The aim of this paper is to develop an algorithm for calculating and plotting payoff of option strategies for a portfolio of path independent vanilla and exotic options. A general algorithm for calculating the vector matrix for any arbitrary combination strategy is also developed for some of the commonly option trading strategies.option trading strategy, payoff, vector, vanilla and exotic option

Johar ARCHIT

Sinha PANKAJ

English

Research Papers in Economics

  273 
ALGORITHM FOR PAYOFF CALCULATION FOR  
OPTION TRADING STRATEGIES USING VECTOR TERMINOLOGY 
 
Sinha PANKAJ 
University of Delhi, Delhi, India 
pankaj sinha@fms.edu 
Johar ARCHIT 
Institute of Technology, New Delhi, India 
Department of Computer Engineering Netaji Subhas 
 architjohar@gmail.com 
 
Abstract: 
The aim of this paper is to develop an algorithm for calculating and plotting payoff of option strategies 
for a portfolio of path independent vanilla and exotic options. A general algorithm for calculating the vector 
matrix  for  any  arbitrary  combination  strategy  is  also  developed  for  some  of  the  commonly  option  trading 
strategies. 
 
Keywords: option trading strategy, payoff, vector, vanilla and exotic option. 
 
JEL-Classification: C63, C88, C02, G00 
 
1. Introduction 
Hull (2009) discusses the payoffs for long and short positions in Call and Put options by using 
algebraic  techniques.  J.S. Chaput  and  L.H.  Ederington  (2003), Natenberg (1994)  and  Hull (2009) 
contain the bibliographies and survey of literature on the theoretical background of option strategies 
for  the  path  independent  vanilla  and  exotic  options  such  as  European,  Bermuda,  Forward  Start, 
Digital/Binary and Quanto options. There are various open source option strategy calculators like 
“Option” [4] that only rely on algebraic, analytical and graph superposition techniques to plot graphs 
for overall profit/loss. We in this paper develop an algorithm using vector terminology to plot the final 
profit/loss graph of various option strategies. 
 
2.1. Option strategies using vector notation 
For a spot price ST  at time T and a strike price K, the payoff for a long position in call option is 
given by Max(ST K,0) and the payoff is Min(ST K,0) for the short position in the call option. Similarly 
the payoff for a long position in put is Max (K ST, 0) whereas it is Min (ST K, 0) for a short position in 
the put option. We can represent a vector payoff matrix for any option strategy as a 2xN matrix. 
 
Vector  V1  V2  ….  Vn 
Strike 
Price 
K1  K2  …..  Kn 
 
In  the  above  matrix  the  strike  prices  K1,K2,  …..Kn      for  combination  of  options  are  in  the 
ascending order, i.e., K1<K2<…..<Kn.  .  The vector  i V  can be interpreted as slope of the payoff graph 
of option strategy. By default the smallest strike price is always taken to be zero i.e. K1=0.The vector 
is always an integer in the interval (–∞, ∞) .We can interpret the above matrix in terms of slope of the 
profit/loss curve obtained for option strategies. 
 



= >
< < <
=
+
n i   and   K K for    , V
n i   and   for    , V
i i
1    i i i K K K
slope  
 
Vector matrix for long and short position is given by    274 
Long Position 
V1  V2  ….  Vn 
K1  K2  …..  Kn 
Short Position 
 V1   V2  ….   Vn 
K1  K2  …..  Kn 
 
Using the above vector notation we can represent long and short position in call option as 
under:
Long call 
0  +1 
0  K1 
 
Short Call 
0   1 
0  K1 
For long position in call, profit/loss curve has two slopes 0 and +1 whereas for a short position 
the slope of profit/loss curve has two slopes 0 and –1. 
 
 
 
Figure 1. Long Position in Call Option 
 
 
Figure 2. Short position in Call Option 
   275 
 
Similarly, the vector matrix for long and short position in put options are: 
Long Put 
 1  0 
0  K1 
 
Short Put 
+1  0 
0  K1 
For long position in stock, the slope of profit/loss curve is +1 and strike price is assumed to 
be zero whereas for short position in stock, the slope of profit/loss curve is  1 and strike price is 
assumed to be zero. The vector matrix notation is given as: 
 
Long Stock 
+1 
 0 
    
Short Stock 
 1 
0 
When we trade in n units of options using a particular option strategy, the entire vector row 
is multiplied by n. 
 
n*V1  n*V2  ….  n*Vn 
K1  K2  …..  Kn 
 
The data set for a portfolio using n option strategies can be represented as 
 
Strategy 1 
V11  V12 
K11  K12 
 
Strategy 2 
V21  V22 
K21  K22 
… 
… 
… 
Strategy i 
Vi1  Vi2  ….  Vij  …. 
Ki1  Ki2  …..  Kij  …. 
…. 
…. 
…. 
Strategy n 
Vn1  Vn2  …  Vnm 
Kn1  Kn2  ….  Knm 
 
Note that the number of columns in each option strategy can be different. We can use the 
above derived  vector  matrices  to  form  profit/loss  function  for  any  combination  of  option 
strategies using the following algorithm: 
 
Algorithm  
To plot the overall payoff strategy we need the initial Y intercept of the strategy apart from 
the resultant vector matrix. This Y intercept can be calculated using matrices of length greater 
than one using the formula 
Yint = ∑ (  1*Vector(A[j])*Strike_price(A[j+1]) ) 
Yint = Yint + Net_Premium_Paid 
   276 
 
Step 1 
For I ￿ 1 to no_of_options 
  For j ￿ 1 to length_of_option_matrix 
Insert A[j] in Result_matrix in sorted increasing order on the basis of  
Strike_price(A[j]). 
 
Step 2 
For k ￿ 1 to length_of_Result_matrix 
  Vector(B[k])=0 
For I ￿ 1 to no_of_options  
    For j ￿ 1 to length_of_option_matrix 
      If Strike_price(B[k]) = Strike_price(A[j]) 
        Vector(B[k]) = Vector(B[k])+ Vector(A[j]) 
      ElseIf j < length_of_option_matrix 
If Strike_price(A[j]) < Strike_price(B[k]) < Strike_price(A[j+1]) 
        Vector(B[k]) = Vector(B[k])+ Vector(A[j]) 
      Else 
        Vector(B[k]) = Vector(B[k])+ Vector(A[j]) 
 
Step 3 
For I ￿ 1 to no_of_options 
  j=1 
  If length_of_option_matrix > 1 
Yint = Yint +  1 * Vector(A[j]) * Strike_price(A[j+1]) 
Yint = Yint + NetPremium 
 
Step 4  
For k ￿ 1 to length_of_Result_matrix – 1 
Plot line with slope Vector(B[k]) and Y Intercept Yint 
between points Strike_price(B[k]) and Strike_price(B[k+1])  
ypoint=Vector(B[k])*( Strike_price(B[k+1]) – Strike_price(B[k]) ) + 
Yint 
Yint = ypoint    Vector(B[k+1])* Strike_price(B[k+1] 
k = length_of_Result_matrix 
Plot line with slope Vector(B[k]) between points Strike_price(B[k]) and 
infinity 
 
The source code for the above algorithm is written and implemented on VC++.Net 2005 
using open source graph plotting utility Gnuplot. 
 
Illustration 1: An investor buys $3 put with strike price $35 and sells for $1 a put with a 
strike price of $30. (Example 10.2, page 224 given in Hull [1]) 
 
The above data can be represented as 
 
Buy Put                              + 
 1  0 
0  35 
 Sell Put                             =   
+1  0 
0  30 
 Payoff(Bear Spread) 
0   1  0 
0  30  35 
 
Initial Y intercept is  1*( 1*35) +  1*(1*30) – 3 + 1 = 35  30  3 + 1 = 3 
 
One can use the following form to input the data of his/her option strategy: 
   277 
 
 
Figure 3. Input Screen 
 
The following is the output of the final payoff of combination of option strategy in vector 
notation as discussed above. 
 
 
 
Figure 4. Vector Payoff Matrix 
 
The algorithm gives the following resultant profit/loss graph of the above combination of 
option strategies in the form of a bear put spread. 
   278 
 
 
Figure 5. Payoff Graph 
 
The loss is $2 if stock price is above $35 and the profit is $3 if stock price below $30. 
 
2.2. Some More Complex Strategies 
The following are the vector matrices for some of the commonly traded strategies: 
 
Long Combo 
(0<K1<K2) 
Sell Put                           + 
+1  0 
0  K1 
 
Buy Call                              = 
0  +1 
0  K2 
 
Long Combo 
+1  0  +1 
0  K1  K2 
Long Straddle 
Buy Put                           + 
 1  0 
0  K1 
 
Buy Call                              = 
0  +1 
0  K1 
 
Long Straddle 
 1  +1 
0  K1 
Short Straddle 
The vector matrix of short straddle is negative of that of long straddle 
 
+1   1 
0  K1 
 
Strip 
Buy call                             +                                  
0  +1 
0  K1 
 
Buy 2 puts                           = 
 2  0 
0  K1 
 
Strip 
 2  +1 
0  K1 
 
   279 
 
Strap 
Buy 2 calls                          +                               
0  +2 
0  K1 
 
Buy put                               = 
 1  0 
0  K1 
 
Strap 
 1  +2 
0  K1 
Long Strangle 
(0<K1<K2) 
Buy put                               +                             
 1  0 
0  K1 
 
Buy call                             = 
0  +1 
0  K2 
 
Long Strangle 
 1  0  +1 
0  K1  K2 
Short Strangle 
The vector matrix of short strangle is negative of that of short strangle. (0<K1<K2) 
+1  0   1 
0  K1  K2 
 
Collar 
(0<K1<K2) 
Long Stock                          + 
+1 
 0 
 
Buy Put                       + 
 1  0 
0  K1 
 
Sell  call                        = 
0   1 
0  K2 
Collar 
0  +1  0 
0  K1  K2 
 
Box Spread 
(0<K1<K2) 
Buy Call              +          
0  +1 
0  K1 
 
Sell call               +   
0   1 
0  K2 
 
Sell Put            +                      
+1  0 
0  K1 
 
Buy  Put            =             
 1  0 
0  K2 
Box Spread 
0  0  0 
0  K1  K2 
 
Long Call Butterfly 
(0<K1<K2<K3) 
Buy Call                       + 
0  +1 
0  K1 
 
Sell 2 call                 + 
0   2 
0  K2 
 
Buy Call                             =     
0  +1 
0  K3 
Long Call Butterfly 
0  +1   1  0 
0  K1  K2  K3   280 
Short Call Butterfly 
The vector matrix of short call butterfly is negative of that of long call butterfly (0<K1<K2<K3) 
0   1  +1  0 
0  K1  K2  K3 
 
Long Call Condor 
 (0<K1<K2<K3<K4) 
Buy Call              +          
0  +1 
0  K1 
 
Sell call               +   
0   1 
0  K2 
 
Sell Call            +                      
0   1 
0  K3 
 
Buy  Call      =                               
0  +1 
0  K4 
 
Long Call Condor 
0  +1  0   1  0 
0  K1  K2  K3  K4 
 
Short Call Condor 
The vector matrix of short call condor is negative of that of long call condor (0<K1<K2<K3<K4) 
0   1  0  +1  0 
0  K1  K2  K3  K4 
 
Illustration 2: Let a  certain stock be selling at $77. An investor feels that significant 
change in price is un likely in the next 3 months. He observes market price of 3 month calls as 
Strike Price($)  Call Price($) 
75  12 
80  8 
85  5 
 
The investor decided to go long in two calls each with strike price $75 and $85 and writes 
two calls with strike price $80. Payoff for different levels of stock prices is given as: 
 
 
 
Figure 6. Payoff Graph Volume IV/ Issue 2(8)/ Summer 2009      
 
  281 
 
 
 
Figure 7. Vector Payoff Matrix 
 
The profit /loss when stock price is at maturity is  
 
Stock Price($)  Profit/Loss($) 
65   1 
68   1 
73   1 
78  2 
83  1 
 
3. References 
[1] Hull, J.C., (2009), Options, Futures, and Other Derivatives, Prentice Hall 
[2] Natenberg, S., (1994), Option Volatility and Pricing Strategies: Advanced Trading Techniques for 
Professionals McGraw Hill Professional Publishing 
[3] Chaput, J. S. and Ederington L. H., (2003) Option Spread and Combination Trading, in: Journal of 
Derivatives, 10, 4, pp 70 88. 
[4] http://sourceforge.net/projects/option 
  
 

Option Spread and Combination Trading, in:

Option Volatility and Pricing Strategies: Advanced Trading Techniques for Professionals

Options, Futures, and Other Derivatives,

ALGORITHM FOR PAYOFF CALCULATION FOR OPTION TRADING STRATEGIES USING VECTOR TERMINOLOGY

http://www.jaes.reprograph.ro/articles/summer2009/PankajS_ArchitJ.pdf

ALGORITHM FOR PAYOFF CALCULATION FOR OPTION TRADING STRATEGIES USING VECTOR TERMINOLOGY

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