0$$ $$\Gamma = \partial_{SS}V(t = 0, S) = 0$$ $$\rho = \partial_r V(t = 0,S) = \left( e^{-rt}(\frac{K_1}{\tilde{d}})(t)\Phi(d_2) + e^{-rt}(\frac{K_1+K_2}{2\tilde{d}})(t)\Phi(d_2) + e^{-rt}(\frac{K_2}{\tilde{d}})(t)\Phi(d_2)\right) \approx 0$$ $$\mathcal{V} = (S\sqrt{t}\Phi(d_1) + S\sqrt{t}\Phi(d_1) + S\sqrt{t}\Phi(d_1)) \approx 0$$
## Answer by Pandaaaaaaa (score 2, accepted)
https://quant.stackexchange.com/a/25253
In your answer, you don't include dividend. I am sorry to say it is wrong.
Payoff function is $$ g(S_T) = (S_T - K_1)_+ - 2(S_T - \frac{K_1+K_2}{2})_+ + (S_T - K_2)_+ $$
BS pricing formula with dividend gives $$ V(t=0,S) = e^{-r}E(g(\tilde{d}S_T)) = \tilde{d} \left(BS_{call}\left(\frac{K_1}{\tilde{d}}\right) - 2BS_{call}\left(\frac{K_1+K_2}{2 \tilde{d}}\right) + BS_{call}\left(\frac{K_2}{\tilde{d}}\right)\right) $$
Where $$ \tilde{d} = (1-\frac{d}{4})^4 = 0.9037 $$ Plug in all numbers in your question, I get 0.3905 (double check it yourself).
As to the greeks,
$$ \Delta = \frac{\partial V(t=0,S)}{\partial S} = \tilde{d} \left[ N(d_1(K_1/\tilde{d})) - 2N(d_1((K_1+K_2)/(2\tilde{d}))) + N(d_1(K_2/\tilde{d})) \right] $$ where $N$ is the normal c.d.f. and $d_1(K)=\frac{log(S/K)+(r+\frac{1}{2}\sigma)\tau}{sigma\sqrt{\tau}}$.
Rest of the problem is pretty trivial because all greeks are just linear combination of original BS greeks. You just need change the strike price in the original BS greeks (I guess?). I won't go through all the calculation here.
Let me know if anything is not clear.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.