On the approximation of the Black and Scholes call function

https://doi.org/10.1016/j.cam.2020.113154Get rights and content

Abstract

The Black and Scholes call function is widely used for pricing and hedging. In this paper we present a new global approximating formula for the Black and Scholes call function that can be useful for deriving the risk of options i.e. the implied volatility. Lastly we compare, by numerical tests, our results with some popular methods available in literature (which are generally local) and we show, through Monte Carlo analysis, the computation error for extreme cases of both volatility and moneyness.

Section snippets

Preface

In option pricing, finite Taylor expansions, particularly of first and second order, are commonly used for approximations. In fact, given the option’s price, its delta and gamma, a risk manager can derive a quadratic approximation which is easily computed even for exotic options (whose pricing may be cumbersome) and “more importantly, it is linear in the underlying price change and in its squared value and is thus readily aggregated across instruments, portfolios and business units of the firm”

Literature review

“The BSM’s biggest strength is the possibility of estimating market volatility of an underlying asset generally as a function of price and time without, for example, direct reference to expected yield, risk aversion measures or utility functions. The second strongest strength aspect is its self-replicating strategy or hedging: explicit trading strategy in underlying assets and risk-less bonds whose terminal payoff, which equals the payoff of a derivative security at maturity. In other words,

Approximating the call function when SX

Throughout this paragraph we will assume SX.

In [3] we introduced the standardized call function, i.e. a one-parameter family of functions that allows to describe the whole three-parameters family of call functions.

The standardized call function is defined by χα(x)N(α2(x1x))eα22N(α2(x+1x)),x>0,where α is a positive parameter.

Given S, X and T, the relationship between the call function C=C(σ) and the functions {χα}α>0 can be expressed by the formula C(σ)=Sχα(σTα)if X>SSX+Xχα(σTα)if X<S

Approximation of the implied volatility

In order to find the implied volatility we should invert the call function C, i.e. we should be able to solve the equation C(σ)=C. Since Ĉ is a good approximation of C, we approximate the implied volatility by solving the equation Ĉ(σ)=C.or, equivalently, by (2.18): χ̂α(σTα)=C,whereCCSif X>S,(CS+X)Xif X<S.

Thus we need to invert χ̂(x). We consider separately the two cases C>χ(1) and C<χ(1).

The case S=X

In the special case S=X, we have d1(σ)=σ2T and d2(σ)=σ2T=d1(σ), and C(σ) can be written in the form C(σ)=SN(σ2T)SN(σ2T)=S2πσT2σT2et22dt=Serf(σT8),where erf is the error function defined by erf(z)2π0zet2dt.

As shown in [3], the call function C(σ) can be approximated by the function Ĉ1(σ)=Stanh((σT2π)+4π12(σT2π)3),which has the same Taylor expansion of order 3 in 0 of the error function.

In order to get the implied volatility we have to solve the equation Ĉ1(σ)=C, that is: (σT2π)+4π

Numerical test

The rationale of the suggested approximation is to overcome the issues in terms of computations and loss of flexibility caused by calculating the exact formula, and has the advantage to provide an acceptable level of accuracy. From Fig. 1(b), we can see that the approximating function χ̂α(x) replicates well the standardized call function χα(x).

Notice that all calculations have been performed with MATLAB Version: 9.4.0.813654 (R2018a) with proprietary routines developed by the authors. Data used

Conclusions

In this work we have illustrated the importance of calculating the value of the call for pricing as well as for derivingthe implied volatility. Even though software is available for computing the implied volatility numerically, among practitioners it is common to use spreadsheets with approximated solutions which are good enough. This approach also avoids the inconvenience of setting up iterating routines and macros that should run permanently to keep up with price changes in the market. Notice

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