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

  • $\begingroup$ 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

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.


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*}

  • $\begingroup$ 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

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.


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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