How would you price a derivative using the risk-neutral approach?

The risk-neutral approach, also known as risk-neutral valuation or risk-neutral pricing, is a technique used to price derivatives such as options and futures. This approach involves adjusting probabilities of future outcomes to eliminate market risk and then discounting these outcomes at the risk-free interest rate. The main idea behind risk-neutral pricing is that the price of a derivative should be equal to its expected payoff under the risk-neutral measure, discounted at the risk-free rate.

To price a derivative using the risk-neutral approach, we usually follow these general steps:

1. Determine the underlying asset’s probability distribution under the risk-neutral measure.

2. Calculate the expected payoff of the derivative under this risk-neutral measure.

3. Discount the expected payoff at the risk-free interest rate to obtain the present value.

Let’s illustrate the risk-neutral approach with an example. Suppose we want to price a European call option on a non-dividend-paying stock. The option has a strike price of K and expires in time T.

1. Risk-neutral measure: Assume the stock price follows the geometric Brownian motion:

dS_t = rS_tdt + σS_tdW_t

Here, S_t is the stock price at time t, r is the risk-free interest rate, σ is the stock price volatility, and dW_t is a Wiener process with mean 0 and variance dt.

Under the risk-neutral measure, the expected return on the stock becomes equal to the risk-free interest rate. Therefore, the stock price dynamics can be written as:

dS_t = rS_tdt + σS_tdW̃_t

where dW̃_t is a Wiener process under the risk-neutral measure.

2. Expected payoff: The payoff of the European call option at time T is given by (S_T − K)⁺, where (x)⁺ = max (x, 0). To find the expected payoff under the risk-neutral measure, we need to calculate the expectation of this payoff:

E_Q[(S_T − K)⁺]

First, we determine the distribution of S_T under the risk-neutral measure. By solving the stochastic differential equation, we get:

$$S_T = S_0 \exp \left((r - \frac{1}{2} \sigma^2)T + \sigma \sqrt{T}\widetilde{Z} \right)$$

where Z̃ ∼ N(0, 1) under the risk-neutral measure.

Now, we can calculate the expected payoff:

$$E_Q[(S_T - K)^+] = \int_{-\infty}^{\infty} (S_0 \exp \left((r - \frac{1}{2} \sigma^2)T + \sigma \sqrt{T}\widetilde{z} \right) - K)^+ \phi(\widetilde{z}) d\widetilde{z}$$

Here, ϕ( ⋅ ) is the standard normal probability density function.

3. Discounting at the risk-free rate: To find the present value of the option, discount the expected payoff at the risk-free interest rate:

P₀ = e^− rTE_Q[(S_T − K)⁺]

By solving the integral in step 2 and applying the discount factor, we obtain the Black-Scholes-Merton formula for European call options:

P₀ = S₀N(d₁) − Ke^− rTN(d₂)

where

$$d_1 = \frac{\ln \left(\frac{S_0}{K}\right) + \left(r + \frac{1}{2} \sigma^2 \right)T}{\sigma\sqrt{T}}$$

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

and N( ⋅ ) is the cumulative distribution function of the standard normal distribution.

In summary, the risk-neutral approach is a powerful method for pricing derivatives. By adjusting the probabilities of future outcomes to eliminate market risk and discounting the expected payoff at the risk-free rate, we can obtain the present value of a derivative. For European options, this approach leads to the well-known Black-Scholes-Merton formula.