The real P&L in Black-Scholes and Dupire Delta hedging

We derive the real Profit&Loss(P&L) that accrues in what option traders do every day: continuously Delta (∆)-hedge a call, by substituting the option’s running implied volatility (generally stochastic) into the Black-Scholes(BS) ∆-formula. The result provides formal justification for a heuristic rule-of-thumb that a trader’s P&L on continuously ∆-hedged positions is his vega times times the daily change in implied vol. For Digital options, we show that if we ∆-hedge them at the running implied vol of the associated vanilla, then we have hidden exposure to the volatility skew dynamics. We go on to derive the P&L incurred when we periodically re-calibrate a local volatility model (LVM) to the smile, and ∆-hedge under the premise that the most recently fitted LVM is the correct model. As asides, we demonstrate how a Investment Bank can trade implied volatility directly (in the home currency of the bank), using At-The-Money(ATM) FX forward starting options, and derive a new forward equation for Vanillas which contains localized vol-of-vol and vol skew terms, which extends the results of Dupire ∗This article was completed during an internship as a Quantitative Analyst in FX Derivatives at HSBC Bank plc, 8 Canada Square, London E14 5HQ The real P&L in continuously ∆hedging Vanillas and Digitals 1. Summary of relevant literature Schonbucher considered the problem of exogenously specifying Markovian dynamics for the BS implied volatility of one call option. Under this framework, the call option value is just a function of three variables (the Stock price St, calendar time t, and the BS implied volatility σ̂t), so we can easily apply Itō’s lemma to obtain an SDE for the call option value. Using Itō, he derived a no-arbitrage condition for the drift of σ̂t, which ensures that the discounted call value evolves as a martingale. The condition that the call have zero drift also gives rise to an interesting pde linking the call option’s vanna ( ∂St ) and volga ( ∂Vega ∂σ̂t ) to the risk neutral drift of σ̂t, and the correlation between St and σ̂t. Brace,Goldys,vanderHoek&Womersley extend this idea to exogenously imposing dynamics for the evolution of the entire smile. The drift of σ̂t, and the stock’s instantaneous volatility process σt, and thus the St process itself, are endogenously determined by imposing that the discounted call price evolve as a martingale, and the feedback condition that the implied volatility squared times the time to expiration (the dimensionless implied variance) tend to zero as we approach maturity. σt is equal to the just maturing ATM implied vol, as we might expect. This forces the call to converge to its intrinsic value at maturity. If the endogenously determined St process is a bona fide martingale, as opposed to just a local martingale, then arbitrage is precluded. Unfortunately, the largest class of vol-of-vol (the lognormal volatility of σ̂t) processes for which this is the case is not known. They go on to show that their general stochastic implied volatility framework encompasses all the standard existing stochastic volatility models (for which the discounted stock price is a-priori a true martingale) as special cases. They subsequently demonstrate that while the initial smile determines the marginal stock price distribution from Dupire’s results, the vol-of-vol process determines the joint stock price distribution, to which the price of forward starting options are very sensitive. In Appendix 2, we exploit the fact that the ∂vega ∂vol of an ATM vanilla is negligible, to show how to more or less replicate a linear contract which pays us ATM EUR/USD implied vol at some time in the future (in EUR), by buying an ATM forward starting option. In section 1 of this paper, we use Schonbucher’s methodology to derive the P&L that a trader makes when he buys a call, and continuously ∆-hedges it with the future, by plugging the running (stochastic) σ̂t into the BS ∆-formula. We go on to show that if we continuously ∆-hedge a Digital at the running implied vol of the associated Vanilla, we incure a hidden Delta and volatility skew risk. Carr&Madan derived the P&L that accrues when a trader sells an European style contingent claim for a constant BS implied volatility of σ̂, and dynamically BS ∆-hedges it with the future, by always plugging the constant value σ̂ into the The real P&L in continuously ∆hedging Vanillas and Digitals BS ∆-formula. Their expression for the BS ∆-hedging error (which they obtained using only Itō’s lemma and the BS pde) is as follows: