Why HJM Forward-Rate Averages Differ from Initial Forwards

# Convergence of HJM model with constant volatility


# Convergence of HJM model with constant volatility












In HJM is expected that for each step in time the mean of the tenors will be equal the corresponding tenors in the initial forward curve.

I saw a divergence in my model so I give some steps back and try in a simpler model: constant volatility.

But I see the same divergence here. I follow Glasserman to build the code but I cant figure out the error.

Here is a code sample:

```
import numpy as np

def construct_random_numbers(seed_value: int, number_of_simulations: int, number_of_risk_factors: int, number_of_timesteps: int) -> np.array:
    np.random.seed(seed_value)
    if number_of_simulations % 2 == 1:
        raise ValueError("Number of paths must be even")
    sn = np.random.standard_normal((int(number_of_simulations / 2), number_of_risk_factors, number_of_timesteps))
    sn = sn.astype(np.float32)
    sn = np.concatenate((sn, -sn), axis=0)
    sn = (sn - sn.mean()) / sn.std()
    return sn

if __name__ == "__main__":
    number_of_simulations = 10
    number_of_timesteps = 300
    number_of_tenors = 300
    dt = 1/12.0
    sqrt_dt = np.sqrt(dt)

    vol = 0.1
    z = construct_random_numbers(42, number_of_simulations, 2, number_of_timesteps)

    initial_curve = [0.13916995924421227, 0.13927470149892207, 0.13931937944665784, 0.13954950876573402, 0.1394163953124534, 0.13922719184309001, 0.13940934362894178, 0.13920600960110618, 0.13946900910538873, 0.1393136275348182, 0.1393513763811261, 0.13949331966475423, 0.13923202364314158, 0.1392210281814887, 0.13930238006736542, 0.13946457788510194, 0.13920880539245234, 0.13907752444018895, 0.1393156241141591, 0.1394159140799911, 0.1394807661749739, 0.13931442386379148, 0.13931959733928842, 0.13947613007342557, 0.13917376583003047, 0.1393218879224817, 0.1393797823889925, 0.13940566187212852, 0.13930975933068093, 0.1393641841011706, 0.13911190726465872, 0.13929491515876663, 0.13940609087125547, 0.13926251059657455, 0.13936663842215896, 0.1394054272500803, 0.13932648507991424, 0.13937778637709966, 0.13926446674347154, 0.1393400788720304, 0.13942176161695893, 0.13937643729609686, 0.13930208687304363, 0.13938316164042908, 0.13943220097906367, 0.13915574587214227, 0.1394766474064956, 0.1393271145261319, 0.13940328899332402, 0.13948046469236453, 0.13922357263027696, 0.13928979589975726, 0.13933188513041245, 0.13928236464788757, 0.1395342336575837, 0.13927506279061733, 0.13930518532205133, 0.13938117890316032, 0.1392405250473924, 0.13932117192882437, 0.13939248532612503, 0.13932745395956175, 0.13935140131411003, 0.13930780678875518, 0.13920929296908194, 0.13939289932113394, 0.13953623330716475, 0.13936258016307834, 0.13938326641172477, 0.1394300122755117, 0.13923380695034535, 0.13953099146581296, 0.1393809728796423, 0.13952392568990893, 0.1392874367004476, 0.13954254619225578, 0.13943825554940348, 0.13941604400858526, 0.13938414244543532, 0.13933588713819256, 0.1394807931658474, 0.1393747759869848, 0.13916403302964128, 0.1392490213538293, 0.13934215542478073, 0.13939173989831657, 0.13935508115111003, 0.13948263707897937, 0.13916562981359615, 0.13937509879365054, 0.1391789420774613, 0.13943671122954843, 0.13935129412855582, 0.13940270099720242, 0.13927435170394104, 0.1394886864931658, 0.13946491246640588, 0.1394190944303012, 0.13934778827531366, 0.13941872572852954, 0.13929005074320128, 0.13944629556590207, 0.13952568808943241, 0.13946780105599793, 0.13928777975301926, 0.13939688173172732, 0.13923142584562415, 0.13941797903111758, 0.13940580034128627, 0.13917375836604678, 0.13924456161560878, 0.13933242611032565, 0.13933065402528594, 0.13923979230645814, 0.13939963787293522, 0.13938588736089405, 0.13934131308680217, 0.13928516084998419, 0.13946561071264166, 0.13952628071130885, 0.13931238385161535, 0.13929952330828452, 0.13940986076654122, 0.1393249370620253, 0.13934680027993368, 0.1394347053151725, 0.1391698043912493, 0.1394925702324853, 0.13936800805243293, 0.13931959068554378, 0.13935181658457788, 0.13935698293550206, 0.13918761986965553, 0.13951023382274533, 0.13936487256322708, 0.13940504021304476, 0.1394558949770574, 0.13925862375424397, 0.13940808718990352, 0.13933397589091012, 0.13944541707752167, 0.13914801725796486, 0.13935083056997588, 0.1393004579344216, 0.13940396629506344, 0.13940774441873682, 0.1392899774105178, 0.1392895226135824, 0.13934967346314428, 0.1392710757772685, 0.13944493836954522, 0.13932381927995527, 0.13907004117806784, 0.13932325779433782, 0.1392568856179156, 0.1391976124184539, 0.13923699733617775, 0.13937168084718332, 0.13949360408282482, 0.1393920584473865, 0.13930008966117843, 0.13940633041951708, 0.13931483096897812, 0.1392912799630837, 0.1392453037331767, 0.13918555711932484, 0.13932174762934407, 0.13937654534899202, 0.13935107763299853, 0.13931914330720527, 0.13926616539393033, 0.13942156461916852, 0.13922902680800536, 0.13943834279138437, 0.13926051829810734, 0.1392854964762647, 0.13921406148093846, 0.13931059978531143, 0.1394582712458478, 0.1394123028088444, 0.1394585822112028, 0.1394129542545934, 0.13925037301714696, 0.13932766107756725, 0.13928272008708825, 0.13949379749040391, 0.1392410611353574, 0.13945010976893893, 0.1394500353870109, 0.1395437020201716, 0.13931977134073706, 0.13943197221595802, 0.13958890518365388, 0.13929897583230663, 0.13920723597524678, 0.13917273899660923, 0.13943201120288615, 0.13917242497919416, 0.13932350809470623, 0.1394044713187039, 0.139379257553683, 0.13940328448641842, 0.13938296898558009, 0.13948862658133135, 0.13926550836192614, 0.13929304048615415, 0.13928347405547292, 0.13944375918118626, 0.13945077985883952, 0.13940586253193185, 0.139479089366689, 0.13924293591326237, 0.13952746241115513, 0.13951306386217124, 0.13943430219972458, 0.1395389198908771, 0.13919507767923223, 0.13930829518637375, 0.13928720965393182, 0.13943288793237574, 0.1392557259260398, 0.13943648951493376, 0.13949779361439968, 0.1394302602900084, 0.13938911510236615, 0.13926195266590993, 0.1392760549033875, 0.13948570189052967, 0.13943867490039055, 0.13935683804806226, 0.1393189490454234, 0.13915815921268887, 0.13925365475813156, 0.13955188190871398, 0.13923269444289976, 0.1393796879821221, 0.13946148439731954, 0.13923524773319979, 0.1393792110474781, 0.13945453155375107, 0.13936198993219073, 0.13964758030556668, 0.13919388007411218, 0.1394666128529884, 0.13934345982626037, 0.13943793201548368, 0.13940487071516994, 0.13928742438028902, 0.13938363963924594, 0.13917494985486017, 0.13942106475005714, 0.13939055584659907, 0.13942460601686432, 0.13946655183250872, 0.13939750198402098, 0.13937554894197984, 0.13927263316633387, 0.13928951792363511, 0.13947941346201775, 0.1395043735405706, 0.13933692895636882, 0.13936534027472533, 0.13927027856250063, 0.13947801568452675, 0.13942728168236918, 0.1395548617436464, 0.13927290769228204, 0.1393229392100982, 0.13938388459225257, 0.1394253889537726, 0.13933917323624526, 0.13927344389095464, 0.13944888075289236, 0.13943285353625456, 0.13926911258994365, 0.13949628428011057, 0.13947775987091968, 0.1393741696497248, 0.13936869978623762, 0.1394161457771116, 0.1392745696929744, 0.1393637431479164, 0.1391455016455391, 0.1394483855645503, 0.1392045906777558, 0.1395110819828429, 0.13932772799060336, 0.13933957330180688, 0.13929796660939156, 0.1393319361126078, 0.13950259313451482, 0.1391119591343465, 0.13948843653860882, 0.1393945423481102, 0.1392400378352143, 0.13925320876069508, 0.13952837930272083, 0.13926618931200574, 0.13924540494894042, 0.13947447145423647]

    paths = np.zeros(shape=(number_of_simulations, number_of_timesteps + 1, number_of_tenors))
    paths[:, 0, :] = initial_curve
    for tenor_index in range(number_of_tenors):
        for timestep_index in range(1, number_of_timesteps + 1):
            if timestep_index <= tenor_index:
                T = dt * tenor_index
                t = dt * timestep_index
                random_numbers = z[:, 0, timestep_index - 1]
                drift_integral = 0.5 * (vol ** 2) * (T ** 2 - (T - t) ** 2) * np.ones(shape=(number_of_simulations,))
                volatility_multiplied_by_random_numbers = vol * np.sqrt(t) * random_numbers * np.ones(shape=(number_of_simulations,))
                paths[:, timestep_index, tenor_index] = paths[:, 0, tenor_index] + drift_integral + volatility_multiplied_by_random_numbers

    # running over time the mean in the short rate must match the corresponding tenor from the initial curve
    for tenor_index in range(1, number_of_tenors):
        diff = np.mean(paths[:, tenor_index, tenor_index]) - paths[0, 0, tenor_index]
        print(f"{tenor_index}: {diff}")
    print("@")

    # running over time the mean in each tenors must match the corresponding tenor in the initial curve
    for timestep_index in range(number_of_timesteps + 1):
        for tenor_index in range(number_of_tenors):
            if timestep_index + tenor_index < number_of_tenors:
                diff = np.mean(paths[:, timestep_index, timestep_index + tenor_index]) - paths[0, 0, timestep_index + tenor_index]
                print(f"({timestep_index}, {tenor_index}): {diff}")
        print("@")
```

Could you help me find out what I'm missing>

Actually this question is already solved. The misunderstanding was that forward curves are not martingales in HJM, and I was testing for this.

Glasserman appointed this misunderstanding to me. in this model Bond prices are martingales, when I tested then the condition was satisfied.

For high values of volatility the error increases, but now everything is matching as expected.

Thanks!

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/82353

Are you asking why in HJM the simulated path average for some rate is not equal to the forward rate ? So for example the five year rate observed one year from now is such that the path average is say 4% whereas the forward rate is say 3.95% ?

If so this is well known. The path average is in the money market measure whereas the forward rate is in the forward measure. The difference is akin to a convexity adjustment.