As a precursor to studying market risk, VaR estimation, and yield curve constructions in future posts, we discuss the basics of mapping risk-factors and estimating loss/change in portfolio value. I will be following Quantitative Risk Management (McNiel-Frey-Embrechts) and An Overview of Value at Risk (Duffie-Pan). Nothing in this post is original and the purpose is more so for future reference.
Mapping Risks
The uncertainty about future states of the world is represented by a probability space \((\Omega,\mathcal{F}, P)\). The value of a portfolio at time \(t\) is denoted by \(V_t\).
Further assumptions:
- The random variable \(V_t\) is known or can be determined from information available, at time \(t\).
- The portfolio composition remains fixed over a time horizon
- There are no intermediate payments of income during the time period
Choice of time horizons \(\Delta t\): one day or ten days in market risk, or one year in credit, insurance or enterprise-wide risk management
For short time intervals, the change in value of the portfolio is \(\Delta V_{t+1} = V_{t+1} - V_t\) and loss is defined to be \(L_{t+1}:=-\Delta V_{t+1}\). This doesn’t account for the time value of money, so for longer time intervals, the loss is \[L_{t+1}:=V_t - \frac{V_{t+1}}{1+r_{t,1}},\] where \(r_{t,1}\) is the simple risk-free interest rate that applies between times \(t\) and \(t+1\).
How do we model \(V_t\)? Typically, it is modeled as a function of time and a \(d\)-tuple random vector \(\mathbf{Z}_t\) of risk factors \(Z_{t,j}\) for \(j=1,\dots,d\). So, \[\begin{align} V_t=f(t,\mathbf{Z}_t), \tag{1} \end{align}\] for some measurable function \(f:\mathbb{R}_{+}\times\mathbb{R}^d\to \mathbb{R}.\) This is what we mean by mapping of risks, i.e., the representation of the portfolio value as a function of time and risk factors. The rv \(\mathbf{X}_{t+1}:=\mathbf{Z}_{t+1}-\mathbf{Z}_t\) is useful for approximating loss; we will refer to it as the vector of risk-factor changes. So, with this notation, the loss from \(t\) to \(t+1\) is \[\begin{align} L_{t+1} = f(t,\pmb{z}_t) - f(t+1, \pmb{z}_t+\mathbf{X}_{t+1}), \tag{2} \end{align}\] where \(\mathbf{z}_t\) is the realized value of \(\mathbf{Z}_t\) at time \(t\) (similarly \(f(t,z_t)\) is the realized value of \(V_t\)). Since \(\mathbf{Z}_t\) is known at time \(t\), \(L_{t+1}\) is determined by the distribution of the risk factor change at \(t+1.\)
Example. If we have a fixed portfolio of \(d\) stocks and \(n_i\) denotes the number of shares of stock \(i\) in the portfolio at time \(t\), let \((S_{t,i})_{t\in\mathbb{N}}\) denote the price process of stock \(i\). In this case the vector \(\mathbf{Z}_t\) of risk factors is just 1-dimensional for each stock, namely it’s the logarithmic price \(\ln(S_{t,i})\). Hence, the value of the portfolio at time \(t\), in terms of the risk factors, is \[\begin{align} V_t&=n_1\exp(Z_{t,1})+n_2\exp(Z_{t,2})+\cdots+n_5\exp(Z_{t,5})\\ &=\sum_{i=1}^d n_i \exp(Z_{t,i}) \end{align}\]
The portfolio loss from time \(t\) to \(t+1\) is then given by \[\begin{align} L_{t+1} &= -(V_{t+1} - V_t)\\ & = \sum_{i=1}^d n_i \left(S_{t,i}- \exp(z_t + X_{t+1})\right)\\ & = \sum_{i=1}^d n_i S_{t,i}\left(1- \exp(X_{t+1})\right). \end{align}\] We will see how we can approximate this loss in the next section.
Example. Consider a portfolio of a standard European call on a non-dividend paying stock with maturity time \(T\) and strike price \(K\). The value of this call can be calculated using the Black-Scholes formula \[\begin{align} C(t,S;r,\sigma,K,T) = C^{\operatorname{BS}}(t,S;r,\sigma,K,T):= S N(d_1) - K\exp(-r(T-t))\,N(d_2), \end{align}\] where \(N(-)\) denotes the standard normal (cumulative) distribution function, \(r\) the risk-free interest rate, \(\sigma\) the volatility of the underlying stock, and where \[\begin{align} d_1 &= \frac{1}{\sigma\sqrt{T-t}} \left(\ln(S/K) + (r+\frac 12\sigma^2)(T-t)\right)\\ d_2 &= d_1 - \sigma\sqrt{T-t}. \end{align}\] To model the (changes in) value of our portfolio, we have to pick a set of risk factors. The first obvious factor is the log price \(\ln(S_t)\) of the underlying stock. Contrary to the assumption of the Black-Scholes formula that the interest rates and volatility be kept constant, in reality, these quantities change and so, we have to account for that. So, the vector of risk factors is \(\mathbf{Z}_t= (\ln S_t, r_t, \sigma_t)^T\), where \(\sigma_t\) is the implied volatility.
Approximating Loss
To transform the problem of analyzing loss distributions to a problem in financial time-series analysis, we measure time in units of \(\Delta t\) (the time horizon, usually 1 day = 1/365 or 10 days for us) and introduce the notation for \((Y_t)_{t\in\mathbb{N}}\) so that \[Y_t:=Y(\tau_t),\] where \(\tau_t:=t\cdot \Delta t\). So, \(V_t:=g(\tau_t, \mathbf{Z}_t)\), where \(g\) is an appropriate measurable function as \(f\) above (1).
Motivated by (2), we can define an operator \[\begin{align} \ell_{[t]}:\mathbb{R}^d&\to \mathbb{R}\\ \pmb{x} &\mapsto -(g(\tau_{t+1},\pmb{z}_t+\pmb{x})-g(\tau_t,\pmb{z}_t)), \end{align}\] where, as earlier, \(\pmb{z}_t\) is the realized value of \(\mathbf{Z}_t\) at time \(t\). We call this operator the loss operator at time \(t\). So, we have that \(L_{t+1}=\ell_{[t]}(\mathbf{X}_{t+1})\).
Linear (Delta) Approximation
Assuming that the mapping function \(g\) above is differentiable, and for a small \(\Delta t\), we have a first-order Taylor approximation: \[\begin{align} g(\tau_t+\Delta t, \pmb{z}_t+\pmb{x})\approx g(\tau_t,\pmb{z}_t) + g_{\tau}(\tau_t,\pmb{z})\Delta t + \sum_{i=1}^d g_{z_i}(\tau_t,\pmb{z}_t)x_i, \end{align}\] where \(g_\tau\) denotes the partial \(\partial_\tau g\) and likewise \(g_{z_i}=\partial_{z_i}g\). So, we get the linear loss operator at time \(t\): \[\begin{align} \ell^{\Delta}_{[t]}(\pmb x)= -\left(g_{\tau}(\tau_t,\pmb{z}_t)\Delta t + \sum_{i=1}^d g_{z_i}(\tau_t,\pmb{z}_t)x_i\right) \tag{3} \end{align}\]
Quadratic (Delta-Gamma) Approximation
First let \[\begin{align} \delta(\tau_t,\pmb{z}_t) &= \left[g_{z_i}(\tau_t,\pmb{z}_t)\right]_{i=1,\dots,d}\\ \omega(\tau_t,\pmb{z}_t) &= \left[\partial_\tau\,\partial_{z_i}g(\tau_t,\pmb{z}_t)\right]_{i=1,\dots,d}\\ \Gamma(\tau_t,\pmb{z}_t) &= \left[\partial_{z_j}\partial_{z_i}g(\tau_t,\pmb{z}_t)\right] = [g_{z_i\,z_j}(\tau_t,\pmb{z}_t)]_{i,j}, \end{align}\] and the diagonal entries of the matrix \(\Gamma(\tau_t,\pmb{z}_t)\) are referred to as gamma sensitivities of the risk factors while the off diagonal entries are the cross gamma sensitivities.
The second-order Taylor approximation of \(g(\tau_t,\pmb{z}_t)\) is given by \[\begin{align} g(\tau_t+\Delta t, \pmb{z}_t+\pmb x) \approx g + g_\tau\cdot\Delta t + \delta\cdot \pmb x + \frac 12 \left[g_{\tau\tau}(\Delta t)^2 + 2\omega\pmb x \Delta t + \pmb{x}^T\Gamma\pmb{x}\right]. \tag{4} \end{align}\]
At this point, terms that are \(o(\Delta t)\), i.e., \((\Delta t)^k\) for \(k>1\) can be omitted since \(\Delta t\) is small. Terms with \(\pmb x\Delta t\) can also be omitted if we assume the risk factors follow a standard continuous-time process. See Appendix below.
Hence, from (4), we obtain the quadratic loss operator \[\begin{align} \ell^{\Delta\Gamma}_{[t]}(\pmb{x}) := -\left(g_\tau(\tau_t,\pmb{z}_t)\Delta t +\delta(\tau_t,\pmb{z}_t)^T\pmb{x} + \frac 12 \pmb{x}^T\Gamma(\tau_t,\pmb{z}_t)\pmb{x}\right). \tag{5} \end{align}\]
Examples
We return to the simple example of stock portfolio we discussed earlier. Here, \(\Delta t = 1/365\) and so drop the \(g_\tau(\tau_t,\pmb{z}_t)\) term in (3). The linearized loss \(L_{t+1}\approx \ell^{\Delta}_{[t]}(X_{t+1})\) would be \[\begin{align} -\sum_{i=1}^{d}n_i S_{t,i} X_{t+1,i}, \end{align}\] which, by introducing the weight \(w_{t,i}:=(n_i S_{t,i})/V_t\) becomes \[\begin{align} -V_t\sum_{i=1}^d w_{t,i} X_{t+1,i}. \end{align}\] The weight gives the proportion of the portfolio value invested in stock \(i\) at time \(t.\) One can then use information about the mean vector and covariance matrix of the distribution of the risk-factor changes \(X_t\) to calculate the expectation and variance (first two moments) of the linearized loss.
Example. Now, for another example, suppose a bank writes/shorts a European call option and delta-hedges it to remove some of the risk. That is, the portfolio is long on \(h_t\) (delta of the option) shares of the underlying.
Let \(\Delta t = 1/250\) and \(\tau_t = t/250\). So, the value of the portfolio \(V_t\) at time \(t\) is: \[\begin{align} V_t &= h_tS_t - C^{\operatorname{BS}}(\tau_t, S_t; r_t,\sigma_t, K, T) \end{align}\] Assume, for simplicity, that \(r_t\) is constant and we will just write \(C\) for \(C^{\operatorname{BS}}(\cdot)\). The risk-factors we consider are \(\sigma_t\) (implied market volatility), log-prices \(\ln S_t\). With \(g(\tau_t, \mathbf{Z}_t) = V_t\), a linear approximation of the loss (3) the next day is \[\begin{align} \ell_{[t]}^{\Delta} &= -\left((h_tS_t)_{\tau}-C_{\tau}\right)\Delta t - g_{z_1}x_1 - g_{z_2}x_2\\ &= C_{\tau}\Delta t - ((h_tS_t)_{z_1} - C_{z_1})x_1 - ((h_tS_t)_{z_2} - C_{z_2})x_2\\ &= C_{\tau}\Delta t - (h_tS_t)_{z_1} + C_{z_1}x_1 - (h_tS_t)_{z_2} + C_{z_2}x_2. \end{align}\] Since \(z_1=\ln S\) and \(z_2=\sigma_t\), we have \((h_tS_t)_{z_2}=0\) and \[\begin{align} (h_tS_t)_{\ln S} &= h_t\cdot \frac{\partial S}{\partial S} \frac{\partial S}{\partial(\ln S)}\\ &= h_t\cdot \frac{\partial \exp(\ln S)}{\partial(\ln S)}\\ &= h_t\cdot S_t \end{align}\] while \[\begin{align} C_{z_1} = C_{\ln S} & = \frac{\partial C}{\partial S} \frac{\partial S}{\partial(\ln S)}\\ & = C_S \cdot S \\ & = h_t S_t. \end{align}\] Thus, \[\begin{align} \ell_{[t]}^{\Delta} = C^{\operatorname{BS}}_\tau\Delta t + C^{\operatorname{BS}}_{\sigma} x_2. \end{align}\] We observe that for this delta-approximation of the loss, the stock return makes no contribution and only depends on the theta \(C^{\operatorname{BS}}_\tau\) and the vega \(C^{\operatorname{BS}}_{\sigma}\) of the option at a given time.
Now, if the risk-factor changes \(\pmb{x} = (0.05, 0.02)^T\), i.e, a stock return of around 5% and an increase in implied volatility of 2% the next day, and we determined that theta \(C_\tau\approx -4.83\) and vega \(C_{\sigma}\approx 34.91\), then \[\begin{align} \ell_{[t]}^{\Delta} &= C^{\operatorname{BS}}_\tau\frac{1}{250} + C^{\operatorname{BS}}_{\sigma} {0.05}\\ & \approx -0.019 + 0.698 = 0.679, \end{align}\] whereas a direct evaluation of \(V_{t+1}\) gives \(\ell_{[t]}(\pmb{x})\approx 0.812\). It underestimates by 16% for this order of risk-factor change \(\pmb{x}.\) We can do better with a Delta-Gamma (quadratic) approximation.
A similar calculation shows that the quadratic loss (5) to be \[\begin{align} \ell_{[t]}^{\Delta \Gamma} &= C^{\operatorname{BS}}_\tau\Delta t + C^{\operatorname{BS}}_\sigma x_2 + \frac 12 C^{\operatorname{BS}}_{SS}S_t^2x_1^2 + C^{\operatorname{BS}}_{S\sigma}S_t x_1x_2 + \frac 12 C^{\operatorname{BS}}_{\sigma\sigma}x_2^2 \\ & = \ell^{\Delta}_{[t]}(\pmb{x}) + \frac 12 C^{\operatorname{BS}}_{SS}S_t^2x_1^2 + C^{\operatorname{BS}}_{S\sigma}S_t x_1x_2 + \frac 12 C^{\operatorname{BS}}_{\sigma\sigma}x_2^2 \\ & \approx 0.679 + 0.218 - 0.083 + 0.011 = 0.825, \end{align}\] which is less than 2% over the true loss. This is great, but it would under perform for longer time horizons and a bigger risk-factor change \(\pmb{x}\) (third-derivative not zero unaccounted for).
Appendix
Given risk factors that are modeled on a continuous-time process, such as the log-stock price in the Black-Scholes model, we will show that \(\pmb{x}\Delta t\) is actually \(o(\Delta t)\), where \(\pmb{x}\) is the vector of realized risk factor changes. For this case, consider \(\mathbf{Z}_t=(\ln S_t).\)
First, the underlying assumption (which will suffice for approximations) is that the stock price \(S_t\) follows the stochastic differential equation \(dS_t = \mu S_t\,dt + \sigma S_t\, dW_t\), where \(\mu\) is the drift, \(\sigma\) the volatility and \(W_t\) a Wiener process. By an application of Ito’s lemma on \(\ln(S_t)\), \[\begin{align} d(\ln S_t) = \frac{1}{S_t} dS_t + \frac 12 \left(-\frac{1}{S_t^2}\right) (d S_t)^2. \end{align}\] Then, substituting the SDE with the rules that \(dW_t^2 = dt, (dt)^2 =0,\) and that \(dW_t\,dt =0\), we get \[\begin{align} d(\ln S_t) = \frac{1}{S_t}(\mu S_t dt + \sigma S_t dW_t) + \frac 12 \left(-\frac{1}{S_t^2}\right) \sigma^2 S_t^2\, dt, \end{align}\] which after simplifying and rewriting yields \[\begin{align} d(\ln S_t) = \left(\mu - \frac 12\sigma^2\right)\,dt + \sigma\,dW_t. \end{align}\] So, \(\ln(S_t) = (\mu - \frac 12\sigma^2)t + \sigma W_t\), and the risk-factor change satisfies \[\begin{align} X_{t+1} = \ln\left(\frac{S_{t+1}}{S_t}\right) = \left(\mu - \frac 12\sigma^2\right)\Delta t + \sigma(W_{t+\Delta t}-W_t). \tag{6} \end{align}\] Here, \(W_{t+\Delta t}-W_t\) is a Brownian motion increment over the interval \(\Delta t\), which is normally distributed with mean 0 and variance \(\Delta t\). Therefore, if \(Y\) is a standard normal variable \(\mathcal{N}(0,1)\), then \[W_{t+\Delta t}-W_t = \sqrt{\Delta t}\cdot Y.\] Back to \(X_{t+1},\) this implies that \(X_{t+1}\) is normally distributed with mean \((\mu-\frac 12\sigma^2)\Delta t\) and variance \(\sigma^2\Delta t.\)
Now, as \(\Delta t\) approaches \(0\), \(X_{t+1}\) is converges in distribution to \(\sigma(W_{t+\Delta t}-W_t)\), so then \(X_{t+1}/(\sigma\sqrt{\Delta t})\) approaches \(\mathcal{N}(0,1).\) Hence, risk factor changes in this model are of order \(O(\sqrt{\Delta t})\). It follows then that the term \(x\Delta t\) tends to zero at the same rate as \((\Delta t)^{3/2}\) implying that it’s of order \(o(\Delta t).\) A similar argument works for risk factor differences following other continuous-time processes.