Four Things You Might Not Know about the Black-scholes Formula

I demonstrate four little-known properties of the BlackScholes option pricing formula: (1) An easy way to find delta. (2) A quaint relation between calland put-prices. (3) Why vega-hedging though non-sensical will help. (4) What happens if you take vegahedging too far. Introduction The Black-Scholes formula is the mother of all option pricing formulas. It states that under perfect market conditions and Geometric Brownian motion dynamics, the only arbitrage-free time-t price of a strike-K expiry-T call-option is ) , , , ), ( ( ) ( σ r K t T t S BS t Call call − = where S(t) is the time-t price of a dividend-free stock, r is the risk-free rate, σ is volatility (i.e. the standard deviation of appropriately time-scaled returns), and the function is given by call BS )) , , , , ( ( )) , , , , ( ( ) , , , , ( 2 1 σ τ σ τ σ τ τ r K S d N K e r K S d N S r K S BS r call − − = where N denotes the standard normal distribution function and 1 Rolf Poulsen ([email protected]), Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark. 2 A “no dividends” assumption is not always without loss of generality; it may change “standard results”. For instance, with a positive dividend yield call-prices may decrease at long expiries. However, a dividend yield does not alter the results I present, so to ease the exposition, it has been left out. 2 . , ) 2 / ( ) / ln( 1 2 2 1 τ σ τ σ τ σ − = + + = d d r K S d The Black-Scholes formula can be derived in a number of ways. Andreasen, Jensen and Poulsen (1998) is an account of some of them; Derman and Taleb (2005) is a recent (although debatable, see Ruffino and Treussard (2006)) addition. In this note I show some less-know results related to the Black-Scholes formula. You may see them just as “cute”, “quaint” or “a nice exercise”. But they go deeper; as I briefly outline, they are special cases of more general results or techniques. Or poetically, they are the shadows cast from higher dimensions onto the walls of the Black-Scholes cave. Deriving delta – correctly – without lengthy calculations A central quantity for hedging and risk-management is the call-(or any other)option’s sensitivity to changes in the stock-price; its delta: S BS call ∂ ∂ = Δ . A tempting way to a show this is to ignore/forget that S enters inside the N(...)expressions which makes the differentiation very easy: ) ( 1 d N = Δ . Rather un-pedagogically this happens to be the correct result. To derive it properly you must use the chain rule when differentiating. This gives two extra terms that cancel after tedious calculations. A simpler derivation that does not appear to be well-known is this: Remember that a function f (defined on some cone-shaped domain of ) is said to be n R 3 homogenous (of degree one) if ) ( ) ( x f x f α α = for all + R ∈ α and all x (in f’s domain). Euler’s Theorem (that is otherwise predominantly used in microeconomics) says that a differentiable function f is homogenous if an only if it has the form i n i i x f x x f ∂ ∂ = ∑ =1 ) ( . Now observe that the Black-Scholes call-price is homogenous in stock-price and strike. Then Euler’s Theorem tells us that the term that “multiplies S” in the formula is indeed the partial derivative with respect to S; the delta. This homogeneity property (known in the financial engineering literature as “sticky moneyness regime”) holds not just in the Black-Scholes model, but as discussed in Joshi (2003; chapter 15) in a more general class where the return distribution is independent of the current stock-price level. That involves affine jump-diffusions as well as some infinite intensity Levy-driven processes. Before we get carried away, Lee (2004) shows that although call-prices in these models can be written such that they look a lot like the Black-Scholes formula (from which delta is then recognized), that is far from the best representation for numerical calculations. Not all models are homogeneous. The Bachelier-model (where S is arithmetic Brownian motion), the constant elasticity of variance model, the SABR stochastic volatility model as well as Dupire-Derman type local volatility models are inhomogeneous. Devising an empirical methodology that is powerful enough for testing (i.e. for rejecting) homogeneity remains an open problem. Put-call-duality If you plug in σ − into the call-price formula, you get minus the put-price: 3 For instance, Cont and da Fonseca (2002) impose homogeneity from the outset in their empirical analysis.