Replicating Bull and Bear Spreads with Options and Cash

# Linear combination of payoffs of bull and bear spreads


# Linear combination of payoffs of bull and bear spreads












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.

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:

## Answer by Gordon (score 1, accepted)

https://quant.stackexchange.com/a/24933

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*}

## Answer by SmallChess (score 0)

https://quant.stackexchange.com/a/24929

Think like this:

- Start with a long call option for K1. This would give you a payoff reflected at K1.

- Short some cash to move the payoff vertically down.

- 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.

## Answer by Pandaaaaaaa (score 0)

https://quant.stackexchange.com/a/24942

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 $$ C_{\text{bull}} = -\frac{(K_2 - K_1)}{2} + (S_T - K_1)_+ - (S_T - K_2)_+ $$