Options 1

This notebook shows payoff functions and pricing bounds of options.

Load Packages and Extra Functions



In [1]:

    
using Dates

include("jlFiles/printmat.jl")









    Out[1]:





printyellow (generic function with 1 method)



In [2]:

    
using Plots

#pyplot(size=(600,400))
gr(size=(480,320))
default(fmt = :svg)

Payoffs and Profits of Options

Let $K$ be the strike price and $S_m$ the price of the underlying at expiration of the option contract.

The call and put profits (at expiration) are

$\text{call profit}_{m}\ = \max\left( 0,S_{m}-K\right) - e^{my}C$

$\text{put profit}_{m}=\max\left( 0,K-S_{m}\right) - e^{my}P $,

where $C$ and $P$ are the call and put option prices (paid today).

The profit of a straddle is the sum of those of a call and a put.



In [3]:

    
Sₘ = 0:0.1:10          #possible values of underlying at expiration
K  = 5                 #strike price
C  = 0.4               #call price (just a number that I made up)
P  = 0.4               #put price 
(y,m) = (0,1)          #zero interest to keep it simple

CallProfit = max.(0,Sₘ.-K) .- exp(m*y)*C
PutProfit  = max.(0,K.-Sₘ) .- exp(m*y)*P


plot( Sₘ,[CallProfit PutProfit],
      linecolor = [:red :green],
      linestyle = [:dash :dot],
      linewidth = 2,
      label = ["call" "put"],
      ylim = (-1,5),
      legend = :top,
      title = "Profits of call and put options, strike = $K",
      xlabel = "Asset price at expiration" )









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In [4]:

    
StraddleProfit = CallProfit + PutProfit

plot( Sₘ,StraddleProfit,
      linecolor = :blue,
      linewidth = 2,
      label = "call+put",
      ylim = (-1,5),
      legend = :top,
      title = "Profit of a straddle, strike = $K",
      xlabel = "Asset price at expiration" )









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Put-Call Parity for European Options

A no-arbitrage condition says that $ C-P=e^{-my}(F-K) $ must hold, where $F$ is the forward price. This is the put-call parity.

When the underlying asset has no dividends (until expiration of the option), then $F=e^{my}S$.



In [5]:

    
S = 42                     #current price of underlying etc
K = 38
m = 0.5
y = 0.05

C = 5.5                   #assume this is the price of a call option(K)

F = exp(m*y)*S
P = C - exp(-m*y)*(F.-K)

printlnPs("C and P (for the same strike price) are: ",C,P)









    



C and P (for the same strike price) are:      5.500     0.562

Pricing Bounds

The pricing bounds for (American and European) call options are

$\begin{align} C & \leq e^{-my}F\\ C & \geq\max[0,e^{-my}(F-K)] \end{align}$

The pricing bounds for (European) put options are

$\begin{align} P_{E} & \leq e^{-my}K\\ e^{-my}(K-F) & \leq P_{E} \end{align}$



In [6]:

    
S = 42
m = 0.5
y = 0.05

F = exp(m*y)*S

K = 38

C_Upper = exp(-m*y)*F
C_Lower = max.(0,exp(-m*y)*(F.-K))

printlnPs("Pricing bounds for European call option with strike $K",C_Lower,C_Upper)









    



Pricing bounds for European call option with strike 38     4.938    42.000



In [7]:

    
K_range = 30:0.5:50             #pricing bounds for many strike prices
n = length(K_range)

C_Upper = exp(-m*y)*F
C_Lower = max.(0,exp(-m*y)*(F.-K_range))

P_Upper = exp(-m*y)*K_range
P_Lower = max.(0,exp(-m*y)*(K_range.-F));



In [8]:

    
plot( K_range,[C_Upper*ones(n) C_Lower],
      linecolor = [:green :blue],
      linewidth = 2,
      linestyle = [:dash :dot],
      label = [" C <= exp(-m*y)F " "C >= max[0,exp(-m*y)(F-K)]"],
      ylim = (-1,S+1),
      legend = :right,
      title = "Price bounds on European call options",
      xlabel = "Strike price" )









    Out[8]:



In [9]:

    
plot( K_range,[P_Upper P_Lower],
      linecolor = [:green :blue],
      linewidth = 2,
      linestyle = [:dash :dot],
      label = [" P <= exp(-m*y)K " " P >= max[0,exp(-m*y)(K-F)]  "],
      ylim = (-1,50),
      legend = :right,
      title = "Price bounds on European put options",
      xlabel = "Strike price" )









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