Forward Contracts

This notebook introduces forward payoffs and applies the forward-spot parity to different assets.

Load Packages and Extra Functions



In [1]:

    
using Dates

include("jlFiles/printmat.jl")
include("jlFiles/printTable.jl")









    Out[1]:





printTable2



In [2]:

    
using Plots

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

Present Value



In [3]:

    
y  = 0.05
m  = 3/4
Z  = 100
PV = exp(-y*m)*Z

printlnPs("PV of $Z: ",PV)









    



PV of 100:     96.319

Payoff of Forward Contract

Recall: the payoff of a forward contract (at expiration, $m$ periods ahead) is $S_{m}-F$



In [4]:

    
Sₘ = 0:15   #possible values of the underlying price at expiration
F  = 5
ForwardPayoff = Sₘ .- F

plot( Sₘ,ForwardPayoff,
      linecolor = :red,
      linewidth = 2,
      legend = false,
      title = "Payoff of forward contract (F = $F)",
      xlabel = "Asset price at expiration" )

hline!([0],linecolor=:black,line=(:dash,1))
vline!([F],linecolor=:black,line=(:dash,1))









    Out[4]:

Forward-Spot Parity

Recall: for an asset without dividends (at least until expiration of the forward contract), $F=e^{my}S$

For an asset with continuous dividends at the rate $\delta$, $F=e^{m(y-\delta)}S$



In [5]:

    
y   = 0.05         #interest rate
m   = 3/4          #time to expiration (in years)
S   = 100          #spot price now
F_A = exp(m*y)*S   #forward price

δ   = 0.01             #dividend rate 
F_B = exp(m*(y-δ))*S   #forward price

printblue("Forward prices:")
printTable([F_A F_B],["no dividends";"with dividends"],[""],width=16)









    



Forward prices:
    no dividends  with dividends
         103.821         103.045

Forward Price of a Bond

Recall: the forward price (with expiration $m$ of the forward) of a bond that matures in $n$ is $F=e^{my(m)}B(n)$, where $y(m)$ denotes the interest for an $m$-period loan.

By definition, $1/B(m)= e^{my(m)}$. Combine to get

$F=B(n)/B(m)$



In [6]:

    
m  = 5              #time to expiration of forward
n  = 7              #time to maturity of bond 
ym = 0.05           #interest rates 
yn = 0.06
Bm = exp(-m*ym)     #bond price now
Bn = exp(-n*yn)
F  = Bn/Bm          #forward price a bond maturing in n, delivered in m<n
                                             

printblue("bond and forward prices: ")
xx = [Bm,Bn,F]
printTable(xx,["price"],["$m-bond","$n-bond","$m->$n forward"])









    



bond and forward prices: 
                 price
5-bond           0.779
7-bond           0.657
5->7 forward     0.844

Covered Interest Rate Parity

The "dividend rate" on foreign currency is the foreign interest rate $y^*$ (since you can keep the foreign currency on a foreign bank account). The forward-spot parity then gives $F=e^{m(y-y^*)}S$.

We also calculate the return on a "covered" strategy: (a) buy foreign currency (in $t=0$); (b) lend it abroad (in $t=0$); (c) enter a forward on the domestic currency (in $t=0$); (d) pay forward price and get domestic currency (in $t=m$).



In [7]:

    
m     = 1            #time to expiration
y     = 0.0665       #domestic interest rate
ystar = 0.05         #foreign interest rate 
S     = 1.2          #exchange rate now

F = S*exp(m*(y-ystar))

printlnPs("Forward price of foreign currency: ",F)

TradingReturn = exp(ystar)*F/S - 1
logTR         = log(TradingReturn+1)

printlnPs("\nlog return on covered strategy, %:     ",100*logTR)
printlnPs("compare with domestic interest rate, %:",y*100)









    



Forward price of foreign currency:      1.220

log return on covered strategy, %:          6.650
compare with domestic interest rate, %:     6.650



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