Option Pricing
1 Option Pricing
A derivative or contingent claim has a payoff \(h\) that depends on the price process \((S_t)\) of an underlying asset, either at maturity, \(h(S_T)\), as for a European option, or along the path, \(h(S_{[0,T]})\), as for an Asian option. See Derivatives for the definitions.
The key idea of pricing is the principle of no arbitrage, which states that we cannot make a profit without risk. To determine the fair price of the derivative we describe the dynamics of \((S_t)\) and compute the expected payoff, adjusted for the time value of money and for risk. There are two equivalent approaches, both developed below for the Black-Scholes-Merton model:
- the risk-neutral approach, which prices the claim as a discounted expectation under a change of measure,
- the PDE approach, which prices the claim by replicating it with a self-financing portfolio.
2 Black-Scholes-Merton Model
The Black-Scholes-Merton (BSM) model considers a market consisting of two assets: a risk-free asset (e.g. a bond) \((B_t)\) and a risky asset (e.g. a stock) \((S_t)\), with dynamics given by Geometric Brownian Motion (GBM)
\[ \begin{align} dB_t & = rB_t\,dt, \tag{1}\\ dS_t & = \mu S_t\,dt + \sigma S_t\,dW_t, \tag{2} \end{align} \]
where \(r\) is the risk-free interest rate and \(\mu\) and \(\sigma\) are the drift and diffusion components of the stochastic differential equation (SDE).
Equation \((1)\) is an ODE with solution
\[ B_t = B_0 e^{rt}, \tag{3} \]
and the solution to GBM \((2)\) is
\[ S_t = S_0 \exp\left( \left( \mu - \frac{\sigma^2}{2} \right)t + \sigma W_t \right). \tag{4} \]
3 Risk-Neutral Approach
Under the real-world measure \(\mathbb{P}\), the drift \(\mu\) of \((S_t)\) reflects the preference of investors for larger returns on riskier assets. The arbitrage-free price of a derivative is therefore not its discounted expected payoff under \(\mathbb{P}\), since buyers must be compensated for risk.
Instead we change to the risk-neutral measure \(\mathbb{Q}\), constructed with Girsanov Theorem, under which the discounted stock price \(e^{-rt}S_t\) is a martingale. The drift of \((S_t)\) becomes \(r\),
\[ dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}, \tag{5} \]
where \(W^{\mathbb{Q}}\) is a Brownian motion under \(\mathbb{Q}\). The construction, the density \(d\mathbb{Q}/d\mathbb{P}\) and the conditions for it to exist are in Risk-Neutral Pricing. The price at time \(0\) of a claim with payoff \(h(S_T)\) is the discounted expected payoff
\[ V_0 = e^{-rT}\,\mathbb{E}_{\mathbb{Q}}[h(S_T)]. \tag{6} \]
4 PDE Approach
The idea is to build a self-financing replicating portfolio from the stock and the bond that has exactly the same payoff as the claim. Assuming no arbitrage, the portfolio and the claim must have the same price.
Let the portfolio hold \(a_t\) units of stock and \(b_t\) units of bond (with \(B_0=1\)), so its value is \(V_t = a_tS_t + b_te^{rt}\). Self-financing means \(dV_t = a_t\,dS_t + rb_te^{rt}\,dt\). Writing \(V_t = u(t,S_t)\) and comparing this with Itô Formula applied to \(u\) gives
\[ a_t = \partial_x u(t,S_t), \qquad b_te^{rt} = u(t,S_t) - S_t\,\partial_x u(t,S_t), \]
and the function \(u\) must solve the Black-Scholes PDE
\[ \partial_t u + rx\,\partial_x u + \frac{1}{2}\sigma^2x^2\,\partial_{xx}u - ru = 0, \qquad u(T,x) = h(x). \]
The price of the claim at time \(0\) is \(V_0 = u(0,S_0)\). The full derivation by delta-hedging, and an interpretation of the terms, is in the Black-Scholes PDE section of the Black-Scholes-Merton note. By the Feynman-Kac Formula, the solution of this PDE is the risk-neutral expectation \((6)\), so the two approaches agree.
5 Pricing by Option Type
- no-arbitrage bounds.
- Barrier Options
- Parametric Option Pricing