There are call and put options on the same underlying asset, with the same expiry, $T$, and with strikes $K_c=(k_c^1, k_c^2, \ldots, k_c^m)$ and $K_p=(k_p^1, k_p^2, \ldots, k_p^m)$, $S_t$ is a price of the underlying asset at calendar time, $0\leq t \leq T$, $\hat{S}_T$ is a forecasting price of the underlying asset at the time $T$.

Suppose you would like to construct a negative cost portfolio at the start time $t=0$ by combining long and short positions in put and call European-style contracts based on the same underlying asset with different strikes.

Let $X_c=(x_1^c, x_2^c, \ldots, x_m^c)$, $X_p=(x_1^p, x_2^p, \ldots, x_m^p)$ be the number of unit of call and put options, with $x_i^c, x_i^p>0$ for buying, $x_i^c, x_i^p<0$ for selling. Denote by $C=(c_1, c_2, \ldots, c_m)$ and $P=(p_1, p_2, \ldots, p_m)$ the market prices for buying and selling of call and put options respectively.

Then the cost portfolio, $M$, at the time of purchase, $t=0$, is expressed by: $$M=\sum_{i=1}^{m} x_i^c \cdot c_i+ x_i^p \cdot p_i.$$

For the negative cost portfolio $M$ should be negative, i.e. $M<0$.

Denote by $V$ the payoff function $V(S_T, K_c, K_p, T, X)=\sum_{i=1}^{m} x_i^c (\hat{S}_T-k_c^i)^+ + x_i^p (k_p^i - \hat{S}_T)^- - M$

How to show that the payoff $V(T)$ at time $T$ is non-negative ($V≥0$)?

The example of the portfolio and payoff at $t=0$ and $t=T$ in R is below

X_c <- c(-3, -7, 2, 0, -2, 10);
X_p <- c( 1,  1, 7, 4, -5, -8);
K_c <- c(8050, 8150, 8250, 8350, 8400, 8500);
K_p <- c(7850, 7950, 8050, 8150, 8250, 8350);
S_hat_T <- 8400;
C <- c(48.0, 10.0,  0.9,   0.5,   0.3,   0.2); 
P <- c( 2.2, 10.5, 35.0, 100.0, 183.0, 343.0);

M <- sum(X_c*C) + sum(X_p*P);  
# [1] -3212.1
V <- sum(X_c*max(S_hat_T-K_c,0)) + sum(X_p*max(K_p - S_hat_T,0))  
#[1] 0
#[1] 3212.1

$M=-3212.1<0$, therefore, the portfolio having negative cost at time $t=0$, and $V=0$, at the time $T$.

My question is what is wrong with what I am doing?

Update. After noob2's comment. I tried to find the rule of the cost portfolio calculation, and the book by Eliezer Z Prisman (2000) Pricing derivative securities: an interactive, dynamic environment with Maple V and Matlab was found.

  • 2
    $\begingroup$ "negative cost portfolio (i.e. the portfolio is arbitrage)" is not true. If the portfolio offers a negative payout in at lest one of the future state of the world, then its "cost" (or rather its value today) may well be negative - depending on the likelihood of this negative payout and its size - without it being an arbitrage opportunity. $\endgroup$ – Quantuple Oct 13 '16 at 6:35
  • 1
    $\begingroup$ In general there are two kinds of arbitrage: Type I - you create a portfolio having negative cost today and the payoff in every future state is $\ge 0$, Type II - you create a portfolio having positive cost K today and in every future state the payoff is $\ge K$ with at least one state having payoff strictly $> K$. $\endgroup$ – noob2 Oct 13 '16 at 17:27
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    $\begingroup$ @Nick what you are doing wrong is that what you call the payoff is not a payoff but rather à P&L (payoff - premium paid). Consider a European call option for illustration purpose. Its price, or premium today is $C_t$ (this you will pay if you are to go long). The payoff at maturity $T$ is $(S_T-K)^+$ (this you will receive assuming you are long). Thus the overall profit and loss of going long the option at $T$ will be $(S_T-K)^+ - C_t$ (payout - premium). Looking at your code, $M$ is the premium, $V$ the payoff, $V-M$ the PnL $\endgroup$ – Quantuple Oct 14 '16 at 7:23
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    $\begingroup$ Let the Strikes be $\{K_1,K_2,\cdots,K_n\}$ in this order, the payoff function is piecewise linear. You can investigate the $n+1$ intervals $0\le S\le K_1$, $K_1\le S \le K_2$, ..., $K_n \le S \le \infty$ separately. (The last, semi-infinite, interval is a little trickier than the others. Generally in a finite linear segment just looking at the two endpoints tells you all you need to know about the segment (the max, the min, whether it crosses 0, etc.)). $\endgroup$ – noob2 Oct 14 '16 at 12:28
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    $\begingroup$ To determine the slope of the payoff function in $K_n \le S \le \infty$ where $K_n$ is the largest strike, you could evaluate the payoff function at $K_n$ and $K_n + 1$. Alternatively you could count the number of calls in your portfolio (a long call contributes +1 to the slope and a short call contributes -1, the puts contribute nothing). $\endgroup$ – noob2 Oct 14 '16 at 12:55

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