APPROXIMATING OPTION PRICES UNDER LARGE CHANGES OF UNDERLYING ASSET PRICES

1. Introduction

In risk management, the delta-gamma approximation (DGA) is often used to reduce the risk associated with the underlying asset price. The performance of this strategy mainly depends on the time-to-maturity (i.e. the difference between maturity time and time horizon), the implied volatility, and the moneyness (which in turn depends on the initial underlying asset price and the strike). As the underlying asset price varies with respect to its initial value (both negatively and positively), we observe that the DGA works well for a small range of the aforementioned parameters, but when this change becomes large, the option price estimated by the DGA can be significantly different from the actual value (or the one that is estimated, for instance, by the Black–Scholes model).

The present work is devoted to European options (in particular, call options^a) for large variations of the underlying asset price (which may be thought as an equity for instance). Recall that a call option is a contract signed between a writer and a buyer that gives the buyer the right to purchase the underlying asset at a fixed price (the strike price) at a given maturity, at the expense of a fee given to the writer, at the time of signing the contract. Such an option (as a derivative) is a financial product that depends on at least five elements: the value of the underlying asset, time-to-maturity, implied volatility, the interest rate and the strike price. This dependency on five elements shows that it is not a simple task to faithfully estimate the behavior of options.

The DGA method is often used both for the derivative position management and risk management (Raju (2012), Castellacci & Siclari (2003)). This method approximates the option-price changes (between a given time horizon and the current time) using simple functions (often polynomials) that depend on changes in underlying asset prices. The usage of such simple functions offers two main advantages. It allows us first to efficiently consider a large number of scenarios of option prices using the DGA (thanks to its simplicity) for the risk management. These scenarios can be subsequently used in the calculations of value-at-risk (VaR). Along this line, the DGA has also the potential to be used in credit value adjustment (CVA) and the solvency capital requirement (SCR). Further, the simple functions used in the DGA method allow us to replicate the option position using both the underlying asset and other financial instruments in the position management perspective.

In fact, the limitations of the DGA with respect to the changes of the underlying asset prices are already hinted in Hull (2017), Wilmott (2006), Rouah & Vainberg (2007) (among other references). Because the DGA uses up to the second-order polynomial function with respect to the changes of the underlying asset price, one might be tempted to add higher-order Greeks (such as theta, speed, etc.) with the expectation to improve the performance obtained using the DGA. However, the usage of additional sensitivity terms comes with some associated costs (such as time and memory consumption, and the requirement of hedging instruments). Further, the consideration of higher-order terms in the Taylor series does not necessarily improve the accuracy of the option-price approximation (Estrella (1995), Estrella & Kambhu (1997)).

Hence, before the change of the underlying asset price becomes large, rebalancing operations are demanded (to reduce the losses produced due to the inaccuracy of the DGA). The frequency of rebalancing may be high when the rate of the change of the underlying asset price is significant. But, frequent rebalancing may be unattainable, as there are associated transaction costs. Hence, there is a trade-off between losses resulting from the inaccurate performance of the DGA and the transaction costs incurring from frequent hedging operations.

We realize that this drawback is mainly due to the fact that the DGA method approximates the (call or put) option price by considering the initial underlying asset price as its reference value. In this work, we first argue that this reference value is not the most appropriate one (by the evidence observed from the poor approximation achieved using the DGA method), and we should consider another reference value for a better approximation performance. A good candidate for this reference value might be the forecasted underlying asset price at the time horizon.^b This value may be estimated using various methods such as regression, simple historical approaches, a neural network, and the functional data analysis (FDA) (Ramsay & Silverman (2007)). We coin this modified version of the DGA method as the extended delta-gamma approximation (EDGA). We then propose another approach to approximate the option prices. This time, it is an algorithmic approach known as the locally weighted regression (LWR) (Cleveland & Devlin (1988)), but always considering the new reference value (as we do with the EDGA). We compare the results obtained using the two approaches that we propose in the present work to those that are obtained using some existing methods.

Our work is presented as follows. We first review the literature to define the Black–Scholes option price, the delta approximation (DA), and the delta-gamma approximation (DGA) in Sec. 2. We also present the performance of the DA and DGA in the same section. Afterwards, we propose the extended delta-gamma approximation (EDGA) and the locally weighted regression (LWR) in Secs. 3 and 4, respectively. Moreover, we define the relative approximation error of call option prices in Sec. 5. We then present and discuss in Sec. 6 the results obtained from both methods proposed in this work (EDGA and LWR), as well as those obtained from existing methods (DA and DGA). We finish the presentation of our work with concluding remarks and possible future work in Sec. 7.

2. Literature Review

2.1. Black–Scholes call option price

The European call option price at time horizon t expiring at T with the initial underlying asset price $S_0$, and the strike price K can be modeled using the Black–Scholes pricing model

where $S_t$ is the underlying asset price, T is the maturity time, ${\sigma }_t$ is the implied volatility of the price$C_t$, $r_t>0$ is the prevailing interest rate (free of credit risk), K is the strike price, $\mathrm{\Phi }$ is the cumulative distribution function of the standard Gaussian law, and $d_{\pm }$ are defined as

Notice that the original Black–Scholes formula assumes in (2.1) that the implied volatility and the interest rate are constant. That is, for the original Black–Scholes formula (2.1), ${\sigma }_t=\sigma$ and $r_t=r$, respectively. Theoretically, the Black–Scholes formula underlies the assumption that the asset price follows a log-normal law. However, in practice, the implied volatility changes with time (${\sigma }_t$) as well as the interest rate ($r_t$).

2.2. Changes of call option prices

In position management or in risk management, the change of the call option price at time t with respect to the initial call option price can be defined as

For simplicity, the following assumption is often made in the literature: That is, the prevailed interest rate is often supposed that does not change largely with respect to the interest rate initially considered. Further, as the original Black–Scholes model does, for simplicity, the same implied volatility can be used for all time horizons assuming that it does not change largely Next, we can normalize (2.5) by $C_0$ to obtain the relative change of call option prices:

Further, in Secs. 2.3 and 2.4, we review the methods that replicate the option prices using both the underlying asset and other financial instruments.

2.3. Delta approximation

With the purpose to hedge (i.e. to invest to reduce the risk of adverse price movements), the change of call option price ($C_t-C_0$) is often approximated using the delta approximation (DA)

where $v_t$ is the relative change (or linear return) of the underlying asset prices during the time period $(0,t)$ as $S_t$ is the underlying asset price at time t, $S_0$ is the underlying asset price at time 0, and ${\mathrm{\Delta }}_0$ is defined as ${\mathrm{\Delta }}_0$ is the first-order sensitivity of the option price to the underlying asset price.

2.4. Delta-gamma approximation

One often considers a higher-order term to approximate the change of call option price, as this is the case in the delta-gamma approximation (DGA)

where $v_t$ is the relative change (or linear return) of the underlying asset prices during the time period $(0,t)$ defined in (2.8). $S_t$ is the underlying asset price at time t, $S_0$ is the underlying asset price at time 0, and ${\mathrm{\Gamma }}_0$ is defined as $\rho$ is the probability density function of the standard Gaussian law. ${\mathrm{\Gamma }}_0$ is the second-order sensitivity of the option price to the underlying asset price. All the rest of variables and parameters are defined in Sec. 2.3.

Notice that the classic delta-gamma approximation takes as the reference value $S_0$, which is the underlying asset price at time 0. That is, we approximate the option price as a function of the difference between the underlying asset price at time t ($S_t$) and the underlying asset price at time 0 ($S_0$). But, we could consider a reference value different from $S_0$, which could be more suitable for approximating the option price. In fact, in the present work, we propose such a reference value that gives better approximation performance, as indicated in Secs. 3 and 4.

2.5. Value-at-risk

The value-at-risk (VaR) can be defined as a maximum potential loss suffered by a portfolio considered over a future horizon. For a probability level fixed in advance, the determination of the VaR is closely linked to the models representing the changes of underlying asset prices (Jorion (2006)).

In the literature, we can find several approaches to estimate the risk based on geometric Brownian motion, with stochastic volatility, including jumps of the underlying asset (Hull (2017), Oosterlee & Grzelak (2019), Cont & Tankov (2003)).

The stochastic or econometric models have the advantage of modeling the evolution of single type of asset, but they often face calibration difficulties with the market data. This is the case especially in the context of high frequency data because the assumptions made in these models often do not align with the considered data, and this misalignment may be more notorious with high frequency data.

On the other hand, in the literature, there are research works along the data-driven approaches (Homescu (2014), Fabozzi et al. (2021), Bernard et al. (2021)).

In the present work, we first propose a novel method to approximate call option prices, which is based on the Practitioner Black–Scholes (PBS) model (Berkowitz (2001)). Then, we consider a data-driven approach (i.e. the locally weighted regression (LWR)) to approximate call option prices. Both of these approaches are described in Secs. 3 and 4, respectively. We then use these methods to compute the VaR in Sec. 6.6, which are compared with those values that are obtained with existing methods (i.e. DA and DGA).

3. Extended Delta-Gamma Approximation

As mentioned previously, the DGA uses as its reference value the initial underlying asset price. But, if the time horizon considered to estimate the option price is relatively far from the time at which the option contract is signed, then this reference value may not be the most appropriate one. In the present work, we propose to use as the reference value the underlying asset price forecasted for the time horizon t (see Fig. 1). We denote such a value as $S_t^{\ast }$.

In the sequel, for simplicity, we suppose that ${\sigma }_0,r_0,K$ are fixed. Then, we can rewrite the notation $C^{(BS)}(S_t,T-t,{\sigma }_0,r_0,K)$ as $C^{(\mathrm{\text{BS}})}(S_t,T-t)$. Now, we approximate $C^{(\mathrm{\text{BS}})}(S_t,T-t)$ around $S_t^{\ast }$ using a similar approach to the DGA. We coin such an approach as the extended delta-gamma approximation (EDGA)

with where $d_{\pm }$ is defined in (2.2).

4. Approximating Call Option Prices using the Locally Weighted Regression

Up to now, we addressed the problem of approximating call option prices within the DA/DGA framework for the purpose of hedging. In this section, we address the same problem but this time within the algorithmic framework. A very first approach within this framework is the ordinary least square (OLS) method for its simplicity. But, the drawback with the OLS is that (precisely due to its simplicity) it requires a high-order of approximation (due to the nonlinearity of call option prices with respect to changes of underlying asset price depending on implied volatility, moneyness, maturity time and time horizon). This drawback of the OLS may be overcome with a nonparametric model such as the locally weighted regression (LWR) because it captures better this nonlinearity of call option prices by concentrating the choice of the parameter values based on the data points that are close to a point of interest.

Hence, we regress call option prices using the locally weighted regression (LWR) (Cleveland (1979)). LWR can be seen as a moving regression, which is a generalization of moving average. One of the most common LWR method is the LOESS (LOcal regrESSion). LOESS (Cohen (1999)) is a nonparametric regression method that combines local regression models. LOESS is also referred as the Savitzky–Golay filter (Schafer (2011)). It is an alternative method to solve nonlinear regression problems, for which classical regression methods such as the OLS do not often perform well.

The LOESS can be defined as follows. Let I be the number of observations, P be the number of the neighboring relative changes of asset prices to be considered for LWR, and N be the regressor order. In our case, the observations are the pairs of call option prices ($C_i^{(\mathrm{\text{BS}})}$) and the relative changes of underlying asset prices ($v_i$), for $i\in \{1,\dots ,I\}$. For each $v_i$ of interest, we determine the P nearest neighboring distances as

Then, we define the distance between the relative change of underlying asset prices of interest ($v_i$) and the furthest relative change of underlying asset prices in its neighborhood ($v_{i_P}$) Next, we normalize (4.1) with (4.2) to obtain Now, using (4.3), we define the weights for the LWR as The above function is known as the tri-cube weight function. Then, we can formulate the LWR problem as where The solution for the problem (4.5) using the LWR approach has the expression where $W\in {ℝ}^{I\times I}$ is the weight matrix with weights defined in (4.4). Then, one can approximate the $C^{(\mathrm{\text{BS}})}$ by the LWR as where $C^{\mathrm{\text{(LWR)}}}$ represents the call option prices estimated using the LWR approach.

5. Relative Approximation Error of Call Option Prices

So far, we have introduced various methods to approximate call option prices: DA, DGA, EDGA and LWR. In order to compare and evaluate their performance for a time horizon t, we define the relative approximation error of call option prices as

where approach should be replaced by the approximation technique employed for each case (among the DA, DGA, EDGA and LWR). In the numerator of the expression in (5.1), we have the difference between the call option price at time t estimated using an approximation technique and the call option price obtained using the Black–Scholes model (which serves as the reference value and is introduced in Sec. 2.1). The resulting value is normalized by the call option price computed using the Black–Scholes model for time $t=0$ (time at which the option contract is signed).

In fact, (5.1) can be seen also as the difference of profit and loss (P&L) obtained using between the Black–Scholes and our approach (normalized by the initial option price):

6. Evaluation and Comparison of the Approximation Performance Achieved by Various Methods: DA, DGA, EDGA and LWR

Once defined various methods for approximating call option prices (see Secs. 2–4) and a performance measure to compare between the results obtained from each of these methods (see Sec. 5), we are now ready to show the comparison results of these methods. We do this by comparing them step by step as shown in the subsequent sections. Notice that we take the call option prices obtained from the Black–Scholes model as the reference.

For all the approximation methods, we present the relative approximation error of call option prices as function of five factors: change of underlying asset prices, implied volatility, moneyness, maturity time and time horizon. Each figure has ten sub-figures for five different values of moneyness and two different values of time horizons. The five values of moneyness to cover all three cases concerning the relationship between strike price and the price of the underlying asset. Two values for out of the money (OTM): $-10$% and $-5$%; one value for at the money (ATM): 0%; and two values for in the money (ITM): 5% and 10%. Each sub-figure has the horizontal axis representing the relative change of underlying asset price (defined in (2.8)) and the vertical axis representing the relative approximation error of call option price (defined in (5.1)). In addition, each sub-figure has three graphs corresponding to three implied volatility values: 10%, 30%, 50%.

In the sequel, we illustrate these results for all the considered approximation methods. We also discuss the results and compare them across the considered methods.

6.1. Results of DA

In this section, we illustrate the results of approximating European call option prices using the DA method, as function of change of underlying asset prices, implied volatility, moneyness, maturity time and time horizon. These results are shown in Fig. 2.

In Fig. 2, the sub-figures (a), (c), (e), (g) and (i) correspond to time horizon being 5 days (near the date at which the option contract is signed), while the sub-figures (b), (d), (f), (h) and (j) correspond to time horizon being 25 days (near the maturity time). For both cases, the maturity time (T) is 30 days. This figure contains five rows corresponding to the five moneyness values considered in the present work: $-10$% (OTM), $-5$% (OTM), 0% (ATM), 5% (ITM), and 10% (ITM).

For the OTM cases (i.e. (a), (b), (c), and (d)), we observe that in general the relative approximation errors in absolute value are smaller for negative relative changes of underlying asset prices than for positive relative changes of underlying asset prices. We also observe that, for negative relative changes of underlying asset prices, the relative approximation errors in absolute value are smaller for lower implied volatility values. On the contrary, for the positive relative changes of underlying asset prices, the relative approximation errors in absolute value are larger for lower implied volatility values. Further, the relative approximation errors in absolute value grow as the relative changes of underlying asset prices grow.

For the ATM cases (i.e. (e) and (f)), we observe a symmetric error behavior about the zero change of underlying asset price. In this case as well, the relative approximation errors in absolute value grow as the relative changes of underlying asset prices grow in both directions.

For the ITM cases (i.e. (g), (h), (i) and (j)), the opposite behavior of the relative approximation error is observed with respect to the OTM cases. That is, the error values are smaller for positive relative changes of underlying asset prices than for negative relative changes. For negative relative changes of underlying asset prices, the relative approximation errors in absolute value are smaller for higher implied volatility values, while, for positive relative changes of underlying asset prices, the relative approximation errors in absolute value are smaller for lower implied volatility values.

In the coming sections, we compare the approximation results of the DA method to those of other approximation methods, but we can already say that the relative approximation errors in absolute value obtained using the DA method are the largest among all the considered methods.

Finally, chances are that the volume of option contracts is larger around the ATM (corresponding to (e) and (f) in Fig. 2). However, for these cases, we observe that the range for which the approximation error is small is in fact narrow, and the error rapidly grows in both directions of change values of underlying asset prices. This fact illustrates that the DA method does not perform well in these regions (i.e. the regions of large change values of underlying asset prices).

6.2. Results of DGA

In this section, we illustrate the results of approximating call option prices using the DGA method, as function of relative change of underlying asset prices, implied volatility, moneyness, maturity time and time horizon. These results are shown in Fig. 3.

In general, we observe opposite behavior of the relative approximation error between the OTM and the ITM with respect to relative change of underlying asset prices. Whereas, for the ATM, we observe symmetric graphs of relative approximation error with respect to relative change of underlying asset prices, regardless of implied volatility values.

In Fig. 3, for the OTM cases ((a), (b), (c), and (d)), we observe in general that the larger relative change of underlying asset price is, the larger the relative approximation error in absolute value is, for all the considered volatility values except for moneyness being −10% and for negative relative change of underlying asset prices.

For the ATM cases (i.e. (e) and (f)), we observe a symmetric error behavior about the zero change of underlying asset price. We observe that the lower the implied volatility is, the larger the error values (in absolute value) are, for large relative changes of underlying asset prices. In this case as well, the relative approximation errors in absolute value grow as the relative changes of underlying asset prices grow. Further, we observe that, for small relative changes of underlying asset prices, the range for which the relative approximation errors in absolute value are small is larger for the DGA method than for the DA method. Nonetheless, for large relative changes of underlying asset prices, the relative approximation errors are larger for the DGA method than for the DA method.

For the ITM cases (i.e. (g), (h), (i) and (j)), the opposite behavior of the relative approximation error is observed with respect to the OTM cases. We observe in general that the larger relative change of underlying asset price is, the larger the relative approximation error in absolute value is, for all the considered volatility values except for moneyness being 10% and for positive relative change of underlying asset prices.

6.3. Results of EDGA with 1% and 5% of error for predicting the underlying asset price at time t

In this section, we illustrate the results of approximating the European call option prices using the EDGA method with 1% and 5% of error for predicting the relative change of underlying asset price at time t, as function of relative change of underlying asset prices, implied volatility, moneyness, maturity time and time horizon. Instead of considering the case where the prediction error is null, for more plausible cases (as nearly always there is prediction error), we consider the cases where there are 1% and 5% of prediction error and see how this factor affects the performance of approximating the European call option prices using the EDGA method. Figures 4 and 5 show the cases where the error for predicting the underlying asset price are 1% and 5%, respectively.

Recall that the EDGA uses the predicted value of the underlying asset price at time t as its reference value to approximate the call option price. Hence, the approximation performance of the EDGA inherently depends on the performance of predicting the underlying asset price.

We first consider the case how the 1% error on predicting the underlying asset price at time t influences on approximating the call option price at this time using the EDGA method. Figure 4 shows that the relative approximation error values of call option prices (given in (5.1)) are significantly smaller than for the cases of the DA and DGA, for all the considered values of change of underlying asset prices, implied volatility, moneyness, maturity time and time horizon (see Secs. 6.1 and 6.2, respectively). In general, the smaller the implied volatility is, the larger the peak of the relative approximation error of call option price is. Further, these peaks shift towards negative relative changes of underlying asset price as the moneyness increases. There is not much difference in terms of ${𝜖}_{{\mathrm{\text{rel}}}_t}^{(\mathrm{\text{EDGA}})}$ with respect to time horizon (between $t=5$ days and $t=25$ days). But, we can observe that the range of non-zero ${𝜖}_{{\mathrm{\text{rel}}}_t}^{(\mathrm{\text{EDGA}})}$ values is larger for short time horizon than for large time horizon. Moreover, compared to the DA and DGA methods, for large (negative and positive) values of change of underlying asset price, ${𝜖}_{{\mathrm{\text{rel}}}_t}^{(\mathrm{\text{EDGA}})}$ tends to converge to zero. Notice that this is not the case for both the DA and DGA methods.

Now, let us consider the case where the error for predicting the underlying asset price is 5%. Figure 5 shows similar results to the case of 1% error for predicting the underlying asset price (see Fig. 4). However, due to the inherent relationship between the error for predicting the underlying asset price at time t and the relative error for approximating the call option price (${𝜖}_{{\mathrm{\text{rel}}}_t}^{(\mathrm{\text{EDGA}})}$), this last value is larger in general than for the case where we consider 1% error for predicting the underlying asset price at time t. Nonetheless, ${𝜖}_{{\mathrm{\text{rel}}}_t}^{(\mathrm{\text{EDGA}})}$ are still small even for 5% of prediction error, compared to DA/DGA.

6.4. Results of LWR with 1% and 5% of error for predicting the underlying asset price at time t

In this section, we illustrate the results of approximating the European call option prices using the first-order LWR method (with 1% and 5% of error for predicting the relative change of underlying asset price at time t), as function of relative change of underlying asset prices, implied volatility, moneyness, maturity time and time horizon. Once again, it is usual to have error when predicting the underlying asset price, and, therefore, in order to consider realistic cases, we look at the scenarios where there are 1% and 5% of prediction error.

Recall that the LWR (similar to the EDGA) uses the predicted value of the underlying asset price at time t as its reference value to approximate the call option price (see Sec. 4). Hence, the approximation performance of the LWR inherently depends on the performance of predicting the underlying asset price.

We first consider the case how the 1% error on predicting the underlying asset price at time t influences on approximating the call option price using the first-order LWR method. In fact, Fig. 6 shows that the relative approximation error values of call option prices (given in (5.1)) are much smaller than for the cases of the DA and DGA, for all the considered values of relative change of underlying asset prices, implied volatility, moneyness, maturity time and time horizon (see Secs. 6.1 and 6.2, respectively). However, the performance of the LWR is not as good as that of the EDGA (see Sec. 6.3). In general, we observe one peak of error regardless of implied volatility, moneyness, maturity time and time horizon. This peak corresponds to the change of underlying asset prices for which the nonlinearity of call option prices is more significant. We also observe that the magnitude of these peaks are more significant for lower implied volatility, because the lower the implied volatility is, the more nonlinear the relation between the call option price and the relative change of underlying asset price is. Similar to the cases of DA, DGA and EDGA, the opposite behavior between the OTM ($-10$% and $-5$% of moneyness) and ITM (5% and 10% of moneyness) is observed in terms of the relation between the relative approximation error of call option price and the relative change of underlying asset price.

Finally, Fig. 7 illustrates the results of approximating the call option prices with respect to relative changes of underlying asset price values using the first-order LWR when the prediction error is 5%, as function of implied volatility, moneyness, maturity time and time horizon. We observe that, as the change of underlying asset price grows, the relative approximation error of call option price grows more rapidly than for the case that the error for predicting the relative change of underlying asset price is 1%. Hence, the more error on predicting the underlying asset price is, the more relative approximation error of call option price is. Another aspect to remark is that the opposite behaviors observed between the OTM and the ITM for 1% prediction error is no longer so clear, when the prediction error corresponds to 5%.

6.5. Comparison of the performance results obtained from DA, DGA, EDGA, and LWR approaches

In this section, we compare the performance results obtained from DA, DGA, EDGA, and LWR approaches, presented in Figs. 2–7 and discussed in Secs. 6.1–6.4. For this purpose, we compute and present in Table 1 the maximum absolute value of the relative error of option price (across the change of the underlying asset price) obtained from each approach and from each value of volatility, time horizon and moneyness considered in this work. We also graphically present these values in Fig. 8, which shows these values for each approach (DA, DGA, EDGA (1%), EDGA (5%), LWR (1%), and LWR (5%)) and for each value of volatility (10% (first row: (a), (b)), 30% (second row: (c), (d)), 50% (third row: (e), (f))), time horizon (5 [days] (first column: (a), (c), (e)), 25 [days] (second column: (b), (d), (f))), and moneyness (OTM (−10%, −5%), ATM (0%), ITM (5%, 10%)) considered in this work. In general, we observe significant outperformance of the EDGA (for both forecast errors considered (i.e. 1% and 5%)) and of the LWR (for both forecast errors considered (i.e. 1% and 5%)) over the existing approaches (i.e. DA and DGA) regardless of the values of volatility, time horizon, and moneyness. In general, we observe from Fig. 8 that the maximum absolute value of the relative error of option price is ordered descendingly (i.e. from large to small errors) when the option price is approximated by DA, DGA, LWR (5%), LWR (1%), EDGA (5%) and EDGA (1%), respectively. This is not the case exceptionally when $\sigma =10\%$ for the case of ATM (i.e. when moneyness is zero) regardless of the time horizon. In this case, as shown in Figs. 8(a) and 8(b), the maximum absolute value of the relative error of option price for DGA is larger than the value obtained by DA. We can infer the reason why this happens by observing the relative approximation errors of call option price obtained from the DA (Figs. 2(e) and 2(f)) and from the DGA (Figs. 3(e) and 3(f)). Through these figures, we observe that the DGA introduces larger relative approximation errors of call option price than those obtained from the DA, for large absolute values of relative change of underlying asset price, when $\sigma =10\%$ and at ATM.

6.6. Comparison of value-at-risk $($VaR$)$ computed using DA, DGA, EDGA, and LWR approaches

In this section, we compare the value-at-risk computed using the DA, DGA, EDGA, and LWR approaches. For this purpose, we first synthetically generate realizations of the underlying asset price using the Geometric Brownian Motion (GBM) as

where $S_0$ is the initial asset price, $\delta =1∕360$, $\sigma =0.1$ is the volatility, and $r=0.025$ is risk-free interest rate. In addition, for each realization index j and each time index t, where $s[j,t]\sim {𝒩}(0,{\sigma }^2\delta t)$. ${𝒩}(0,{\sigma }^2\delta t)$ is the Gaussian distribution with zero mean and variance being ${\sigma }^2\delta t$. Figure 9(a) shows the realizations $S[j,t]$, for $j\in \{1,\dots ,J\},t\in \{1,\dots ,T\}$, with $J=50$ and $T=30$.

Next, the implied volatilities are computed using the Malz model (Meucci (2008)):

where ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$ are heuristically found with some small values (in this case, 0.1 for each of the three parameters),^c and K is the strike. We considered the ATM case and therefore $K=S_0$. The realizations of implied volatility are shown in Fig. 9(b).

Then, we can proceed to compute the realizations of call option price for each of the models considered in this work, as indicated in (2.1) (for BS), (2.7) (for DA), (2.10) (for DGA), (3.1) (for EDGA), and (4.8) (for LWR). Figures 9(c)–9(g) show the realizations of call option prices obtained from these models.

Next, from these realizations of call option price, we can compute the corresponding losses as

In addition, we consider the mean curve of the realizations of the losses obtained from the BS as the reference.

Finally, the value-at-risk (VaR) is computed as the 95% quantile of ${\mathrm{\text{loss}}}^{\mathrm{\text{(approach)}}}$, for each of the considered approaches. The VaR obtained using all the considered approaches are compared to the reference. Because both the EDGA and LWR depend on the suppositions made on the forecast error, we consider 1%, 5%, and 10% forecast errors to compute their respective VaRs. These results are shown in Figs. 9(h)–9(j).

Figures 9(h)–9(j) show six curves: VaR (BS), VaR (DA), VaR (DGA), VaR (EDGA), VaR (LWR), and the reference (the mean curve of loss realizations of BS). Among these six curves, the ones of VaR (BS), VaR (DA), VaR (DGA), and the reference are the same across Figs. 9(h)–9(j). The only trajectories that are different are those of EDGA and LWR with 1% of forecast error (Fig. 9(h)), 5% of forecast error (Fig. 9(i)), and 10% of forecast error (Fig. 9(j)). We first observe that the VaR of DA and DGA are in general below the reference curve. Hence, the DA and DGA are not able to reasonably represent the risk for the scenario considered in this section. Whereas, regardless of the forecast error, the VaR curves of EDGA and of LWR (for different time horizons) are always above the reference curve (even above the VaR curve of BS). On the other hand, as the forecast error becomes larger, these VaR curves (for both EDGA and LWR) are too high with respect to the reference.

In Figs. 9(b), (c), (h), (i) and (j), big jumps are observed in the option price at the expiry date. These jumps correspond to the cases of DA and DGA. This is mainly due to two reasons: the first reason is that both DA and DGA introduce errors due to the fact that they consider the initial underlying asset price ($S_0$) as the reference value to approximate the option prices; the second is that at expiry date the option price is forced to be the price of the underlying asset price at expiry date ($S_T$). Combining these two aspects, as the time advances the approximation errors grow more and more, and this fact is observed by the jumps between the time instant before the expiry date and the expiry date. This is not the case with the methods that we propose (EDGA and LWR) because we forecast underlying asset price for each time horizon as the reference value.

7. Conclusion and Future Work

In the present work, we have addressed the problem of approximating option prices under large changes of underlying asset prices. It is a common practice to use the delta approximation (DA) and the delta-gamma approximation (DGA), among other Greek approaches, to resolve the problem addressed in this work (Raju (2012), Castellacci & Siclari (2003)). However, as pointed out in several references in the literature (such as in Hull (2017), Wilmott (2006), and Rouah & Vainberg (2007)), when the changes of underlying asset prices become large, the (call and put) option prices estimated by these traditional approaches introduce errors that can grow with the magnitude of the changes of underlying asset prices. Hence, in practice, before the change of underlying asset price becomes large, rebalancing operations take place to minimize the possible losses due to the error introduced by the traditional hedging approaches (such as the DA and the DGA). But, such rebalancing operations have transaction costs associated, and, therefore, there is a trade-off between the assumption of losses due to the inaccuracy introduced by hedging approaches and the rebalancing operations to reduce the impact of this inaccuracy. Motivated by this observation, we have proposed two approaches to improve the approximation performance of the option prices: extended delta-gamma approximation (EDGA) and locally weighted regression (LWR). Both the EDGA and LWR inherently depend on the performance of predicting the change of underlying asset price. The EDGA is the extended version of the DGA. The DGA takes as its reference value the initial underlying asset price, while the EDGA takes as its reference value the underlying asset price predicted for a specific time horizon. On the other hand, the LWR regresses the option prices with weights that are specific for a neighborhood of the changes of underlying asset prices. Due to the fact that the relation between option prices and the change of underlying asset price is nonlinear, the LWR seems to be appropriate to capture this nonlinearity even with a low order regressor. In addition, we have showed the benefits of our contributions in terms of the value-at-risk.

In the present work, we have showed that both EDGA and LWR perform better (for the cases of the out of the money (OTM), at the money (ATM) and in the money (ITM)) than the DA and DGA for the values of implied volatility, moneyness, maturity time, time horizon, changes of underlying asset price considered and the prediction error of the change of underlying asset prices. In the present work, we have supposed the implied volatility to be constant from the moment when both the writer and the buyer sign the option contracts until the maturity time. But, in reality, this is not the case. Hence, as a possible future work, we are interested in considering stochastic implied volatility. In addition, in this work, we have only focused on approximating the European call option prices using synthetic data. In the future, we are interested in applying the approximation techniques presented in this work for hedging purpose with real data. Further, beyond the Greeks that we employed in this work, we would like to add the theta term as well in the future with the purpose to consider the loss of time value (1) to see if there is improvement with respect to the results that we have obtained (but this time with real data) or (2) to see whether the results would rather be in alignment to the works of Estrella (1995) and Estrella & Kambhu (1997). Finally, in this work, we have considered only one call option contract with one underlying asset. In the future, we are interested in considering a portfolio of multiple (put and/or call) option contracts with multiple underlying assets.

Acknowledgments

The authors appreciate the ECE Paris research lab for financing the purchase of the Lambda Quad Max Deep Learning server to obtain the results illustrated in the present work.

Notes

^a Similar results can be obtained for put options from call options using, for instance, the put-call parity.

^b The words “forecast” and “prediction” are used interchangeably along the presentation of our work.

^c Usually the parameters ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, are found by performing some regression. In this section, however, with the purpose to show our contributions implemented for an application, we synthetically generate both the underlying asset prices and implied volatilities, from which we can later compute the value-at-risk values.