ergebnis = 2 + 2 * 2
print(f'Ergebnis = {ergebnis:.2f}')

Ergebnis = 6.00

plot_FV()

print(f'FV(m = 100000) = {FV(100000):.4f}')
print(f'EXP(1)         = {np.exp(1):.4f}')

FV(m = 100000) = 2.7183
EXP(1)         = 2.7183

"""Interest rate conversion (translation)
"""
import numpy as np

r_isma = 0.1
r_mm = (r_isma*360/365)
r_sia = ((1+r_isma)**0.5-1)*2
r_cont = np.log(1+r_isma)

print(f'r_isma  = {r_isma:.4f}')
print(f'r_mm    = {r_mm:.4f}')
print(f'r_sia   = {r_sia:.4f}')
print(f'r_cont  = {r_cont:.4f}')

r_isma  = 0.1000
r_mm    = 0.0986
r_sia   = 0.0976
r_cont  = 0.0953

def pv(t, C_t, r_t):
    return C_t / (1 + r_t) ** t

def df(t, r_t):
    return 1 / (1+r_t)**t

print(f'pv(1, 110, 0.1)  = {pv(1, 110, 0.1):.2f}')
print(f'110 * df(1, 0.1) = {110 * df(1, 0.1):.2f}')

pv(1, 110, 0.1)  = 100.00
110 * df(1, 0.1) = 100.00

def dcf(cash_flows):
    P = 0
    for t, C_t, r in cash_flows:
        P += pv(t, C_t, r)
    return P

cash_flows = [(1, 10, 0.1), (2, 10, 0.1), (3, 110, 0.1)]
print(f'cash_flows = {cash_flows}')
print(f'dcf(cash_flows) = { dcf(cash_flows) :.2f}')

cash_flows = [(1, 10, 0.1), (2, 10, 0.1), (3, 110, 0.1)]
dcf(cash_flows) = 100.00

plot_irr_cpn()

def annuity_factor(r, T):
    return (1 - (1 / (1 + r)**T ) ) / r

def pv_fix(r, T=5, Cpn=0.03):
    return annuity_factor(r, T) * Cpn + 1 / (1 + r) ** T

def pv_float(r, T=5):
    return (1 - 1 / (1 + r) ** T) + 1 / (1 + r) ** T

def plot_rrisk():
    rates = np.linspace(-0.05,0.2,25)
    plt.figure('rrisk')
    plt.plot(rates*100, pv_fix(rates)*100, label="Festsatzanleihe 3% 5Y")
    plt.plot(rates*100, pv_float(rates)*100, label="Floater 5Y")
    plt.xlabel("Zins in %")
    plt.ylabel("Barwert in %")
    plt.legend()
    plt.show()

plot_rrisk()

plot_pdf()

def plot_ppe():
    # Defining the range for the x values (encompassing most of the significant probability mass of the distribution)
    x = np.linspace(-4, 4, 1000)
    pdf = norm.pdf(x)

    # Calculate the PPE
    quantile_95 = norm.ppf(0.95)

    # Calculate the EPE
    expected_positive = norm.expect(lambda x: x, lb=0, ub=np.inf)

    # Plotting
    plt.figure("ppe")
    plt.plot(x, pdf, color='black')
    plt.fill_between(x, pdf, where=(x > 0), color='steelblue', alpha=0.5, label='Positive exposure')
    plt.axvline(x=expected_positive, color='green', linestyle='--', label='Expected Positive Exposure (EPE, auch EE)')
    plt.axvline(x=quantile_95, color='red', linestyle='--', label='Potential Positive Exposure (95%)')
    plt.title('Mark to Market Value')
    plt.xlabel('Value')
    plt.ylabel('Probability Density')
    plt.legend(loc='upper left')
    plt.show()

plot_ppe()

def r_t(t, B_t):
    return (1 / B_t) ** (1 / t) - 1

def r_st(s, t, r_s, r_t):
    B_s = 1 / (1 + r_s) ** s
    B_t = 1 / (1 + r_t) ** t
    return (B_s / B_t) ** (1 / (t - s)) - 1

print(f'r_2  = r_t(2, 0.8116)       = { r_t(2, 0.8116)*100 :.2f}%')
print(f'r_3  = r_t(3, 0.7118)       = { r_t(3, 0.7118)*100 :.2f}%')
print(f'r_23 = r_st(2, 3, r_2, r_3) = { r_st(2, 3, r_t(2, 0.8116), r_t(3, 0.7118))*100 :.2f}%')

r_2  = r_t(2, 0.8116)       = 11.00%
r_3  = r_t(3, 0.7118)       = 12.00%
r_23 = r_st(2, 3, r_2, r_3) = 14.02%

def bootstrap(swap_rates):
    discount_factors = []
    for C_t in swap_rates:
        # See Slide 61 
        B_t = (1 - C_t * sum(discount_factors)) / (1 + C_t)
        #B_t = round(B_t, 4)
        discount_factors.append(B_t)
    return discount_factors

swap_rates = [0.10, 0.1095, 0.1185]  # Beispiel

discount_factors = bootstrap(swap_rates)
print(f'Discount factors: {[f"{d:.4f}" for d in discount_factors]}')

Discount factors: ['0.9091', '0.8116', '0.7118']

print(f'r_1  = { r_t(1, discount_factors[0])*100 :.2f}%')
print(f'r_2  = { r_t(2, discount_factors[1])*100 :.2f}%')
print(f'r_3  = { r_t(3, discount_factors[2])*100 :.2f}%')

r_1  = 10.00%
r_2  = 11.00%
r_3  = 12.00%

def MTM_receiver(discount_factors, C_old, s=0):
    discount_factors = [1] + discount_factors  # B_0 = 1!
    P_Fix = C_old * sum(discount_factors[(s+1)::])
    P_Var = discount_factors[s] - discount_factors[-1]
    return P_Fix - P_Var

print(f'MTM at inception = {MTM_receiver(discount_factors, C_swap(discount_factors)):.2f}%')

MTM at inception = 0.00%

def annuity_factor(r, T):
    return (1 - (1 / (1 + r)**T ) ) / r

def r2df(r, T):
    return 1 / (1 + r) ** T

print(f"Annuity factor = {annuity_factor(0.1, 5):.4f}")
print(f"Sum of the DFs = {sum(r2df(0.1, np.array([1,2,3, 4, 5]))):.4f}")

def pf_fix(r, Cpn=0.03):
    discount_factors = r2df(r, np.array([1,2,3,4,5]))
    return Cpn * sum(discount_factors) + discount_factors[-1]

def mtm_swap(r, original_swap_rate=0.03):
    discount_factors = r2df(np.array(r), np.array([1,2,3,4,5]))
    return original_swap_rate * sum(discount_factors) - (1 - discount_factors[-1])

def plot_synthetic_floater():
    rates = np.linspace(-0.05,0.2,25)
    pv_bond_position = pv_fix(rates)*100
    pv_swap_positions = [-mtm_swap(r)*100 for r in rates]
    pv_asset_swap = pv_bond_position + pv_swap_positions
    plt.figure('synthetic_floater')
    plt.plot(rates*100, pv_bond_position, label="Festsatzanleihe 3% 5Y")
    plt.plot(rates*100, pv_swap_positions, label="Payer swap 3% 5Y")
    plt.plot(rates*100, pv_asset_swap, label="Asset swap package")
    plt.axvline(x=3, color='r', linestyle='--')  # Add vertical line at x = 3%
    plt.xlabel("Zins (flach) in %")
    plt.ylabel("Barwert in %")
    plt.legend()
    plt.title("Asset Swap Package (Synthetic Floater)")
    plt.show()

Annuity factor = 3.7908
Sum of the DFs = 3.7908

plot_synthetic_floater()

def C_fwdswap(discount_factors, s=0):
    discount_factors = [1] + discount_factors  # B_0 = 1!
    return (discount_factors[s] - discount_factors[-1]) / sum(discount_factors[(s+1)::])

# this is the example from your ppt Slide 157
print(f'C_fwdswap in 0 for 3 = {C_fwdswap(discount_factors)*100 :.2f}%')
print(f'C_fwdswap in 1 for 2 = {C_fwdswap(discount_factors, 1)*100 :.2f}%')

C_fwdswap in 0 for 3 = 11.85%
C_fwdswap in 1 for 2 = 12.95%

def delta_vector(swap_rates, s=0, nominal=100e+6):
    C_market = C_fwdswap(bootstrap(swap_rates), s)
    shifted = np.array([swap_rates, ]*len(swap_rates))
    shifted = shifted + np.eye(len(swap_rates)) * 0.0001
    deltas = []
    for rates in shifted:
        delta_t = MTM_receiver(bootstrap(rates), C_market, s)
        delta_t = round(delta_t, 5) * nominal
        deltas.extend([delta_t])
    return deltas

# 1Y bis 8Y Swap Rates
swap_rates = np.array([0.0164, 0.0156, 0.0167, 0.0185, 0.0205, 0.0223, 0.0238, 0.025])
show_delta_vector()

forward_rate = ( (1 + 0.07 * 365/360) / (1 + 0.065 * 183/360) - 1 ) * 360 / 182
print(f'Forward Rate (FRA Satz) = {forward_rate*100:.2f}%')

Forward Rate (FRA Satz) = 7.26%

settlement_payment = (10000000 * (0.08 - 0.0726)*182/360) / (1 + 0.08 * 182/360)
print(f'Ausgleichszahlung in d3 = {settlement_payment:.2f} EUR')

Ausgleichszahlung in d3 = 35956.86 EUR

# Set the seed for reproducibility
np.random.seed(422)

# Generate the random process values with the correct standard deviation of 0.05
random_process = np.random.normal(0.05, 0.05, 8) * 100  # Standard deviation correctly set as 0.05

def plot_capfloor(is_cap=True):
    # Prepare the plot
    plt.figure()
    plt.plot(np.arange(6, 54, 6), random_process)
    
    # Adding a label "Hello" to grey scatter dots
    if is_cap:
        plt.title('Interest Rate Cap')
        # Plot the strike
        plt.axhline(y=5, color='r', linestyle='--', label='Strike (5%)')
        # Plot the rates
        first_label = True  # Helper to add the label only once
        for i, value in enumerate(random_process):
            if value > 5:
                if first_label:
                    plt.scatter(np.arange(6, 54, 6)[i], 5, color='red', s=100, marker='v', label='Hedged rate')
                    plt.scatter(np.arange(6, 54, 6)[i], value, color='grey', s=100, marker='o', label='Market rate')
                    first_label = False
                else:
                    plt.scatter(np.arange(6, 54, 6)[i], 5, color='red', s=100, marker='v')
                    plt.scatter(np.arange(6, 54, 6)[i], value, color='grey', s=100, marker='o')
    else:
        plt.title('Interest Rate Floor')
        # Plot the strike
        plt.axhline(y=3, color='r', linestyle='--', label='Strike (5%)')
        # Plot the rates
        first_label = True  # Helper to add the label only once
        for i, value in enumerate(random_process):
            if value < 3:
                if first_label:
                    plt.scatter(np.arange(6, 54, 6)[i], 3, color='red', s=100, marker='^', label='Hedged rate')
                    plt.scatter(np.arange(6, 54, 6)[i], value, color='grey', s=100, marker='o', label='Market rate')
                    first_label = False
                else:
                    plt.scatter(np.arange(6, 54, 6)[i], 3, color='red', s=100, marker='^')
                    plt.scatter(np.arange(6, 54, 6)[i], value, color='grey', s=100, marker='o')

    # Add labels and title
    plt.xlabel('Time in Months')
    plt.ylabel('Interest Rate (%)')
    plt.xticks(np.arange(6, 54, 6))
    plt.legend()

    # Display the plot
    plt.show()

plot_capfloor(is_cap=True)

plot_capfloor(is_cap=False)

# Beispiele:
EURUSD = 0.84226 * 1.6136  # EUR in GBP in USD
EURGBP = 1.3591  / 1.6136  # EUR in USD in GBP (Beachte die Richtung!)
print(f'EURUSD Cross Rate = {EURUSD:.4f}')
print(f'EURGBP Cross Rate = {EURGBP:.4f}')

EURUSD Cross Rate = 1.3591
EURGBP Cross Rate = 0.8423

r_EUR, r_USD = [0.04, 0.02]
EURUSD_forward = 1.05 * ((1 + r_USD*365/360) / (1 + r_EUR*365/360))
print(f'Forward = {EURUSD_forward:.4f} EURUSD')

Forward = 1.0295 EURUSD

# Export slides
from pathlib import Path
import subprocess
import sys

local_dir = Path.cwd()
subprocess.run([
    sys.executable, "-m", "jupyter", "nbconvert", str(local_dir / "vdt.ipynb"),
    "--to", "slides", "--SlidesExporter.reveal_scroll=True",
], check=True)

# Generate PDF without the default mathrsfs dependency
from pathlib import Path
import os
import subprocess
import sys

local_dir = Path.cwd()
env = os.environ.copy()
env["TEXMFVAR"] = "/tmp/texmf-var"

subprocess.run([
    sys.executable, "-m", "jupyter", "nbconvert", str(local_dir / "vdt.ipynb"),
    "--to", "latex", "--output", "vdt",
], check=True, env=env)

tex_path = local_dir / "vdt.tex"
tex_path.write_text(
    tex_path.read_text(encoding="utf-8").replace("\\usepackage{mathrsfs}\n", ""),
    encoding="utf-8",
)

for _ in range(3):
    subprocess.run([
        "xelatex", "-interaction=nonstopmode", tex_path.name,
    ], check=True, cwd=local_dir, env=env)

	1Y	3Y	8Y	Total
8Y			-73000.0	-73000.0
1Y7Y	10000.0		-73000.0	-63000.0
3Y5Y		29000.0	-73000.0	-44000.0

Einleitung¶

Zinsen (Usancen / 'Sprachen')¶

Discount factor (Barwertfaktor / Abzinsungsfaktor)¶

Discounted cash flow¶

Rendite¶

Zero/spot rate¶

Internal rate of return (IRR)¶

Referenzzinsen¶

LIBOR Skandal¶

Risikoübersicht¶

Trennung Zins/Liquidität!¶

Risiko bedeutet Ungewissheit¶

Derivate zur Risikosteuerung¶

Kontrahentenrisiko (Credit Risk)¶

Arbitrage¶

Zinsmanagement und Zinsderivate¶

Zinsswap¶

Plain vanilla swap pricing¶

Exkurs: Bootstrapping¶

Swap mark-to-market¶

Forward swap pricing¶

Delta vector¶

Forward Rate Agreement¶

Cap / Floor¶

Währungsmanagement und FX Derivate¶

FX Quotierung (Spot)¶

FX Termingeschäft¶

Devisenoptionen¶

Exkurs: Cross Currency Swap¶