Derive Dupire Formula by Conditional Expectation
Assumption
Assume the underlying stock follows the geometric brownian motion under Risk-neutral world:
\[dS_t = \mu_tS_tdt+ \sigma_tS_tdw\]Since it’s under RN world, the $\mu_t$ can be expressed as:
\[\mu_t = r_t - q_t\]Where, $r_t$ is instantaneous risk free rate and $q_t$ is divident.
Derivation
Now, define a function $f$
\[f(S_T,T) = P(0,T)(S_T-K)^+\]Note, $S_T$ is a random number, so as $f$.
Apply Ito’s Lemma to function $f$:
\[df = [\frac{\partial f}{\partial T} + \mu_T S_T\frac{\partial f}{\partial S_T} + \frac{1}{2}\sigma_T^2S_T^2\frac{\partial^2 f}{\partial S_T^2}]dT + [\sigma_TS_T\frac{\partial f}{\partial S_T}]dw\]\[\frac{\partial f}{\partial T} = -r_TP(0,T)(S_T-K)^+\]
\[\frac{\partial f}{\partial S_T} = P(0,T)1_{(S_T>K)}\]
\[\frac{\partial^2 f}{\partial S_T^2} = P(0,T)\delta_{(S_T-K)}\]
Note that the $\delta$ is Dirac Delta Function.
Plug in the above items
\[df = P(0,T) * [-r_T(S_T-K)^+ + \mu_T S_T1_{(S_T>K)} + \frac{1}{2}\sigma_T^2S_T^2\delta_{(S_T-K)}]dT + P(0,T)*[\sigma_TS_T\delta_{(S_T-K)}]dw\]\[(S_T-K)^+ = (S_T-K)1_{(S_T>K)}\]
\[\mu = r_T - q_T\]
Take expectation on $df$, and $E[dw] = 0$. The second term can be ignored. The expected value of $f$ can be viewed as $dC$ (Because $f$ is discounted payoff).
\[dC = E(df) = P(0,T) * E[-r_T(S_T-K)^+ + \mu_T S_T1_{(S_T>K)} + \frac{1}{2}\sigma_T^2S_T^2\delta_{(S_T-K)}]dT \\ = P(0,T) * E[r_T K1_{(S_T>K)} - q_TS_T1_{(S_T>K)} + \frac{1}{2}\sigma_T^2S_T^2\delta_{(S_T-K)}]dT\]Rearrange
\[\frac{dC}{dT} = P(0,T) * E[r_T K1_{(S_T>K)} - q_TS_T1_{(S_T>K)} + \frac{1}{2}\sigma_T^2S_T^2\delta_{(S_T-K)}]\]Recall the results from last article
\[C = P(0,T)E[(S_T-K)^+] = P(0,T)E[S_T1_{(S_T>K)}] - P(0,T)KE[1_{(S_T>K)}]\]
\[\frac{dC}{dK} = -P(0,T)E[1_{(S_T>K)}]\]\[\frac{dC}{dT} = -K(r_T-q_T)\frac{dC}{dK}-q_TC + \frac{1}{2}P(0,T)E[\sigma_T^2S_T^2\delta_{(S_T-K)}]\]
Since $\delta_{(S_T-K)}$ is the derivative of $1_{(S_T>K)}$, it only has value at the point $S_T = K$, 0 otherwise .
\[E[\delta_{(S_T-K)}] = p(K,T)\]
\[E[\sigma_T^2S_T^2\delta_{(S_T-K)}] = E[\sigma_T^2S_T^2|S_T = K]E[\delta_{(S_T-K)}] \\ = K^2E[\sigma_T|S_T = K]E[\delta_{(S_T-K)}]\]
Replace $P(0,T)E[\delta_{(S_T-K)}]$ with $\frac{\partial ^2 C}{\partial K^2}$
\[\frac{\partial C}{\partial T} = -K(r_T-q_T)\frac{\partial C}{\partial K}-q_TC + \frac{1}{2}K^2E[\sigma_T^2|S_T = K]\frac{\partial ^2 C}{\partial K^2}\]Rearrange, we will get to the final result:
\[E[\sigma_T^2|S_T = K] = \frac{\frac{\partial C}{\partial T} + K(r_T-q_T)\frac{\partial C}{\partial K} + q_TC}{\frac{1}{2}K^2\frac{\partial ^2 C}{\partial K^2}}\]Comparing with the results in last article, the right hand site is exactly the same. So we conclude that local variance is the risk-neutral expectation of the instantaneous variance conditional on the final stock price $S_T$ being equal to the strike price K.
References
Fokker Planck Equation Derivation: Local Volatility, Ornstein Uhlenbeck, and Geometric Brownian
The Dupire Formula
Derivation of Local Volatility
DerivationofFokker–Planck EquationfortheLocal VolatilityModel
