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American Options (qfn)

Background

American Options

Exploration of American options, covering stopping times, the perpetual American put, the fixed-expiration American put (with Monte Carlo pricing via backward induction), and American calls on dividend-paying and non-dividend-paying assets. The theoretical discussion follows chapter 8 of Stochastic Calculus for Finance II: Continuous-Time Models (Shreve, 2008).

Source: "quantitative-finance-notebooks" collection (Notebook 5.4 American Options).

Code

```python """ American Options =================

Exploration of American options, covering stopping times, the perpetual American put, the fixed-expiration American put (with Monte Carlo pricing via backward induction), and American calls on dividend-paying and non-dividend-paying assets. The theoretical discussion follows chapter 8 of Stochastic Calculus for Finance II: Continuous-Time Models (Shreve, 2008).

Source: "quantitative-finance-notebooks" collection (Notebook 5.4 American Options). """

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Imports

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import numpy as np import scipy.stats as ss

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0. Introduction

============================================================================

The purpose of this script is to explore American options with reference to

chapter 8 from Stochastic Calculus for Finance II Continuous-Time Models

(Shreve, 2008).

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1. Stopping Times

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European option contracts specify an expiration date, and if the option is to

be exercised at all, the exercise must occur on the expiration date. An

American option allows the owner to choose to exercise at any time up to and

including the expiration date. Because of this early exercise feature, such an

option is at least as valuable as its European counterpart. Sometimes the

difference in value is negligible or even zero, and then American and European

options are close or exact substitutes. We shall see that the early exercise

feature for a call on a stock paying no dividends is worthless; American and

European calls on such a stock have the same price. In other cases, most

notably put options, the value of this early exercise feature, the so-called

early exercise premium, can be substantial. An intermediate option between

American and European is Bermudan, an option that permits early exercise but

only on a contractually specified finite set of dates.

Because an American option can be exercised at any time prior to its

expiration, it can never be worth less than the payoff associated with

immediate exercise. This is called the intrinsic value of the option. In

contrast to the case for a European option, whose discounted price process is

a martingale under the risk-neutral measure, the discounted price process of

an American option is a supermartingale under this measure. The holder of this

option may fail to exercise at the optimal exercise date, and in this case the

discounted option price has a tendency to fall; hence, the supermartingale

property. During any period of time in which it is not optimal to exercise,

however, the discounted price process behaves as a martingale.

To price an American option, just as with a European option, we could imagine

selling the option in exchange for some initial capital and then consider how

to use this capital to hedge the short position in the option. In this case,

we would need to be ready to pay off the option at all times prior to the

expiration date because we do not know when it will be exercised.

A stopping time tau is a random variable taking values in [0, inf] satisfying

{tau <= t} in F(t) for all t >= 0.

Thus a stopping time tau has the property that the decision to stop at time t

must be based on information available at time t.

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2. American Put

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2.1 Perpetual American Put

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The simplest interesting American option is the perpetual American put. It

is interesting because the optimal exercise policy is not obvious, and it is

simple because this policy can be determined explicitly. Although this is not

a traded option, we begin our discussion with it in order to present in a

simple context the ideas behind the subsequent analysis of more realistic

options.

The underlying asset has the price process S(t) given by

dS(t) = r S(t) dt + sigma S(t) dW~(t), (2.1.1)

where the interest rate r and the volatility sigma are strictly positive

constants and W~(t) is a Brownian motion under the risk-neutral probability

measure P~. The perpetual American put pays K - S(t) if it is exercised at

time t. This is its intrinsic value.

Let T be the set of all stopping times. The price of the perpetual American

put is defined to be

v_*(x) = max_{tau in T} E~[ e^{-r tau} (K - S(tau)) ], (2.1.2)

where x = S(0) is the initial stock price. In the event that tau = inf, we

interpret e^{-r tau} (K - S(tau)) to be zero.

The idea behind this is that the owner of the perpetual American put can

choose an exercise time tau, subject only to the condition that she may not

look ahead to determine when to exercise. The mathematical formulation of

this "not look ahead" restriction is that tau must be a stopping time. The

price of the option at time zero is the risk-neutral expected payoff of the

option, discounted from the exercise time back to time zero. If the option is

never exercised, its payoff is zero. The owner of the option should choose the

exercise strategy that maximizes this expected payoff, discounted back to time

zero, and thus we define the price of the option to be the maximum over

tau in T of the discounted expected payoffs.

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2.2 Fixed-Expiration American Put

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Let's now consider an American put on a stock whose price is the same

geometric Brownian motion in equation (2.1.1), but now the put has a finite

expiration time T.

Let 0 <= t <= T and x >= 0 be given. Assume S(t) = x. Let F_u^(t),

t <= u <= T, denote the sigma-algebra generated by the process S(v) as v

ranges over [t, u], and let T_{t,T} denote the set of stopping times for the

filtration F_u^(t), t <= u <= T, taking values in [t, T] or taking the value

infinity. In other words, {tau <= u} in F_u^(t) for every u in [t, T]; a

stopping time in T_{t,T} makes the decision to stop at a time u in [t, T]

based only on the path of the stock price between times t and u. The price at

time t of the American put expiring at time T is defined to be

v(t, x) = max_{tau in T_{t,T}} E~[ e^{-r(tau - t)} (K - S(tau)) | S(t) = x ].

In the event that tau = inf, we interpret e^{-r tau} (K - S(tau)) to be zero.

This is the case when the put expires unexercised.

Let's consider an example in Python.

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Define parameters

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if name == "main": S0 = 100.0 # initial asset price K = 100.0 # strike price T = 1.0 # time to maturity r = 0.04 # risk-free rate per unit T sigma = 0.2 # volatility per unit T

# ---------------------------------------------------------------------------
# Monte Carlo method -- backward induction for American put pricing
# ---------------------------------------------------------------------------
N = 252
dt = T / N
M = 100000
S = np.zeros((N + 1, M))
S[0, :] = S0

Z = ss.norm.rvs(loc=0, scale=1, size=(N, M), random_state=42)
for t in range(1, N + 1):
    S[t, :] = S[t-1, :] * np.exp((r - 0.5 * sigma**2) * dt + sigma * np.sqrt(dt) * Z[t-1, :])

payoffs = np.maximum(K - S, 0)
option_values = payoffs[-1, :]

# Backward induction to determine optimal stopping
for i in range(N - 1, -1, -1):
    discounted_payoff = np.exp(-r * dt) * option_values
    option_values = np.maximum(payoffs[i, :], discounted_payoff)

american_put = np.mean(option_values)
american_put_std_err = ss.sem(option_values)

print(f"The price of the American put option is: {american_put:.3f}"
      f"\nwith standard error: {american_put_std_err:.3f}")

# ============================================================================
# 3. American Call
# ============================================================================

# ----------------------------------------------------------------------------
# 3.1 Underlying Asset Pays No Dividends
# ----------------------------------------------------------------------------
# In the case of a non-dividend-paying asset, American and European calls have
# the same price. The details can be found in Shreve (2008, pp. 361 - 363), but
# are not presented here.

# ----------------------------------------------------------------------------
# 3.2 Underlying Asset Pays Dividends
# ----------------------------------------------------------------------------
# If the asset pays dividends, the prices of American and European calls may
# differ.
#
# Let's consider an American call on an asset that pays dividends whose price
# process is a geometric Brownian motion governed by equation (2.1.1). We assume
# there are times 0 < t_1 < t_2 < ... < t_n < T, and at each time t_j the
# dividend paid is a_j * S(t_j-), where S(t_j-) denotes the asset price just
# prior to the dividend payment. The asset price S(t_j) after the dividend
# payment is the asset price before the dividend payment less the dividend
# payment:
#
#     S(t_j) = S(t_j-) - a_j * S(t_j-) = (1 - a_j) * S(t_j-).
#
# We assume that each a_j, j = 1, ..., n, is a number between 0 and 1. We set
# t_0 = 0, but this is not a dividend payment date. We also assume that T is
# not a dividend payment date, although it is not difficult to modify the
# analysis to handle the case when T is a dividend payment date.
#
# We will not go through the details here, but Shreve (2008, pp. 363 - 368)
# shows that it is not optimal to exercise an American call on this asset except
# possibly immediately before a dividend payment. The price of the call is shown
# to satisfy the Black-Scholes-Merton PDE between dividend payment dates. At
# dividend payment dates, the price of the call is the maximum of the call's
# intrinsic value and the price of the call after the dividend is paid and the
# stock price is reduced by the amount of the payment. These observations lead
# to a recursive algorithm for determining the price.

```

Exercises

Exercise 1. State the optimal stopping problem for an American put. What is the smooth pasting condition?

Solution to Exercise 1

\(V(S,t) = \sup_{\tau} E^Q[e^{-r(\tau-t)}(K - S_\tau)^+ | S_t = S]\). At the exercise boundary \(S^*(t)\): \(V(S^*,t) = K - S^*\) (value matching) and \(V_S(S^*,t) = -1\) (smooth pasting). Smooth pasting ensures the value function transitions smoothly from the continuation to the exercise region.

Exercise 2. Derive the value and optimal boundary for the perpetual American put.

Solution to Exercise 2

The ODE \(\frac{1}{2}\sigma^2 S^2 V'' + rSV' - rV = 0\) has solution \(V = AS^\alpha\) with \(\alpha < 0\). Smooth pasting gives \(S^* = \frac{\alpha}{\alpha-1}K\) and \(V(S) = (K - S^*)(S/S^*)^\alpha\).

Exercise 3. Why does an American call on a non-dividend-paying stock equal the European call?

Solution to Exercise 3

Early exercise means paying \(K\) now to receive \(S\). But \(C \ge S - Ke^{-r(T-t)} > S - K\) for \(r > 0\), so the continuation value always exceeds the exercise value. Therefore early exercise is never optimal and \(C_A = C_E\).

Exercise 4. Explain the low bias of the LSM estimator and how to mitigate it.

Solution to Exercise 4

LSM finds a suboptimal exercise strategy (limited by the regression basis), yielding a price \(\le\) the true American price. Mitigation: use rich basis functions, separate regression and pricing path sets, or combine with an upper-bound estimator for a confidence interval.