Explain the difference between European and American options, and how their pricing differs using stochastic calculus.?

European and American options are types of financial derivatives that give their holders the right, but not the obligation, to buy or sell an underlying asset at a predetermined price (the strike price) on or before a predetermined time (the expiration date). The main difference between European and American options lies in the exercise policy:

1. European Options: These can only be exercised at the expiration date. 2. American Options: These can be exercised at any time on or before the expiration date.

Due to this difference in exercise policy, pricing European and American options requires different approaches using stochastic calculus. In the case of European options, the Black-Scholes-Merton (BSM) model is most commonly used, while the pricing of American options often involves solving a more complex optimal stopping problem.

European Option Pricing:

The BSM model assumes that the logarithm of the underlying asset price follows a geometric Brownian motion (GBM) with a constant drift and volatility:

dS_t = μS_tdt + σS_tdW_t

Here, dS_t represents the change in the asset price S_t in an infinitesimally small time interval dt, μ is the drift (the expected return per unit time), σ is the asset’s volatility, and dW_t denotes a Wiener process (also called Brownian motion), which is the continuous-time analogue of a random walk.

Applying Ito’s lemma to the GBM of the underlying asset price and using risk-neutral valuation, we can derive the Black-Scholes equation for the value of a European call option C(S_t, t) and a European put option P(S_t, t):

$$\frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} + rS\frac{\partial C}{\partial S} - rC = 0$$

$$\frac{\partial P}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 P}{\partial S^2} + rS\frac{\partial P}{\partial P} - rP = 0$$

where r is the risk-free interest rate. These are parabolic partial differential equations that need to be solved with boundary conditions given by the payoff functions of the European call and put options to obtain their pricing formulas:

C(S_t, t) = S_tN(d₁) − Ke^{− r(T − t)}N(d₂)

P(S_t, t) = Ke^{− r(T − t)}N( − d₂) − S_tN( − d₁)

where N(x) is the cumulative distribution function of the standard normal distribution,

$$d_1 = \frac{\ln{\frac{S_t}{K}} + (r + \frac{1}{2} \sigma^2)(T-t)}{\sigma \sqrt{T-t}}$$

and

$$d_2 = d_1 - \sigma \sqrt{T - t}$$

American Option Pricing:

Compared to European options, the pricing of American options is more challenging due to the early exercise feature. For American options, we need to find the optimal stopping time t^* for exercising the option in such a way as to maximize its value. This leads to a free boundary problem, which can be solved using stochastic calculus methods, such as dynamic programming or martingale techniques.

An example of such a technique is the Least Square Monte Carlo (LSM) method, which estimates the value of an American option by solving a dynamic programming problem using regression analysis on simulated option price paths. The LSM method approximates the value of the option V(S_t, t) as the expected continuation value in each time step, using a polynomial regression function that depends on the underlying asset price S_t:

$$V(S_t,t) \approx E[V_t(S_{t+1}, t+1) | S_t] = \sum_{k=0}^{p} a_k(t) S_t^k$$

Here, E[ ⋅ ] denotes the conditional expectation, V_t( ⋅ ) represents the continuation value of the option, S_t + 1 is the asset price at the next time step, and a_k(t) are the regression coefficients. The optimal stopping time can then be found by comparing the current option value with the continuation value at each time step and choosing the time when the current option value is maximized.

In summary, the main difference between European and American options lies in their exercise policies, which leads to different pricing methodologies based on stochastic calculus. While the BSM model, formulated as a parabolic partial differential equation, is used to price European options, pricing American options typically involves solving an optimal stopping problem, such as the free boundary problem, using more complex stochastic calculus techniques.