1 Option Greeks
References: [[pstat-222-option-notes-pdf|Haosheng PSTAT222 Notes]]. [[bjork-pdf]]
In mathematical finance, specifically derivatives pricing, we study methods of determining the prices \(V(t,S_{t})\) of derivative contracts. Options have several factors affecting their price: (1) time which is constantly eroding their value, (2) underlying asset price, and (3) underlying asset volatility; and it is important for traders to be able to measure the sensitivity of the option price process to these factors. These sensitivity measures are known as the option greeks.
Suppose a derivative on asset \(S_{t}\) has value at time \(t\) given by \(V(t,S_{t})\). Furthermore, denote the volatility of the asset price process by \(\sigma\) and the risk-free rate by \(r\). The option greeks are defined as follows: 1. Delta: \(\Delta= \frac{\partial V}{\partial S}\); 2. Gamma: \(\Gamma = \frac{\partial^2V}{\partial S^2}\); 3. Theta: \(\Theta=\frac{\partial V}{\partial t}\); 4. Vega: \(\mathcal{V}=\frac{\partial V}{\partial \sigma}\); and 5. Rho: \(\rho = \frac{\partial V}{\partial r}\).
Intuitively, the greeks are a collection of first and second order partial derivatives of the derivative value process with respect to asset price, asset volatility, time and the risk-free rate. For notes on the use of PDEs in derivatives pricing see our note Black-Scholes PDE (BSPDE).
1.1 Option Greeks of Common Options
The derivations of the analytical results for the option Greeks for common options are detailed in our notes for each specific option type:
- Black-Scholes Model Options:
- Black-Scholes Model European Options
- Black-Scholes Model American Options
- Black-Scholes Model Corridor Options
- Black-Scholes Model Asian Options
- Black-Scholes Model Compound Options
- Heston Model Options
1.1.0.1 Example - BS Model Euro-Call Greeks
Consider a Euro-Call with strike \(K\) at maturity \(T\) on a stock \(S_{t}\) with dynamics given by the Black-Scholes-Merton Model i.e. Geometric Brownian Motion GBM) \[ \begin{cases} dS_{t} = \mu S_{t}dt+\sigma S_{t}dW_{t} \\ dB_{t} = rB_{t}dt, \end{cases} \] with drift and diffusion \(\mu,\sigma\) and BM \(W_{t}\). The payoff function for the option is \[ \Phi(S_{T})=(S_{T}-K)_{+}. \] The solution of GBM is given by \[ S_t = S_0\exp\left(\left(\mu-\frac{1}{2}\sigma^2\right)t+\sigma W_t\right), \] and hence under the risk-free measure, the price of the option is given by \[ V(t,S_{t})=\mathbb{E}_{\mathbb{Q}}[e^{-r(T-t)}(S_{T}-K)_{+}]. \] Although this can be computed to a high accuracy using [[sde-monte-carlo-methods]] to compute the expli
2 Option Greek Estimation
In this section we explore the various classic empirical methods for the estimation of option Greeks. Many of the results in this section build upon our study of numerical methods for derivatives pricing.
2.1 Bump-and-Revalue
The easiest way to approximate option Greeks is the bump-and-revalue method whereby we approximate option Greeks using estimations for option prices \[ \begin{align} \widehat{\Delta} & = \lim_{ h \to 0 } \frac{\widehat{V}(0,S_{0}+h)-\widehat{V}(0,S_{0}-h)}{2h} \\ \widehat{\Gamma} & = \lim_{ h \to 0 } \frac{ \widehat{V}(0,S_{0}+h)+\widehat{V}(0,S_{0-h})-2\widehat{V}(0,S_{0})}{h^2}, \end{align} \] where we have employed Numerical Option Pricing Methods such as Finite-Difference Methods or Monte-Carlo Methods. Since we have already estimated the option price, this estimation does not require any significant additional work (no more simulation).
We note however that there is a trade-off here between accuracy and efficiency since an accurate numerical differentiation scheme requires evaluations of option prices for multiple values of \(S_{0}\) but each estimated option price has its own error and it is time-consuming to run Monte-Carlo multiple times.
2.2 Path-wise Estimation
We can perform path-wise estimation of option Greeks by taking the derivative with respect to the option price in the analytic formula before performing our Monte-Carlo simulations. To provide some context, assume for example that we have the BS Model and a Euro-Call so the option price at \(t=0\) is given under risk-neutral measure \(\mathbb{Q}\) by \[ \mathbb{E}_{\mathbb{Q}}[e^{-rT}(S_{T}-K)_{+}]. \] From standard results we details in European Call Option Price we know that under \(\mathbb{Q}\) \[ S_{T}=S_{0}\exp \left\{ \left( r- \frac{\sigma^2}{2} \right)t+\sigma W_{t} \right\} \] and so the option \(\Delta\) can be computed through \[ \begin{align} \Delta & = \frac{\partial}{\partial S_{0}}\mathbb{E}[e^{-rt}(S_{T}-K)_{+}] \\ & =\mathbb{E}\left[ \frac{\partial}{\partial S_{0}}e^{-rt}(S_{T}-K)_{+} \right] \\ & =\mathbb{E}\left[ e^{-rt}\exp \left\{ \left( r- \frac{\sigma^2}{2} \right)t+\sigma W_{t} \right\}\mathbb{1}_{\{ S_{T}>K \} } \right] \\ & =\mathbb{E} \left[ e^{-rt} \frac{S_{T}}{S_{0}}\mathbb{1}_{\{ S_{T}>K \}} \right]\end{align}, \] if we assume that the interchange of the expectation and the derivative is justified. Accordingly, one has to simulate the trajectories of \(S_{t}\), calculate this value and take the sample average. This methodology is also known as infinitesimal perturbation for the payoff function and is significantly more efficient and accurate than the bump-and-revalue method.
One drawback of this methodology is that we may not be able to deal with more exotic options or higher-order option Greeks. Specifically, this method falls down for non-smooth payoff functions including those with discontinuities or kinks (which strangely is a very large number of option types). For example, this methodology cannot help us estimate \(\Gamma\) under Black-Scholes since taking the derivative for the indicator function causes problems. More generally, any option payoff structure that includes payoff functions are off the table.
Despite this drawback we note the following interesting result. Denote the option payoff as \(\Phi(S_{T})\) and assume it has a nice structure (i.e. does not contain an indicator function) so that the option price under risk-neutral measure \(\mathbb{Q}\) is given by \(\mathbb{E}_{\mathbb{Q}}[e^{-rt}\Phi(S_{T})]\). Path-wise estimation always needs to know \[ \frac{\partial}{\partial S_{0}}h(S_{T})= h'(S_{T}) \frac{{\partial S_{T}}}{\partial S_{0}}, \] and so the problem turns into finding \(\frac{{\partial S_{T}}}{\partial S_{0}}\). However, this partial derivative has already been provided within the SDE dynamics. Assume that the stock price follow the SDE \[ dS_{t}=a(S_{t})dt+b(S_{t})dW_{t} \] under the risk-neutral measure. Using the Euler-Maruyama Method we perform the simulation \[ S_{t_{k+1}}=S_{t_{k}}+a(S_{t_{k}})(t_{k+1}-t_{k})+b(S_{t_{k}})\sqrt{ t_{k+1}-t_{k} }Z_{k+1} \] where \(Z_{k+1}\stackrel{i.i.d}\sim\mathcal{N}(0,1)\). Denoting \(\Delta t_{k}:= \partial S_{t_{k}} /\partial S_{0}\) and differentiating both sides with respect to \(S_{0}\) we obtain \[ \Delta t_{k+1}=\Delta t_{k}+a'(S_{t_{k}})\Delta t_{k}(t_{k+1}-t_{k})+b'(S_{t_{k}})\Delta t_{k}\sqrt{ t_{k+1}-t_{k} }Z_{k+1}, \] with \(\Delta_{0}=1\). This gives rise to the tangent SDE of \(S_{t}\) \[ \begin{cases} d\Delta t =a'(S_{t})\Delta_{t}dt + b'(S_{t})\Delta_{t}dW_{t} \\ \Delta_{0}=1. \end{cases} \] The path-wise estimation can be generalized to path-dependent option Greeks. For example, consider an Asian-Call with payoff \[ \begin{align} \frac{\partial}{\partial S_{0}}\mathbb{E}[e^{-rT}(\overline{S}-K)_{+}] & = \mathbb{E}\left[ e^{-rT} \frac{\partial}{\partial S_{0}}(\overline{S}-K)_{+} \right] = \mathbb{E}\left[ e^{-rT}\mathbb{1}_{\{ \overline{S}>K \}} \frac{{\partial \overline{S}}}{\partial S_{0}} \right] \\ & =\mathbb{E}\left[ e^{-rT}\mathbb{1}_{\{\overline{S}>K\}} \frac{1}{m}\sum_{i=1}^{m} \frac{S_{t_{i}}}{S_{0}} \right]=\mathbb{E}\left[e^{-rT}\mathbb{1}_{\{\overline{S}>K\}} \frac{\overline{S}}{S_{0}}\right]. \end{align} \] We note that this estimator requires negligible extra work to compute after trajectory simulations and has practical use due to the lack of a closed form formula for the price of Asian options.
2.3 Likelihood Ratio Method (LRM)
The Likelihood Ratio method is the proposed methodology that is applicable for non-smooth functions. The idea is to represent the option price as an integral and differentiate the density of stock price instead of the payoff function.
For European option price with payoff \(\Phi\) we have \[ \mathbb{E}[e^{-rT}\Phi(S_{T})]=e^{-rT}\int_{}^{}{\Phi(x)f_{S_{T}}}~d{x}, \] where \(f_{S_{T}}\) denotes the density of \(S_{T}\). To obtain \(\Delta\) we differentiate with respect to \(S_{0}\) and get \[ \begin{align} \Delta & = \frac{\partial}{\partial S_{0}}e^{-rT}\int_{}^{}{\Phi(x)f_{S_{T}}}(x)~d{x} \\ & =e^{-rT}\int_{}^{}{\Phi(x) \frac{\partial}{\partial S_{0}}f_{S_{T}}(x)}~d{ x} \\ & =e^{-rT}\int_{}^{}{\Phi(x) \left( \frac{\partial}{\partial S_{0}}f_{S_{T}}(x)\right) \frac{1}{f_{S_{T}}(x)}f_{S_{T}}(x) }~d{x} \\ & = e^{-rT}\int_{}^{}{\Phi(x) \left( \frac{\partial}{\partial S_{0}} \log f_{S_{T}}(x) \right)f_{S_{T}}(x)}~d{x} \\ & = e^{-rT}\mathbb{E}\left[ \Phi(S_{T})\left( \frac{\partial \log f_{S_{T}}}{\partial S_{0}}(S_{T}) \right) \right], \end{align} \] again assuming the interchange between the integral and the derivative is justified. We notice here that \(g(x)\)is free of \(S_{0}\) while \(f_{S_{T}}\) depends on \(S_{T}\) thus being a function of \(S_{0}\). This clever log-likelihood trick coming from Statistical Inference circumvents the non-smoothness of \(\Phi\) and Monte-Carlo is again application to estimate \(\Delta\).
Under the Black-Scholes-Merton Model we have \[ S_{T}=S_{0}\exp \left\{ \left( r- \frac{\sigma^2}{2} \right)T + \sigma W_{T} \right\} \] which is a function of a gaussian random variable, and so some simple computations we obtain the density of \(S_{T}\) as \[ f(x)=\frac{1}{\sqrt{ 2\pi T }\sigma x} \exp \left\{ -\frac{{ \left( \log \frac{x}{S_{0}}-\left( r- \frac{\sigma^2}{2} \right)T \right)^2}}{2\sigma^2T} \right\} \] and so the likelihood ratio method estimates \(\Delta\) as \[ \begin{align} e^{-rT}\mathbb{E}\left[ \Phi(S_{T})\left( \frac{\partial \log f}{\partial S_{0}} (S_{T}) \right) \right] & = e^{-rT}\mathbb{E}\left[ \Phi(S_{T}) \frac{1}{\sigma^2 TS_{0}}\left( \log \frac{S_{T}}{S_{0}}-\left( r- \frac{\sigma^2}{2} \right)T \right) \right] \\ & =e^{-rT}\mathbb{E}\left[ \Phi(S_{T}) \frac{W_{T}}{\sigma T S_{0}} \right]. \end{align} \] We notice that the expression above works for any payoff function \(\Phi\) under the Black-Scholes model so we need just calculate the derivative of the log-likelihood (score function) once and for all for each model. This method works well for European options but is complicated for path dependent payoff functions (e.g. Asian options) where one has to consider the joint distribution of \((S_{t_{1}}, \dots, S_{t_{m}})\) and compose it into the product of conditional densities together with the Markovian structure of GBM.
For \(\Gamma\) LRM works where path-wise estimation may fail to work. A similar derivation shows \[ \begin{align} \Gamma & = \frac{\partial^2}{\partial S_{0}^2}e^{-rT}\int_{}^{}{g(x)f_{S_{T}}(x) }~d{x} \\ & = e^{-rT}\int_{}^{}{g(x) \frac{\partial^2}{\partial S_{0}^2}f_{S_{T}}(x)}~d{x} \\ & =e^{-rT}\int_{}^{}{g(x) \left( \frac{\partial^2 f_{S_{T}}(x)}{\partial S_{0}^2}\cdot \frac{1}{f_{S_{T}}(x)} \right)f_{S_{T}}(x)}~d{x} \end{align} \] Notice that