1 Implied Volatility from Local Volatility

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1 Abstract We try to understand the Berestycki, Busca, and Florent () (BBF) result in the context of the work presented in Lectures and. Implied Volatility from Local Volatility. Current Plan as of March 3rd. First we rederive the BBF result. Then we revisit the extension of Blacher s work which gives us a general formula for implied volatility in terms of local volatility. Then, we will attempt to show that the BBF solution is consistent with the extended Blacher formula (not yet done).. The Zero Time to Expiration Limit We start with the equation w T = v L x w w x + ( 4 4 ) ( w w + x w x ) + } w x () In the notation of BBR, we have: w = ϕ T v L = σ T In the limit T, w and equation becomes ( w T = v L ) x w () w x We assume that ϕ is slowly varying in T so that as T, w T w T = ϕ. Then equation becomes ( ϕ = σ x ) ϕ (3) ϕ x As BBR point out, the solution to equation 3 is ϕ(x, ) = ds σ(sx, ) (4)

2 To see this, substitute φ = ϕ into equation 3. Then we get + x φ φ x = φσ Then Integrating this gives (xφ) x = σ ϕ(x) = φ(x) = x x ds σ(s ) =.3 No Local Volatility Skew If the skew σ x is zero, we must have σ = (ϕ T ) T ds σ(sx) So the local variance in this case reduces to the forward Black-Scholes implied variance. The solution to this is of course ϕ (T ) = T σ (t) dt = σ (st ) ds So in one case, we have that the implied volatility is the harmonic mean of the local volatility and in the other, we have that the implied volatility is the root-mean square local volatility. We now review our earlier extension of Blacher s work to get a general expression for implied volatility in terms of local volatility..4 Extending Blacher So far, we have shown how to express implied volatility in terms of local volatility in the zero-time-to-expiration and no-skew limits. Now, by extending the work of Blacher (998), we derive a general pathintegral representation of Black-Scholes implied variance. We start by assuming that the stock price S t satisfies the (local volatility) SDE ds t = µ t dt + σ t,st dz t S t

3 and that the market prices contingent claims accordingly so that, in particular, the value V of a contingent claims must satisfy a generalization of the Black-Scholes equation: σ t,s t St V + µs t V S t = V t Path-by-path, for any suitably smooth function f (S t, t) of the random stock price S t, the difference between the initial value and the final value of the function f (S t, t) is obtained by anti-differentiation. Then, applying Itô s Lemma, we get f (S T, T ) f (S, ) = = df f ds t + f S t t dt + σ S t,t S t } f dt (5) In particular, the Black-Scholes (BS) formula C BS (S t, t, T ) for a call option expiring at time T with some arbitrary time-dependent volatility parameter is a smooth function of the stock price and must satisfy equation (5). Recall the form of the Black-Scholes formula with C BS (S t, K, t, T ) = F t,t N (d ) K N (d ) d = ln (F t,t /K) σ BS T + σ BS T ; d = d σ BS T where the time -T forward price at time t is denoted by F t,t, the strike price of the option by K and σ BS = σ BS (K, t, T ) is the Black- Scholes implied volatility which is of course a function of calendar time t, strike K and T. C BS (S t, K, t, T ) must satisfy the Black-Scholes equation (assuming zero interest rates and dividends): C BS t = v K,T (t) S t C BS S t (6) where the forward Black-Scholes variance v K,T (s) is given by v K,T (s) = s } σ BS (K, t, s) (s t) 3

4 Under the usual assumptions, the non-discounted value C(S, K,, T ) of a call option is given by the expectation of the final payoff under the risk-neutral measure. Then, applying (5), we obtain: C (S, K,, T ) = E [(S T K) + S = E [C BS (S T, K, T, T ) S [ = C BS (S, K,, T ) + E CBS S t ds t + C BS dt + t σ S t,tst Finally, we use the BS equation to substitute for the time derivative and obtain: C BS t C (S, K,, T ) = C BS (S, K,, T ) [ +E CBS S t ds t + = C BS (S, K,, T ) [ } +E σs t,t v K,T (t) } σs t,t v K,T (t) St S t C BS S t dt S where the second equality uses the fact that S t is a martingale. By definition of implied volatility, C BS (S, K,, T ) = C (S, K,, T ) when v K,T is the Black-Scholes implied forward variance (i.e. the Black-Scholes formula must give the market price of the option). Then the second term in equation (8) must vanish. A sufficient condition for this is to have [ v K,T (t) = E σs t,t S t Γ BS (S t ) S E [ S t Γ BS(S t ) S (9) where we define Γ BS (S t ) := C BS (S t, K, t, T ). Now we have a formula for the Black-Scholes implied volatility of a European option in terms of local volatilities. From the definition of v K,T (t), we have that Then, explicitly σ BS (K, T ) = T σ BS (K, T ) = T v K,T (t)dt E [σs t,t S t Γ BS (S t ) S E [ St Γ dt () BS(S t ) S 4 C BS dt} S (8) C BS dt} S (7)

5 Note however that equations (9) and () are implicit because the gamma Γ BS (S t ) of the option depends on all the forward implied variances v K,T (t). Special Case (Black-Scholes) Suppose σ St,t = σ t, a function of t only. Then v K,T (t) = E [ σ t S t Γ BS (S t ) S E [ S t Γ BS(S t ) S = σ t The forward implied variance v K,T (t) and the local variance σ t coincide. As expected, v K,T (t) has no dependence on the strike K or the option expiration T. Interpretation In order to get better intuition for equation (9), first recall how to compute a risk-neutral expectation: E [f (S t ) = ds t p (S t, t; S ) f (S t ) We get the risk-neutral pdf of the stock price at time t by taking the second derivative of the market price of European options with respect to strike price. p (S t, t; S ) = C (S, K, t) K K=St Then we may rewrite equation (9) as v K,T (t) = E [ St Γ ds t p (S t, t; S ) St Γ BS (S t ) σs BS(S t ) S t,t = ds t q (S t, t; S, K, T ) σs t,t () where we further define q (S t, t; S, K, T ) := p (S t, t; S ) S t Γ BS (S t ) E [ S t Γ BS(S t ) S () q (S t, t; S, K, T ) is a probability density which looks like a Brownian Bridge density for the stock price given that the initial stock price is S and the time-t stock price is K. 5

6 ( For convenience, we rewrite Equation () in terms of x t log St S ). In terms of x t, v K,T (t) = dx t q (x t, t; x T, T ) σx t,t (3) Figure : Graph of the pdf of x t conditional on x T = Log(K) for a year European option, strike.3 with current stock price = and % volatility Figure shows how q (x t, t; x T, T ) looks in the case of a year European option struck at.3 with a flat % volatility. We see that q (x t, t; x T, T ) peaks on a line (which we will denote by x t ) joining the stock price today with the strike price at expiration. Moreover, the density looks roughly symmetric around the peak. This suggests an expansion around the peak x t (at which the derivative of q (x t, t; x t, T ) with respect to x t is zero). Then we write: q (x t, t; x T, T ) q( x t, t; x T, T ) + (x t x t ) q x t (4) xt = x t In practice, the local variance σ x t,t is typically not so far from linear in x t in the region where q (x t, t; x T, T ) is significant so we may further 6

7 write σ x t,t σ x t,t + (x t x t ) σ x t,t x t (5) xt = x t Substituting (4) and (5) into the integrand in equation (3) gives v K,T (t) σ x t,t and we may rewrite equation () as σ BS (K, T ) T σ x t,tdt (6) In words, equation (6) says that the Black-Scholes implied variance of an option with strike K is given by the integral from valuation date (t = ) to the expiration date (t = T ) of the local variances along the path x t that maximizes the Brownian Bridge density q (x t, t; x T, T ). Of course, in practice, it s not easy to compute the path x t. However, we now have a very simple picture for the meaning of Black- Scholes implied variance of a European option with a given strike and expiration - it is approximately the integral from today to expiration of local variances along the most probable path for the stock price conditional on the stock price at expiration being the strike price of the option. References Berestycki, H, J Busca, and I Florent,, Asymptotics and calibration of local volatility models, Quantitative Finance, Blacher, Guillaume, 998, Local volatility, RISK Conference Presentation. 7