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Write the following payoffs as linear combination of call options with different strikes and possibly some cash and give the closed form formula for them.enter image description here

Attempted solution: The payoff for the bear spread is $$(K_2 - S_T)^{+} - (K_1 - S_T)^{+}$$ Therefore our closed form solution for the B-S price is $$V(\tau,S) = P(\tau,K_2,S) - P(\tau,K_1,S) $$ The payoff for the bull spread is $$(S_T - K_1)^{+} - (S_T - K_2)^{+}$$ Therefore our closed form solution for the B-S price is $$V(\tau,S) = C(\tau,K_1,S) - C(\tau,K_2,S)$$

I was told the closed form B-S price is incorrect but my professors lecture notes say otherwise:

professors example

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  • $\begingroup$ Not yet correct. Write each term using the BS formula. $\endgroup$ – Gordon Mar 17 '16 at 22:18
  • $\begingroup$ What part is not correct, I don't understand your statement $\endgroup$ – Wolfy Mar 18 '16 at 0:10
  • $\begingroup$ @Gordon I made an edit, is my solution correct now? $\endgroup$ – Wolfy Mar 18 '16 at 0:43
  • $\begingroup$ No. Please double check what the BS formula look like. Your formula is not the BS formula at all. Can you please write out the formulas for payoffs $(S_T-K_1)^+$ and $(S_T-K_2)^+$. Any book has the formula or you can google it $\endgroup$ – Gordon Mar 18 '16 at 1:14
  • $\begingroup$ In addition, I have specifically written out the formula in my answer to your question quant.stackexchange.com/questions/24927/…. $\endgroup$ – Gordon Mar 18 '16 at 1:21
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Here, we assume that the bottom is zero and the top is $K_2-K_1$. Then, in mathematical form, the ${\color{blue} {blue}}$ option payoff is given by \begin{align*} & \ (K_2-K_1)\pmb{1}_{S_T \le K_1} + (K_2-S_T)\pmb{1}_{K_1 < S_T \le K_2} \\ =& \ (K_2-K_1)\pmb{1}_{S_T \le K_1} + (K_2-S_T)\left(\pmb{1}_{S_T \le K_2} - \pmb{1}_{S_T \le K_1}\right)\\ =& \ (K_2-S_T)\pmb{1}_{S_T \le K_2}+\big[(K_2-K_1) - (K_2-S_T) \big]\pmb{1}_{S_T \le K_1}\\ =& \ (K_2-S_T)^+ - (K_1-S_T)^+ ,\tag{1} \end{align*} that is, a put spread. Note that, this option can also be replicated with a zero-coupon bond and a call spread: \begin{align*} & \ (K_2-K_1)\pmb{1}_{S_T \le K_1} + (K_2-S_T)\pmb{1}_{K_1 < S_T \le K_2} \\ =& \ (K_2-K_1)\left(1-\pmb{1}_{S_T \ge K_1} \right)+ (K_2-S_T)\left(\pmb{1}_{S_T \ge K_1} - \pmb{1}_{S_T \ge K_2}\right)\\ =& \ (K_2-K_1) + \big[(K_2-S_T) - (K_2-K_1) \big]\pmb{1}_{S_T \ge K_1} +(S_T-K_2)\pmb{1}_{S_T \ge K_2}\\ =& \ (K_2-K_1) +(S_T-K_2)^+ - (S_T-K_1)^+. \end{align*}

For the ${\color{red} {red}}$ option payoff, the approach is the same.

EDIT:

The price of Payoff (1) is given by \begin{align*} put(K_2) - put(K_1). \end{align*} Note that this price is 'not necessarily the Black-Scholes' price', as Black-Scholes' price has a particular form. In particular, in the Black-Scholes' pricing framework, we assume that the underlying equity price process $\{S_t \mid t \ge 0\}$ satisfies, under the risk-neutral probability measure, an SDE of the form \begin{align*} dS_t/S_t = rdt + \sigma dW_t, \end{align*} where $\{W_t \mid t \ge 0\}$ is a standard Brownian motion. Then, \begin{align*} put(K_1) &= K_1 e^{-rT} \Phi(-d_2^1) - S_0 \Phi(-d_1^1)\\ put(K_2) &= K_2 e^{-rT} \Phi(-d_2^2) - S_0 \Phi(-d_1^2), \end{align*} where \begin{align*} d_1^1 &= \frac{\ln \frac{S_0}{K_1} + (r+\frac{1}{2}\sigma^2)T}{\sigma \sqrt{T}},\\ d_2^1 &= d_1^1 - \sigma \sqrt{T},\\ d_1^2 &= \frac{\ln \frac{S_0}{K_2} + (r+\frac{1}{2}\sigma^2)T}{\sigma \sqrt{T}},\\ d_2^2 &= d_1^2 - \sigma \sqrt{T}. \end{align*}

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  • $\begingroup$ I made an edit to my post, can you check to see if I am correct? $\endgroup$ – Wolfy Mar 17 '16 at 21:26
  • $\begingroup$ Also, I liked how you included the replication do you have a reference material to know how to replicate all of the different option strategies i.e. bull spread, bear spread, straddle, strangle, etc...? $\endgroup$ – Wolfy Mar 17 '16 at 21:41
  • $\begingroup$ No reference. But the technique is the same. First express it using indicator functions and then decompose. $\endgroup$ – Gordon Mar 17 '16 at 22:15
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Think like this:

  1. Start with a long call option for K1. This would give you a payoff reflected at K1.
  2. Short some cash to move the payoff vertically down.
  3. Short a call option for K2. The payoff of this short option will offset the payoff of your long call option for >K2.

You can visualize this portfolio should give you the payoff for bull spread call. The other payoff is similar.

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Let's look at the blue case, bear spread call. Since it starts with a constant, then you need some cash. Also we need to assume, say bear call spread, the constant parts are $(K_1 + K_2)/2$ and $-(K_1 + K_2)/2$ at both ends (technically makes the slope be 1 in middle of the function).

Consider the call option as following $$ C_{\text{bear}} = \frac{(K_2 - K_1)}{2} - (S_T - K_1)_+ + (S_T - K_2)_+ $$

The red curve is pretty much the same as blue one, just as following $$ C_{\text{bull}} = -\frac{(K_2 - K_1)}{2} + (S_T - K_1)_+ - (S_T - K_2)_+ $$

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