Exposure Simulation Part III / CVA for Bermudan Swaptions

Screen Shot 2016-06-26 at 12.16.43

In this post we are going to simulate exposures of a bermudan swaption (with physical settlement) through an backward induction (aka American Monte-Carlo) method. These exposure simulations are often needed at the counterparty credit risk management. We will use this exposures to calculate the Credit Value Adjustment (CVA). One could easily extend this notebook to calculate the PFE or other xVAs. The implementation is based on the paper “Backward Induction for Future Values” by Dr. Antonov et al. .

We assume that the value of a derivate and the default probability of the counterpart are independent (no wrong-way risk). For our example our netting set consists of only one Bermudan swaption (but it can be easily extended). For simplicity we assume a flat yield termstructure.

In our example we consider a 5Y Euribor 6M payer swap at 3 percent. The bermudan swaption allows us to enter this swap on each fixed rate payment date.

Just as a short reminder the unilateral CVA of a derivate is given by:

CVA = (1-R) \int_0^Tdf(t)EE(t)dPD(t),

with recovery rate R, portfolio maturity T (the latest maturity of all deals in the netting set), discount factor df(t) at time t, the expected exposure EE(t) of the netting set at time t and the default probability PD(t).

We can approximate the integral by the discrete sum

CVA = (1-R) \sum_{i=1}^n df(t_i)EE(t_i)(PD(t_{i-1})-PD(t_i)).

In one of my previous posts we calculated the CVA of a plain vanilla swap. We performed the following steps:

  1. Generate a timegrid T
  2. Generate N paths of the underlying market values which influences the value of the derivate
  3. For each point in time calculate the positive expected exposure
  4. Approximate the integral

As in the plain vanilla case we will use a short rate process and assume its already calibrated.

The first two steps are exactly the same as in the plain vanilla case.

But how can we calculate the expected exposure of the swaption?

In the T-forward measure the expected exposure is given by:

EE(t) = P(0,T) E^T[\frac{1}{P(t,T)} \max(V(t), 0)]

with future value of the bermudan swaption V(t) at time t. The npv of a physical settled swaption can be negative at time t if the option has been exercised  before time t  and the underlying swap has a negative npv at time t.

For each simulated path we need to calculate the npv of the swaption V(t_i,x_i) conditional the state x_i  at time $t_i$.

In the previous post we saw a method how to calculate the npv of a bermudan swaption. But for the calculation of the npv we used a Monte Carlo Simulation. This would result again in a nested Monte-Carlo-Simulation which is, for obvious reasons, not desirable.

If we have a look on the future value, we see that is very much the same as the continuation value in the bermudan swaption pricing problem.

But instead of calculate the continuation values only one exercises date we calculate it now on each time in our grid. We use a the same regression based approximation.

We approximate the continuation value through some function of  current state of the stochastic process x_i:

V(t_i, x_i) \approx f(x_i).

A common choice for the function is of the form:

f(x) = a_0 + a_1 g_1(x)+ a_2 g_2(x) + \dots a_n g_n(x)

with g_i (x) a polynom of the degree i.

The coefficients of this function f are estimated by the ordinary least square error method.

We can almost reuse the code from the previous post, only a few extensions are needed.

We need to add more times to our time grid (we add a few more points in time after the maturity to have some nicer plots):

date_grid = [today + ql.Period(i, ql.Months) for i in range(0,66)] + calldates + fixing_dates

When we exercise the option we will enter the underlying swap. Therefore we need to update the swaption npvs to the underlying swap npv for all points in time after the exercise time on each grid:

if t in callTimes:
cont_value = np.maximum(cont_value_hat, exercise_values)
swaption_npvs[cont_value_hat < exercise_values, i:] = swap_npvs[cont_value_hat < exercise_values, i:].copy()

For our example we can observe the following simulated exposures:

Screen Shot 2016-06-26 at 12.16.43

If we compare it with the exposures of the underlying swap, we can see that the some exposure paths coincide. This is the case when the swaption has been exercised. In some of this cases we observe also negative npvs after exercising the option.

Screen Shot 2016-06-26 at 12.16.35

For the CVA calculation we need the positive expected exposure, which can be easily calculated:

swap_npvs[swap_npvs<0] = 0
swaption_npvs[swaption_npvs<0] = 0
EE_swaption = np.mean(swaption_npvs, axis=0)
EE_swap = np.mean(swap_npvs, axis=0)

Screen Shot 2016-06-26 at 12.16.18.png

Given the default probability of the counterparty as a  default termstructure we can now calculate the CVA for the bermudan swaption:

# Calculation of the default probs
defaultProb_vec = np.vectorize(pd_curve.defaultProbability)
dPD = defaultProb_vec(time_grid[:-1], time_grid[1:])

# Calculation of the CVA
recovery = 0.4
CVA = (1-recovery) * np.sum(EE_swaption[1:] * dPD)

As usual the complete notebook is available here.

Exposure Simulation / CVA and PFE for multi-callable swaps Part II

RegressionPlot

Hey Everyone,

today I want to continue my last post and show you today how to calculate the NPV of a bermudan swaption through Monte-Carlo simulation. The techniques I will show you in this post can be easily extended to simulate exposure paths of a multi callable swap. This exposure simulations can be used for a xVA calculation like the Credit Value Adjustment (CVA) or Potential Future Exposure (PFE).

A physical settled Bermudan swaption is a path dependent derivate. The option has multiple exercise dates and the swaption holder has the right to enter the underlying swap at any of this exercise dates.

On each of the exercise dates the holder have to decide whether it is optimal to exercise the option now or continue to hold the option and may exercise it on a later date.

Lets consider a bermudan swaption with two exercise dates.

At the latest exercise date T the payoff is the well known european swaption payoff

V(T) = \max(S(T), 0),

where S(T) is the npv of the underlying swap at time t.

At the first exercise date t_i \le T the npv of the swaption given by

V(t_i) = \max(S(t_i), N(t_i) E [ \frac{V(T)} {N(T)} | F_{t_i}].

and so to the NPV at time 0

V(0) = N(0) E[\frac{V(t_i)}{N(t_i)} | F_0].

One way to solve problem is performing a Monte-Carlo-Simulation. But a naive Monte Carlo approach would require a nested Monte-Carlo Simulation on each path to calculate the continuation value
C(t_i ) = N(t_i) E[\frac{V(T)}{N(T)} | F_{t_i} ] at time t_i.

Lets say we use 100.000 samples in our simulation, so a bermudan swaption with two exercise dates would require 100.000 x 100.000 samples. Which is very time consuming and grows exponential with the number of exercise dates.

Instead of calculate the continuation value also with a Monte-Carlo simulation we will use a approximation. We use the approximation and algorithm developed by Longstaff and Schwarz.

We approximate the continuation value through some function of some state variable (e.g swap rate, short rate, etc…) x_i:

C(t_i) \approx f(x_i).

A common choice for the function is of the form:

f(x) = a_0 + a_1 g_1(x)+ a_2 g_2(x) + \dots a_n g_n(x)

with g_i (x) a polynom of the degree i.

Lets choice g_i (x)= x^{i-1} and n=5 for our example.

The coefficients of this function f are estimated by the ordinary least square error method. Thats why some people call this method also the OLS Monte-Carlo Method.

So how does the the algorithm look like?

We would simulate n paths of our stochastic process (in our case the short rate process) and go backward through time (backward induction).

We start at the last exercise date T and calculate for each path the deflated terminal value of the swaption V_T (exactly like we did in the european case).

Then we go back into time to the next exercise date and approximate the needed continuation value by our ordinary least square estimate.

Our choice for the state variable is the state variable of the short rate process itself.

We choice our coefficient so that the square error of f(x_i) - V(T) is minimized, where x_i is a vector of all simulated states over all paths and V(T) the corresponding vector of the calculated npvs at time T from the previous step.

Now we use this coefficients to calculate the continuation value on each path by inserting the state of the current path into the formula.

On path j the deflated value of the swaption will be

V_{i,j} = \frac{1}{N(t-i, x_{i,j})} max(S(t_i, x_{i,j}), N(t_i, x_{i,j}) C(x_{i,j})

where x_{i,j} is the the state of the process at time i on path j.

The npv at time 0 is then

V(0) = N(0) \frac{1}{n} \sum_j=0^n V_{i,j}.

Some remarks:

  • One could also perform a 2nd MC Simulation (forward in time). First estimate the needed coefficients with a smaller numbers of paths. After train the model with this small set generate a larger sets of paths and go straight forward in time and use the approximation of the continuation value of the first run. On each path you only need to go forward until the holder will exercise the option.
  • The exercise time by this approach is only approximatively optimal, since its uses a approximation of the continuation value. Therefore our price will be an (asymptotic) lower bound of the real price.
  • The choice of appropriate functions g_i and the degree n is not obvious and can be tricky (see below).

Now lets have a look how this algorithm could be implemented in Python and Quantlib.

We use the notebook from my previous post as our starting point. We use the same yield curves, model (Gaussian short rate model) and the same underlying swap. The underlying 5y swap starts in 1Y.

We setup a bermudan swaption with 2 exercise dates, the first exercise date is on the start date of the swap and the 2nd date on the start date of the 2nd fixed leg accrual period.

Lets add both dates in our list of exercise dates:

calldates = [ settlementDate, 
              euribor6m.valueDate(ql.Date(5,4,2017))
            ]

We need to evaluate the value of the underlying swap at both exercise dates on each simulated path. To do that we introduce two new auxiliary functions. The first gives us all future (relative to an arbitrary evaluation time t) payment times and amounts from the fixed rate leg of the swap:

def getFixedLeg(swap, t):
    """
    returns all future payment times and amount of the fixed leg of the underlying swap

    Parameter:
        swap (ql.Swap)
        t (float) 

    Return:
        (np.array, np.array) (times, amounts)

    """
    fixed_leg = swap.leg(0)
    n = len(fixed_leg)
    fixed_times=[]
    fixed_amounts=[]
    npv = 0
    for i in range(n):
        cf = fixed_leg[i]
        t_i = timeFromReference(cf.date())
        if t_i > t:
            fixed_times.append(t_i)
            fixed_amounts.append(cf.amount())
    return np.array(fixed_times), np.array(fixed_amounts)

The 2nd will give us all future payment times, accrual start and end times, notionals, gearings and day count fractions of the floating leg. We need all this information to estimate the fixing of the float leg.

def getFloatingLeg(swap, t):

    float_leg = swap.leg(1)
    n = len(float_leg)
    float_times = []
    float_dcf = []
    accrual_start_time = []
    accrual_end_time = []
    nominals = []
    for i in range(n):
        # convert base classiborstart_idx Cashflow to
        # FloatingRateCoupon
        cf = ql.as_floating_rate_coupon(float_leg[i])
        value_date = cf.referencePeriodStart()
        t_fix_i = timeFromReference(value_date)
        t_i = timeFromReference(cf.date()) 
        if t_fix_i >= t:
            iborIndex = cf.index()

            index_mat = cf.referencePeriodEnd()
            # year fraction
            float_dcf.append(cf.accrualPeriod())
            # calculate the start and end time
            accrual_start_time.append(t_fix_i)
            accrual_end_time.append(timeFromReference(index_mat))
            # payment time
            float_times.append(t_i)
            # nominals 
            nominals.append(cf.nominal())
    return np.array(float_times), np.array(float_dcf), np.array(accrual_start_time), np.array(accrual_end_time), np.array(nominals)

With these two function we can evaluate the the underlying swap given the time and state of the process:

def swapPathNPV(swap, t):
    fixed_times, fixed_amounts = getFixedLeg(swap, t)
    float_times, float_dcf, accrual_start_time, accrual_end_time, nominals = getFloatingLeg(swap, t)
    df_times = np.concatenate([fixed_times, 
                           accrual_start_time, 
                           accrual_end_time, 
                           float_times])
    df_times = np.unique(df_times)
    # Store indices of fix leg payment times in 
    # the df_times array
    fix_idx = np.in1d(df_times, fixed_times, True)
    fix_idx = fix_idx.nonzero()
    # Indices of the floating leg payment times 
    # in the df_times array
    float_idx = np.in1d(df_times, float_times, True)
    float_idx = float_idx.nonzero()
    # Indices of the accrual start and end time
    # in the df_times array
    accrual_start_idx = np.in1d(df_times, accrual_start_time, True)
    accrual_start_idx = accrual_start_idx.nonzero()
    accrual_end_idx = np.in1d(df_times, accrual_end_time, True)
    accrual_end_idx = accrual_end_idx.nonzero()
    # Calculate NPV
    def calc(x_t):
        discount = np.vectorize(lambda T: model.zerobond(T, t, x_t))
        dfs = discount(df_times)
        # Calculate fixed leg npv
        fix_leg_npv = np.sum(fixed_amounts * dfs[fix_idx])
        # Estimate the index fixings
        index_fixings = (dfs[accrual_start_idx] / dfs[accrual_end_idx] - 1) 
        index_fixings /= float_dcf
        # Calculate the floating leg npv
        float_leg_npv = np.sum(nominals * index_fixings * float_dcf * dfs[float_idx])
        npv = float_leg_npv - fix_leg_npv
        return npv
    return calc

This functions returns us a pricing function, which takes the current state of the process to price calculate the NPV of the swap at time t. The technique we use here is called closure function. The inner function is aware of the underlying swap and the ‘fixed’ time t and can access all variables defined in the outer function.

Remark on the pricing function
This is a very simple function. Its not possible to calculate the correct NPV of the swap for a time t between to fixing times of the floating leg. The current floating period will be ignored. So use that function only to evaluate the swap NPV only on fixing dates. We will extend this function to be capable to evaluate the NPV on any time t in the next post.

Use of the function at time t=0

SwapNPV at time t0

The generation of the time grid hasn’t changed from the last time. It just consists of three points.

Timegrid

Also the generation of our sample path is the same as last time. After we generate our path we calculate the deflated payoffs of our swap at time T:

pricer = np.vectorize(swapPathNPV(swap, time_grid[-1]))
cont_value = pricer(y[:,-1]) / numeraires[:,-1]
cont_value[cont_value < 0] = 0

First we generate a vectorised function of the pricing function and use it on the array of our sample paths y and then apply the maximum function on the result.

In the next step we go one step back in time and calculate the deflated exercise value of the swaption at that time:

pricer = np.vectorize(swapPathNPV(swap, time_grid[-2]))
exercise_values = pricer(y[:,-2]) / numeraires[:,-2]
exercise_values[exercise_values < 0] = 0

Now we estimate the coefficients of continuation value function. We use the library statsmodels and fit an OLS model to the data.

states = y[:, -2]
Y = np.column_stack((states, states**2, states**3, states**4))
Y = sm.add_constant(Y)
ols = sm.OLS(cont_value, Y)
ols_result = ols.fit()

With this coefficients we can now calculate the continuation value on each path, given the state:

cont_value_hat = np.sum(ols_result.params * Y, axis=1)

The deflated value of the swaption at the first exercise is the maximum out of exercise value and continuation value:

npv_amc = np.maximum(cont_value_hat, exercise_values)

The npv at time 0 is the mean of the simulated deflated npvs at the first exercise date times the value of the numeraire at time 0:

npv_amc = np.mean(npv_amc) * numeraires[0,0]

NPV Bermudan

To check the quality of our regression function f we can have a look on a scatter plot:

RegressionParameter

RegressionPlot

As we can see the regression function doesn’t seems to fit that good to the left tail. So we could either increase the degree of our function, try other polynomial function, change to another state variable or try piecewise regression functions.

As usual you can download the source code from my github account or find it on nbViewer.

In the next post we are going to use this regression based approach to generate exposure paths for a multi callable swap.

I hope you enjoy the post. Till next time.

Exposure simulation / PFE and CVA for multi-callable swaps / Bermudan swaptions… Part 1 of 3

In my previous posts we have seen a Monte-Carlo method to generate market scenarios and calculate the expected exposure, potential future exposure and credit value adjustment for a netting set of plain vanilla swaps. In the next three posts we will add multi-callable swaps (Bermudan swaptions) to the netting set.

Roadmap to multi callable products

In the first part we will see how to use a Monte-Carlo simulation for a single-callable swap (European Swaption) pricing. We don’t worry about the model calibration in this posts. This post is intend to fix the used notation and provide a simple example about basic Monte-Carlo pricing. The techniques presented here will be used and extend in the following two posts.

In the second part we develop a regression based approach (aka backward induction method or American Monte-Carlo or Longstaff-Schwartz method) to calculate the npv of a Bermudan swaption. Further we will calibrate the Gaussian short rate model to fit to a set of market prices of European swaptions.

At this point we will be able to calculate the npv of a multi-callable swap but the Monte-Carlo pricing is not suitable for an exposure simulation. Since a nested pricing simulation in a exposure simulation is very time consuming.

Therefore we will modify the American Monte-Carlo method in the third part and make it useable for our exposure simulation. The approach in the third post will follow the method presented by Drs. Alexandre Antonov, Serguei Issakov and Serguei Mechkov in their research paper ‘Backward Induction for Future Values’ and the methodology presented in the book ‘Modelling, Pricing, and Hedging Counterparty Credit Exposure: A Technical Guide’ by Giovanni Cesari et al.

But for now let’s start with something easy: European swaptions.

European swaption pricing

Since my first post we have been living in a single curve world and our model parameter have been being exogenous. To make things even more easy we have been using a flat yield curve. For now we don’t leave this comfortable world and apply the same setting for this example.

How to create a swaption with QuantLib?

An European payer/receiver swaption with physical delivery is an option that allows the option holder at option expiry to enter a payer/receiver swap. The rate paid/received on the fixed leg equals the strike of the swaption.

Given a plain vanilla swap, one can create an European swaption in the QuantLib with very few lines of code. All we need is the expiry date and the settlement type (cash settled or physical delivery).

def makeSwaption(swap, callDates, settlement):
    if len(callDates) == 1:
        exercise = ql.EuropeanExercise(callDates[0])
    else:
        exercise = ql.BermudanExercise(callDates)
    return ql.Swaption(swap, exercise, settlement)

settlementDate = today + ql.Period("1Y")

swaps = [makeSwap(settlementDate,
                  ql.Period("5Y"),
                  1e6,
                  0.03047096,
                  euribor6m)
        ]

calldates = [euribor6m.fixingDate(settlementDate)]
 
swaptions = [makeSwaption(swap, 
                          calldates, 
                          ql.Settlement.Physical) 
             for swap, fd in swaps]

Monte-Carlo pricing

At option expiry T_E the npv of the swaption is V(T_E) = \max(Swap(T_E), 0.0) with Swap(T_E) donating the value of the underlying swap at expiry.

In the Gaussian short rate model under the T-forward measure the zerobond with maturity in T years is our numeraire N(t). Under the usual conditions and using the T-forward measure we can calculate the npv at time 0 by

V(0) = N(0) E[\frac{V(T_E)}{N(T_E)} | F_0].

In our model the numeraire itself is not deterministic so we have to simulate it too.

The Monte-Carlo pricing will consist of three steps

– generate M paths of the short rate process and
– evaluate the swap npv V_i and calculate the numeraire price N_i at option expiry T_E for each path i=0\dots,M-1
– and finally approximate the expected value by \frac{1}{M} \sum_{i=0}^{M-1} \max(\frac{V_i}{N_i},0.0).

Implementation

Instead of using the QuantLib swap pricer we will do the path pricing in Python. Therefore we need to extract the needed information from the instrument.

We convert all dates into times (in years from today). We use the day count convention Act/365.

mcDC = yts.dayCounter()

def timeFromReferenceFactory(daycounter, ref):
    """
    returns a function, that calculate the time in years
    from a the reference date *ref* to date *dat* 
    with respect to the given DayCountConvention *daycounter*
    
    Parameter:
        dayCounter (ql.DayCounter)
        ref (ql.Date)
        
    Return:
    
        f(np.array(ql.Date)) -> np.array(float)
    """
    def impl(dat):
        return daycounter.yearFraction(ref, dat)
    return np.vectorize(impl)

timeFromReference = timeFromReferenceFactory(mcDC, today)

In the first step we extract all fixed leg cashflows and payment dates to numpy arrays.

That are all information we need to calculate the fixed leg npv on a path. We calculate the discount factors for each payment time and multiply the cashflow array with the discount factors array element-wise. The sum of this result gives us the fixed leg npv.

fixed_leg = swap.leg(0)
n = len(fixed_leg)
fixed_times = np.zeros(n)
fixed_amounts = np.zeros(n)
for i in range(n):
    cf = fixed_leg[i]
    fixed_times[i] = timeFromReference(cf.date())
    fixed_amounts[i] = cf.amount()

For the floating leg npv we extract all payment, accrual period start and end dates. We assume that the index start and end dates coincide with the accruals start and end dates and that all periods are regular. With this information we can estimate all floating cashflows by estimating the index fixing through

fixing(t_s) = (\frac{df(t_s)}{df(t_e)}-1) \frac{1}{dcf_{idx}(t_s,t_e)},

with the discount factor df(t) at time t, the year fraction dcf_{idx} between accrual start time t_s and accrual end time t_e using the index day count convention.

float_leg = swap.leg(1)
n = len(float_leg)
float_times = np.zeros(n)
float_dcf = np.zeros(n)
accrual_start_time = np.zeros(n)
accrual_end_time = np.zeros(n)
nominals = np.zeros(n)
for i in range(n):
    # convert base classiborstart_idx Cashflow to
    # FloatingRateCoupon
    cf = ql.as_floating_rate_coupon(float_leg[i])
    iborIndex = cf.index()
    value_date = cf.referencePeriodStart()
    index_mat = cf.referencePeriodEnd()
    # year fraction
    float_dcf[i] = cf.accrualPeriod()
    # calculate the start and end time
    accrual_start_time[i] = timeFromReference(value_date)
    accrual_end_time[i] = timeFromReference(index_mat)
    # payment time
    float_times[i] = timeFromReference(cf.date())
    # nominals 
    nominals[i] = cf.nominal()

We could extend this about gearings and index spreads, but we set the gearing to be one and the spread to be zero.

To calculate the swap npv we need the discount factors for all future payment times (fixed and floating leg), accrual period start and end dates. We store all times together in one array. To get the discount factors we apply the method zeroBond of the GSR model on this array element-wise.

# Store all times for which we need a discount factor in one array
df_times = np.concatenate([fixed_times,
				       ibor_start_time,
				       ibor_end_time,
				       float_times])
df_times = np.unique(df_times)

# Store indices of fix leg payment times in 
# the df_times array
fix_idx = np.in1d(df_times, fixed_times, True)
fix_idx = fix_idx.nonzero()
# Indices of the floating leg payment times 
# in the df_times array
float_idx = np.in1d(df_times, float_times, True)
float_idx = float_idx.nonzero()
# Indices of the accrual start and end time
# in the df_times array
accrual_start_idx = np.in1d(df_times, ibor_start_time, True)
accrual_start_idx = accrual_start_idx.nonzero()
accrual_end_idx = np.in1d(df_times, ibor_end_time, True)
accrual_end_idx = accrual_end_idx.nonzero()

Our pricing algorithm for the underlying swap is:

# Calculate all discount factors
discount = np.vectorize(lambda T: model.zerobond(T, t, x_t))
dfs = discount(df_times)
# Calculate fixed leg npv
fix_leg_npv = np.sum(fixed_amounts * dfs[fix_idx])
# Estimate the index fixings
index_fixings = (dfs[accrual_start_idx] / dfs[accrual_end_idx] - 1) 
index_fixings /= float_dcf
# Calculate the floating leg npv
float_leg_npv = np.sum(nominals * index_fixings * float_dcf * dfs[float_idx])
npv = float_leg_npv - fix_leg_npv

Our time grid for the simulation consists of two points, today and option expiry.

The path generation is very similar like the one in the previous posts, but this time we not only simulate the underlying process but also the numeraires, and we calculate all needed discount factors on a path.

M = 100000
m = len(time_grid)
x = np.zeros((M, m))
y = np.zeros((M, m))
numeraires = np.zeros((M,m))
dfs = np.zeros((M, m, len(df_times)))

for n in range(0,M):
    numeraires[n, 0] = model.numeraire(0, 0)
    
for n in range(0,M):
    dWs = generator.nextSequence().value()
    for i in range(1, len(time_grid)):
        t0 = time_grid[i-1]
        t1 = time_grid[i]
        e = process.expectation(t0, 
                                x[n,i-1], 
                                dt[i-1])
        std = process.stdDeviation(t0,
                                   x[n,i-1],
                                   dt[i-1])
        x[n,i] = e + dWs[i-1] * std 
        e_0_0 = process.expectation(0,0,t1)
        std_0_0 = process.stdDeviation(0,0,t1)
        y[n,i] = (x[n,i] - e_0_0) / std_0_0
        df = np.vectorize(lambda T : model.zerobond(T, t1, y[n,i]))
        numeraires[n ,i] = model.numeraire(t1, y[n, i])
        dfs[n,i] = df(df_times)                            

Given the matrix of numeraires and discount factors we can calculate the npv on the path very fast using numpy arrays.

index_fixings = dfs[:,-1, accrual_start_idx][:,0,:] / dfs[:, -1, accrual_end_idx][:,0,:] - 1
index_fixings /= float_dcf
floatLeg_npv = np.sum(index_fixings * float_dcf * dfs[:,-1, float_idx][:,0,:] * nominals, 
                     axis = 1) 
fixedLeg_npv = np.sum(fixed_amounts * dfs[:, -1, fix_idx][:,0,:], axis=1)
npv = (floatLeg_npv - fixedLeg_npv)
# Apply payoff function 
npv[npv < 0] = 0
# Deflate NPV
npv = npv / numeraires[:,-1] 
npv = np.mean(npv) * numeraires[0,0]

Some remarks

To extract the information from the swap we use the method leg. This method is not a part of the QuantLib 1.5, but you could clone my QuantLib fork on GitHub (branch: SwigSwapExtension) and build the Swig binding yourself. I also send a pull request to Luigi. Maybe it will be part of the official QuantLib at a later time.

In the real world there are quotes for European swaptions in terms of implied volatility available and one would like use a model that is consistent with the market quotes. This is done by model calibration (choice the model parameter so that the model give the same premium for the quoted swaptions). Of cause one could use the Monte-Carlo pricing to calibrate the model, but this would be very time consuming process. The Gaussian short rate model provide some faster and very convenient routines for that. In the next part we will see how to calibrate the model and use the calibrated model to price Bermudan swaptions.

As usual you can download the notebook on nbviewer or GitHub.

Stay tuned for the next part coming soon…

CVA Calculation with QuantLib and Python

Discounted_NPV_Paths

Today I am going to present a way to calculate the credit value adjustment (CVA) for a netting set of plain vanilla interest rate swaps. This Monte-Carlo method is based on the code example of my previous post about the expected exposure and PFE calculation and the first steps will be exactly the same.

What is the credit value adjustment (CVA)?

The credit value adjustment is the difference between the risk-free price of a netting set and the the price which takes the possibility of the default of the counterparty into account. A netting set is a portfolio of deals with one counterparty for which you have a netting agreement. That means you are allowed to set positive against negative market values. A netting agreement will reduce your exposure and therefore the counterparty credit risk. If the time of default and value of the portfolio are independent then the CVA is given by

CVA = (1-R) \int_0^Tdf(t)EE(t)dPD(t),

with recovery rate R, portfolio maturity T (the latest maturity of all deals in the netting set), discount factor df(t) at time t, the expected exposure EE(t) at time t and the default probability PD(t).

The Monte Carlo approach

In my previous post we have seen a Monte Carlo method to estimate the expected exposure EE(t) at time t on an given time grid 0=t_0<\dots<T=t_n. We can reuse this estimator to approximate the integral by the discrete sum

CVA = (1-R) \sum_{i=1}^n df(t_i)EE(t_i)(PD(t_{i-1})-PD(t_i)).

As we see in the formula we need two more ingredients to calculate the CVA, the discounted expected exposure and the the default probabilities of the counterparty.

Calculate the discounted expected exposure

To convert the expected exposure at time t in its present value expressed in time-zero dollars we only need to add a few more lines of code.

vec_discount_function = np.vectorize(t0_curve.discount)
discount_factors = vec_discount_function(time_grid)

We use the todays yield curve to calculate the discount factor for each point on our grid. With the numpy function vectorize
we generate a vectorized wrapper of the discount method of the QuantLib YieldTermStructure. This vectorized version can be applied on a array of ql.Dates or times instead of a scalar input. Basicalliy it’s equivalent to the following code snippet:

discount_factors = np.zeros(time_grid.shape)
for i in range(len(time_grid)):
	time = time_grid[i]
	discount_factors[i] = t0_curve.discount(time)

As the numpy documentation states the np.vectorize function is provided primarily for convenience, not for performance.

After the generating the market scenarios and pricing the netting set under each scenario we will calculate the discounted NPVs for each deal in the portfolio:

# Calculate the discounted npvs
discounted_cube = np.zeros(npv_cube.shape)
for i in range(npv_cube.shape[2]):
    discounted_cube[:,:,i] = npv_cube[:,:,i] * discount_factors
# Netting
discounted_npv = np.sum(discounted_cube, axis=2)

Discounted_NPV_Paths

Using the discounted npv cube we can calculate the discounted expected exposure:

# Calculate the exposure and discounted exposure
E = portfolio_npv.copy()
dE = discounted_npv.copy()
E[E&lt;0] = 0
EE = np.sum(E, axis=0)/N
dE[dE&lt;0] = 0
dEE = np.sum(dE, axis=0)/N

discounted_exposure

discounted_expected_exposure

The default probability of the counterparty

To derive the default probability one could either use market implied quotes (e.g. CDS) or use rating information (e.g. based on historical observations). We assume that the survival probability is given by sp(t)=\exp(-\int_0^t \lambda(t)) with a deterministic piecewise-constant hazard rate function \lambda(t). Given a grid of dates d_0<\dots<d_n and the corresponding backward flat hazard rate $latex\lambda_0,\dots,\lambda_n$ we can use the Quantlib HazardRateCurve to build a default curve.

# Setup Default Curve 
pd_dates =  [today + ql.Period(i, ql.Years) for i in range(11)]
hzrates = [0.02 * i for i in range(11)]
pd_curve = ql.HazardRateCurve(pd_dates,hzrates,ql.Actual365Fixed())
pd_curve.enableExtrapolation()

The QuantLib provides a real bunch of different types of DefaultTermStructures. You can either bootstrap a default curve from CDS quotes or you build a interpolated curve like we do here and combine one of the many interpolators (Linear, Backward Flat, etc.) with one of the possible traits (hazard rate, default probability, default density).

defaultCurve

With the default termstructure we can calculate the probability for a default between the times t_i and t_{i+1} for all i in our time grid.

# Calculation of the default probs
defaultProb_vec = np.vectorize(pd_curve.defaultProbability)
dPD = defaultProb_vec(time_grid[:-1], time_grid[1:])

Again we use the numpy function vectorize to apply a scalar function on an array. The method defaultProbability takes two times as input, t and T. It returns the probability of default between t and T.

Now we have all pieces together and the following code snippet gives us the CVA of our netting set:

# Calculation of the CVA
recovery = 0.4
CVA = (1-recovery) * np.sum(dEE[1:] * dPD)

You can download the code as an IPython (Juypter) notebook from here or just clone my repository (IPythonscripts) at GitHub.

If you want to read more about the QuantLib I would recommend to have a look on the blog and book “Implementing QuantLib” by Luigi. Another fantastic blog “Fooling around with QuantLib” by Peter has a very good and detailed post the Gsr model. Actually Peter has implemented this model in C++ and contributed it to the QuantLib.

I hope you have enjoyed reading my post and you will have fun playing around with the notebook. In one of my following posts I will extend this simulation by add a new product class to the netting set: European and bermudan swaptions.

So long.

Expected Exposure and PFE simulation with QuantLib and Python

In this post I will show how to use the Python bindings of the QuantLib library to calculate the expected exposure (EE) for a netting set of interest rate swaps in a IPython notebook. The technique I will present is very simple and works out of the box with standard QuantLib instruments and models. I will use a forward Monte Carlo Simulation to generate future market scenarios out of one-factor gaussian short rate model and evaluate the NPV of all swaps in the netting set under each scenario. The source code to this post (ExpectedExposureSimulation.ipynb) can be found in my repository IPythonScripts on GitHub or at nbviewer .

Methodology

First we define a time grid. On each date/time in our grid we want to calculate the expected exposure. For each date in our time grid we will simulate N states of the market and for each of these states we will calculate the NPV all of instruments in our portfolio / netting set. This results in N x (size of the netting set) simulated paths of NPVs. These paths can be used for EE, CVA (Credit value adjustment) or PFE (potential future exposure) calculations. swapPathsMc In the next step will we will floor each path at zero. This give the exposure of the portfolio on a path at each time. exposureSwap The expected exposure is given by the average of all paths: expExposureSwap The total number of NPV evaluations is (size of time grid) x (size of portfolio) x N. For a big portfolio and a very dense time grid it can be very time consuming task even if the single pricing is done pretty fast.

Assumption made in this example

For simplicity we restrict the portfolio to plain vanilla interest rate swaps in one currency. Further we assume that we live in a “single curve” world. We will use the same yield curve for discounting and forwarding. No spreads between the different tenor curves neither CSA discounting are taken into account. For the swap pricing we will need future states of the yield curve. In our setup we assume the the development of the yield curve follow an one factor Hull-White model. At the moment we make no assumption on how it is calibrated and assume its already calibrated. In our setting we will simulate N paths of the short rate following the Hull-White dynamics. At each time on each path the yield curve depend only on the state of our short rate process. We will use QuantLib functionalities to simulate the market states and perform the swap pricing on each path. The calculation of the expected exposure will be done in Python.

Technical Implementation

1. Setup of the market state at time zero (today)

rate = ql.SimpleQuote(0.03)
rate_handle = ql.QuoteHandle(rate)
dc = ql.Actual365Fixed()
yts = ql.FlatForward(today, rate_handle, dc)
yts.enableExtrapolation()
hyts = ql.RelinkableYieldTermStructureHandle(yts)
t0_curve = ql.YieldTermStructureHandle(yts)
euribor6m = ql.Euribor6M(hyts)

As mentioned above we live in a single curve world, we use a flat yield curve as discount and forward curve. During the Monte Carlo Simulation we will relink the Handle to the yieldTermStrucutre htys to our simulated yield curve. The original curve is stored in yts and the handle t0_curve.

2. Setup portfolio / netting set

# Setup a dummy portfolio with two Swaps
def makeSwap(start, maturity, nominal, fixedRate, index, typ=ql.VanillaSwap.Payer):
    &quot;&quot;&quot;
    creates a plain vanilla swap with fixedLegTenor 1Y
    
    parameter:
        
        start (ql.Date) : Start Date
        
        maturity (ql.Period) : SwapTenor
        
        nominal (float) : Nominal
        
        fixedRate (float) : rate paid on fixed leg
        
        index (ql.IborIndex) : Index
        
    return: tuple(ql.Swap, list&lt;Dates&gt;) Swap and all fixing dates
    
        
    &quot;&quot;&quot;
    end = ql.TARGET().advance(start, maturity)
    fixedLegTenor = ql.Period(&quot;1y&quot;)
    fixedLegBDC = ql.ModifiedFollowing
    fixedLegDC = ql.Thirty360(ql.Thirty360.BondBasis)
    spread = 0.0
    fixedSchedule = ql.Schedule(start,
                                end, 
                                fixedLegTenor, 
                                index.fixingCalendar(), 
                                fixedLegBDC,
                                fixedLegBDC, 
                                ql.DateGeneration.Backward,
                                False)
    floatSchedule = ql.Schedule(start,
                                end,
                                index.tenor(),
                                index.fixingCalendar(),
                                index.businessDayConvention(),
                                index.businessDayConvention(),
                                ql.DateGeneration.Backward,
                                False)
    swap = ql.VanillaSwap(typ, 
                          nominal,
                          fixedSchedule,
                          fixedRate,
                          fixedLegDC,
                          floatSchedule,
                          index,
                          spread,
                          index.dayCounter())
    return swap, [index.fixingDate(x) for x in floatSchedule][:-1]

The method makeSwap create a new QuantLib plain vanilla swap (see my previous post). We use this method to setup a netting set with two swaps:

portfolio = [makeSwap(today + ql.Period(&quot;2d&quot;),
                      ql.Period(&quot;5Y&quot;),
                      1e6,
                      0.03,
                      euribor6m),
             makeSwap(today + ql.Period(&quot;2d&quot;),
                      ql.Period(&quot;4Y&quot;),
                      5e5,
                      0.03,
                      euribor6m,
                      ql.VanillaSwap.Receiver),
            ]

Our netting set consists of two swaps, one receiver and one payer swap. Both swaps differ also in notional and time to maturity. Finally we create a pricing engine and link each swap in our portfolio with it.

engine = ql.DiscountingSwapEngine(hyts)

for deal, fixingDates in portfolio:
 deal.setPricingEngine(engine)
 print(deal.NPV())

In our Monte Carlo Simulation we can relink the handle hyts and use the same pricing engine. So we don’t need to create new pricing engines or relink the the deals to a new engine. We just need to call the method NPV of the instruments after relinking the yield term structure handle.

3. Monte-Carlo-Simulation of the “market”

We select a weekly time grid, including all fixing days of the portfolio. To generate the future yield curves we are using the GSR model and process of the QuantLib.

volas = [ql.QuoteHandle(ql.SimpleQuote(0.0075)),
ql.QuoteHandle(ql.SimpleQuote(0.0075))]
meanRev = [ql.QuoteHandle(ql.SimpleQuote(0.02))]
model = ql.Gsr(t0_curve, [today+100], volas, meanRev, 16.)
process = model.stateProcess()

The GSR model allows the mean reversion and the volatility to be piecewise constant. In our case here both parameter are set constant. For a more detailed view on the GSR model have a look on the C++ examples “Gaussian1dModels” in the QuantLib or here. Given a time t_0 and state x(t_0) of the process we know the conditional transition density for x(t_1) for t_1 > t_0. Therefore we don’t need to discretize the process between the evaluation dates. As a random number generator we are using the Mersenne Twister.

#%%timeit
# Generate N paths
N = 1500
x = np.zeros((N, len(time_grid)))
y = np.zeros((N, len(time_grid)))
pillars = np.array([0.0, 0.5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
zero_bonds = np.zeros((N, len(time_grid), 12))
for j in range(12):
    zero_bonds[:, 0, j] = model.zerobond(pillars[j],
                                         0,
                                         0)
for n in range(0,N):
    dWs = generator.nextSequence().value()
    for i in range(1, len(time_grid)):
        t0 = time_grid[i-1]
        t1 = time_grid[i]
        x[n,i] = process.expectation(t0, 
                                     x[n,i-1], 
                                     dt[i-1]) + dWs[i-1] * process.stdDeviation(t0,
                                              x[n,i-1],
                                              dt[i-1])
        y[n,i] = (x[n,i] - process.expectation(0,0,t1)) / process.stdDeviation(0,0,t1)
        for j in range(12):
            zero_bonds[n, i, j] = model.zerobond(t1+pillars[j],
                                                 t1,
                                                 y[n, i])
                                                 

We also save the zero bonds prices on each scenario for a set of maturities (6M, 1Y,…,10Y). We use this prices as discount factors for our scenario yield curve.

4. Pricing on path & netting

On each date t and on each path p we will evaluate the netting set. First we build a new yield curve using the scenario discount factors from the step before.

date = date_grid[t]
ql.Settings.instance().setEvaluationDate(date)
ycDates = [date, 
           date + ql.Period(6, ql.Months)] 
ycDates += [date + ql.Period(i,ql.Years) for i in range(1,11)]
yc = ql.DiscountCurve(ycDates, 
                      zero_bonds[p, t, :], 
                      ql.Actual365Fixed())
yc.enableExtrapolation()
hyts.linkTo(yc)

After relinking the yield termstructure handle is the revaluation of the portfolio is straight forward. We just need to take fixing dates into account and store the fixings otherwise the pricing will fail.

for i in range(len(portfolio)):
    npv_cube[p, t, i] = portfolio[i][0].NPV()
5. Calculation EE and PFE

After populating the cube of fair values (1st dimension is simulation number, 2nd dimension is the time and 3rd dimension is the deal) we can calculate the expected exposure and the potential future exposure.

# Calculate the portfolio npv by netting all NPV
portfolio_npv = np.sum(npv_cube,axis=2)
# Calculate exposure
E = portfolio_npv.copy()
E[E&amp;amp;amp;lt;0]=0
EE = np.sum(E, axis=0)/N

Expected Expsoure Netting Set With PFE(t) we mean the 95 % quantile of the empirical distribution of the portfolio exposure at time t.

PFE_curve = np.apply_along_axis(lambda x: np.sort(x)[0.95*N],0, E)
MPFE = np.max(PFE_curve)

PFE netting set

Conclusion

With very few lines of code you can build a simulation engine for exposure profiles for a portfolio of plain vanilla swaps. These paths allows us to calculate the expected exposure or potential future exposure. Of course is the problem set in real world applications much more complex, we haven’t covered calibration (historical or market implied) or multi currencies / multi asset classes yet. And its getting even more complicated if you have multi-callable products in your portfolio.

But nevertheless I hope you have enjoyed reading this little tutorial and got an first insight into exposure simulation with QuantLib and Python. In one of my next post I will may extend this example about CVA calculation.

So long!

A brief introduction to the QuantLib in Python…

QuantLib is an open-source framework for quantitative finance  written in C++. There is an active community who develop and extend the library. QuantLib covers a wide range of financial instruments and markets like IR, FX and Equities and provide pricing engines and models, optimization algorithm, a Monte-Carlo framework, business day conventions, day count conventions, holidays calendars and even more… and grows continuously.

There are existing interfaces to different language like Python, Ruby, Java and more via swig. In the following I will describe how to build the QuantLib for Python.

In the following description I assume that you have already installed Python with IPython including all dependencies, numpy, scipy, pandas and matplotlib. A good starting point are distributions like Anaconda or WinPython (for Windows only). These distribution are shipped with many more useful libraries and provide an easy way to install more packages. Its worth to check them out, if haven’t tried them yet.

For windows you can find pre-build executables of the QuantLib at Christoph Gohlke’s website. If you want to have the most recent version you should clone the current master branch from GitHub and build the library and the python package yourself.

To build QuantLib from sources under Windows you need the correct Visual Studio depending on the Python version you are targeting. For Python 3.3 you need Visual Studio 2010 and for Python 2.7 you need Visual Studio 2008. Additional dependencies are the boost library and swig.

I will not cover how to install the dependencies but you can find binary installer for swig under windows on the swig project website and there also exists precompiled binaries for the boost library on source forge. Add the path to swig command to your PATH environment variable.

The following steps are valid for Python 2.7 and 3.3:

0. Clone the git repository either with using git:

cd c:\path\to\quantlib
git clone https://github.com/lballabio/quantlib .

or download as a zipped file from GitHub.

1. Open the QuantLib_vcXX.sln and build it in “Release” or “Release static runtime” configuration. For more details check the install documentation on the QuantLib project site. You will find the solution under

c:\path\to\quantlib\QuantLib

2. Open command line window and set required environment variables

SET QL_DIR=c:\path\to\quantlib\QuantLib
SET INCLUDE=c:\path\to\boost;%INCLUDE%
SET LIB=c:\path\to\boost\lib;%LIB%

3. Create the interface code with swig

cd c:\path\to\quantlib\QuantLib-SWIG\python
python setup.py wrap

4. Build and install the package

python setup.py build
python setup.py install

5. Test your install

python setup.py test

If you use Visual Studio 2010 and target Python 2.7 you will encounter the error message “Unable to find vcversall.bat“. The following “hack” works for me:

Set an additional environment variable at step 2:

SET VS90COMNTOOLS=%VS100COMNTOOLS%

Under unix you can also find some pre-build packages (e.g. for Ubuntu) in your package-manager or you build it from sources yourself with the usual configure, make and make install chain (see here and for Mac OS X here). After you build the library you can continue with step 3 above. Be sure you have installed swig before. Just download the source from the swig project and install via the usual unix install commands.

After we have managed to install QuantLib Python package I want to show you how you can use the QuantLib C++ classes from Python. You can download the following simple example  from my repository “IPythonScripts” on GitHub.

In the first example we will see how to setup an interest rate swap and use a discounting pricing engine to calculate the NPV and the fair rate for this swap.

First we need to import the QuantLib package, since we want also produce some plots we import numpy and matplotlib.

import numpy as np
import matplotlib.pyplot as plt
import QuantLib as ql

In the next step we will setup the yield curve (aka yieldtermstructure). For simplicity we will use an flat forward curve. I will use the same curve for discounting and calculation of the forwards. QuantLib supports multi curve setups, as I will maybe show in a later post or you can see in the examples how to setup more realistic yield curves.


# Set Evaluation Date
today = ql.Date(31,3,2015)
ql.Settings.instance().setEvaluationDate(today)
# Setup the yield termstructure
rate = ql.SimpleQuote(0.03)
rate_handle = ql.QuoteHandle(rate)
dc = ql.Actual365Fixed()
disc_curve = ql.FlatForward(today, rate_handle, dc)
disc_curve.enableExtrapolation()
hyts = ql.YieldTermStructureHandle(disc_curve)

The yieldTermStructure object provides an method which gives us the discount factor for a particular date (QuantLib.Date object) or time in years (with 0 = evaluationDate). This method is called discount.  I am using the numpy method vectorize to apply this function on arrays or list of times and then generate a plot of the discount curve.

discount = np.vectorize(hyts.discount)
tg = np.arange(0,10,1./12.)
plt.plot(tg, discount(tg))
plt.xlabel(&amp;quot;time&amp;quot;)
plt.ylabel(&amp;quot;discount factor&amp;quot;)
plt.title(&amp;quot;Flat Forward Curve&amp;quot;)

discountcurve

In the next step we will setup up a plain vanilla EURIBOR 6M Swap with maturity in 10 years. Therefore we generate an index and using the handle to our yield curve as forward curve and two schedules, one for the fixed rate leg with annual payments and one for the float leg with semi annual payments.

start = ql.TARGET().advance(today, ql.Period(&quot;2D&quot;))

end = ql.TARGET().advance(start, ql.Period(&quot;10Y&quot;))

nominal = 1e7

typ = ql.VanillaSwap.Payer

fixRate = 0.03

fixedLegTenor = ql.Period(&quot;1y&quot;)

fixedLegBDC = ql.ModifiedFollowing

fixedLegDC = ql.Thirty360(ql.Thirty360.BondBasis)

index = ql.Euribor6M(ql.YieldTermStructureHandle(disc_curve))

spread = 0.0

fixedSchedule = ql.Schedule(start, end, fixedLegTenor, index.fixingCalendar(), fixedLegBDC, fixedLegBDC, ql.DateGeneration.Backward, False)
floatSchedule = ql.Schedule(start, end, index.tenor(), index.fixingCalendar(), index.businessDayConvention(), index.businessDayConvention(), ql.DateGeneration.Backward, False)

swap = ql.VanillaSwap(typ, nominal, fixedSchedule, fixRate, fixedLegDC, floatSchedule, index, spread, index.dayCounter())

The last step before we can calculate the NPV we need a pricing engine. We are going to use the discountingSwapEngine. As the name suggests, it will discount all future payments to the evaluation date and calculate the difference between the present values of the two legs.

engine = ql.DiscountingSwapEngine(ql.YieldTermStructureHandle(disc_curve))
swap.setPricingEngine(engine)

print(swap.NPV())
print(swap.fairRate())

That was a very brief demonstration of the QuantLib. The examples which are shipped with the QuantLib give a much more detailed insight. So you may have a look on these.

Hello World!

This is my first entry. I will write in my new blog about using Python in the field of Quantitative Finance.  I want to provide some examples how Python and other open source tools like QuantLib (www.quantlib.org) can be used for pricing and/or risk calculations.

I plan to make all samples downloadable as IPython notebooks from my repository at GitHub.

One may ask why Python and not C++ or some other language? From my point of view Python is easy to learn and its comes with many useful open source libraries which allow fast prototyping like numpy, scipy and pandas. And you can make existing C/C++ libraries accessible from python (using swig or boost::python).

And for me the most important point: Its fun writing Python code.