# Linear combination of Payoffs using Black-Scholes

Write the payoffs in Figure 3.8 as linear combination of call options and derive a closed form formula for the Black-Scholes price, the Delta, and the Gamma of them. All the Greeks of the option are also linear combination of these call option Greeks. For instance, $$\Delta(t,S) = \Phi(d_1(\tau,K_1,S)) - \Phi(d_1(\tau,K_2,S)) - \Phi(d_1(\tau,K_3,S)) + \Phi(d_1(\tau,K_4,S))$$

Partial Solution: For the strangle we have a pay off of $$(K - S_T)_{+} + (S_T - K)_{+}$$ Therefore the closed form solution of B-S price of option is $$V(\tau,S) = P(\tau,K,S) + C(\tau,K,S)$$ and the delta of the position is $$\Delta(\tau,S) = -\Phi(-d_1(\tau,K,S)) + \Phi(d_1(\tau,K,S))$$ Finally our gamma for this position is $$\Gamma(\tau,S) = \frac{\Phi'(d_1(\tau,K,S)) + \Phi'(d_1(\tau,K,S))}{S\sigma \sqrt{\tau}}$$

I guess my professor made a mistake in regards to the B-S closed form price: for the strange it is $$V(\tau,S) = (-S_0\Phi(-d_1) + e^{-rT}K\Phi(-d_2)) + (S_0\Phi(d_1) - e^{-rT}K\Phi(d_2))$$ and similar for the straddle

where $\tau = T - t$ not sure why we use $\tau$ any explanation of that would be great.

• I think the question wants you to replicate the payoffs with some payoff options, only different in strike. You can easily do that by long and short position. Mar 17 '16 at 1:54
• What are the values for the points intercept the X and Y axis? Mar 17 '16 at 12:40
• One thing that is confusing is that the LEFT payoff is a straddle, the RIGHT is a strangle; labels are reversed. Also, these diagrams represent combinations of calls AND PUTS, not just "call options". So the question is somewhat misphrased. Mar 17 '16 at 13:51
• Yea, sorry about that my teacher has a lot of errors in his problems his exam questions are usually not solvable and he can't even answer his own questions Mar 17 '16 at 17:53
• @Gordon I made an edit to the post Mar 17 '16 at 20:52

To express such payoff in mathematical form, it is better to use indicator functions. I assume that the bottom of graphs (i.e., the vertex for the left one and the bottom segment for the right side one) represents zero.

For the left-hand one, the payoff is given by \begin{align*} (K-S_T)\pmb{1}_{S_T \le K} + (S_T-K)\pmb{1}_{S_T \ge K} = (K-S_T)^+ + (S_T-K)^+, \end{align*} that is, a straddle that involves both a European call and put with the same strike price and maturity date.

For the right-hand one, the payoff is given by \begin{align*} (K_1-S_T)\pmb{1}_{S_T \le K_1} + (S_T-K_2)\pmb{1}_{S_T \ge K_2} = (K_1-S_T)^+ + (S_T-K_2)^+,\tag{1} \end{align*} that is, a strangle that involves both a European call and put with the same maturity date, but different strikes.

For valuation, as an example, let's consider (1). According to the Black-Scholes' formula, the value of Payoff (1) is given by \begin{align*} V=\Big[K_1 e^{-rT} \Phi(-d_2^1) - S_0 \Phi(-d_1^1)\Big] + \Big[S_0 \Phi(d_1^2) - K_2 e^{-rT} \Phi(d_2^2)\Big], \end{align*} where the first term is the value of the put option payoff $(K_1-S_T)^+$ and the second is the value of the call option payoff $(S_T-K_2)^+$. Here, \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*} The delta hedge ratio is the sum of deltas of the first put option and the second call options, that is, \begin{align*} \frac{\partial V}{\partial S_0} &= -\Phi(-d_1^1) + \Phi(d_1^2)\\ &=\Phi(d_1^1) + \Phi(d_1^2) - 1, \end{align*} and the gamma hedge ratio is the sum of gammas of the first put option and the second call options, that is, \begin{align*} \frac{\partial^2 V}{\partial S_0^2} &= \frac{\Phi'(d_1^1)}{S_0\sigma \sqrt{T}}+ \frac{\Phi'(d_1^2)}{S_0\sigma \sqrt{T}}\\ &=\frac{\Phi'(d_1^1) + \Phi'(d_1^2)}{S_0\sigma \sqrt{T}}. \end{align*}

• Does this answer provide the "Black-Scholes price, the Delta, and the Gamma of them"? Mar 17 '16 at 17:59
• Those are two vanilla options. You can then compute the price for each of them using Black-scholes' formula. Consequently the delta and gamma can be computed. Mar 17 '16 at 18:08
• I am not sure if I understand completely, unless you think I simply can't see it could you edit your answer to include the Black-Scholes formula and how we compute the delta and gamma? Mar 17 '16 at 18:17
• It is just a sum of two options, please try. Mar 17 '16 at 18:19
• When you say it is the sum of 2 options that sounds like the payoff of the option portfolio. I know how to compute delta if we are given the parameters: interest rate, volatility, price of stock, strike price. But I am not sure how to compute gamma Mar 17 '16 at 19:50