tf_quant_finance.models.longstaff_schwartz.least_square_mc

Last updated: 2023-03-16.

tf_quant_finance.models.longstaff_schwartz.least_square_mc#

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Values Amercian style options using the LSM algorithm.

tf_quant_finance.models.longstaff_schwartz.least_square_mc(
    sample_paths, exercise_times, payoff_fn, basis_fn, discount_factors=None,
    num_calibration_samples=None, dtype=None, name=None
)

The Least-Squares Monte-Carlo (LSM) algorithm is a Monte-Carlo approach to valuation of American style options. Using the sample paths of underlying assets, and a user supplied payoff function it attempts to find the optimal exercise point along each sample path. With optimal exercise points known, the option is valued as the average payoff assuming optimal exercise discounted to present value.

Example. American put option price through Monte Carlo#

# Let the underlying model be a Black-Scholes process
# dS_t / S_t = rate dt + sigma**2 dW_t, S_0 = 1.0
# with `rate = 0.1`, and volatility `sigma = 1.0`.
# Define drift and volatility functions for log(S_t)
rate = 0.1
def drift_fn(_, x):
  return rate - tf.ones_like(x) / 2.
def vol_fn(_, x):
  return tf.expand_dims(tf.ones_like(x), axis=-1)
# Use Euler scheme to propagate 100000 paths for 1 year into the future
times = np.linspace(0., 1, num=50)
num_samples = 100000
log_paths = tf.function(tff.models.euler_sampling.sample)(
        dim=1,
        drift_fn=drift_fn, volatility_fn=vol_fn,
        random_type=tff.math.random.RandomType.PSEUDO_ANTITHETIC,
        times=times, num_samples=num_samples, seed=42, time_step=0.01)
# Compute exponent to get samples of `S_t`
paths = tf.math.exp(log_paths)
# American put option price for strike 1.1 and expiry 1 (assuming actual day
# count convention and no settlement adjustment)
strike = [1.1]
exercise_times = tf.range(times.shape[-1])
discount_factors = tf.exp(-rate * times)
payoff_fn = make_basket_put_payoff(strike)
basis_fn = make_polynomial_basis(10)
least_square_mc(paths, exercise_times, payoff_fn, basis_fn,
                discount_factors=discount_factors)
# Expected value: [0.397]
# European put option price
tff.black_scholes.option_price(volatilities=[1], strikes=strikes,
                               expiries=[1], spots=[1.],
                               discount_factors=discount_factors[-1],
                               is_call_options=False,
                               dtype=tf.float64)
# Expected value: [0.379]

References#

[1] Longstaff, F.A. and Schwartz, E.S., 2001. Valuing American options by simulation: a simple least-squares approach. The review of financial studies, 14(1), pp.113-147.

Args:#

  • sample_paths: A Tensor of either shape [num_samples, num_times, dim] or [batch_size, num_samples, num_times, dim], the sample paths of the underlying ito process of dimension dim at num_times different points. The batch_size allows multiple options to be valued in parallel.

  • exercise_times: An int32 Tensor of shape [num_exercise_times]. Contents must be a subset of the integers [0,...,num_times - 1], representing the time indices at which the option may be exercised.

  • payoff_fn: Callable from a Tensor of shape [num_samples, S, dim] (where S <= num_times) to a Tensor of shape [num_samples, batch_size] of the same dtype as samples. The output represents the payout resulting from exercising the option at time S. The batch_size allows multiple options to be valued in parallel.

  • basis_fn: Callable from a Tensor of the same shape and dtype as sample_paths and a positive integer Tenor (representing a current time index) to a Tensor of shape [batch_size, basis_size, num_samples] of the same dtype as sample_paths. The result being the design matrix used in regression of the continuation value of options.

  • discount_factors: A Tensor of shape [num_exercise_times] or of rank 3 and compatible with [num_samples, batch_size, num_exercise_times]. The dtype should be the same as of samples. Default value: None which maps to a one-Tensor of the same dtype as samples and shape [num_exercise_times].

  • num_calibration_samples: An optional integer less or equal to num_samples. The number of sampled trajectories used for the LSM regression step. Note that only the lastnum_samples - num_calibration_samples of the sampled paths are used to determine the price of the option. Default value: None, which means that all samples are used for regression and option pricing.

  • dtype: Optional dtype. Either tf.float32 or tf.float64. The dtype If supplied, represents the dtype for the input and output Tensors. Default value: None, which means that the dtype inferred by TensorFlow is used.

  • name: Python str name prefixed to Ops created by this function. Default value: None which is mapped to the default name ‘least_square_mc’.

Returns:#

A Tensor of shape [num_samples, batch_size] of the same dtype as samples.

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