Derive Dupire’s formula in Section 5.1.2 in terms of put prices rather than call prices.

Section 5.1.2,

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More interesting is the question of how to recover (St, t) to fit all given market prices. Let us start with f(St, t) = előrudu (St — K)+. Applying Tanaka's formula, we get the equation (*): d [e- Jo rudu (St — K)+] = [-te-forudu (St - K)+ + e¯f Thus, +e- rudu1{St>K}0(St, t) S₁dWt. Taking expectations and noticing that the price of a call option with strike K and expiry t is given by C(K,t) = E[e-Jo rudu (St – K)+], we get + dC(K₁ t) = {E[-rie radu (S₁ – K)+ + e- fordul(5₁>K}(t).Se] 10²C 20K² (K, t) ² (K, 1) K²} dt. ⇒>> dC (K, t) 1 + e -fő rudu SK (St)o² (St, t)S² | dt 2 = E E [-te¯o rudu (St - K)+ + €¯Ső rudu1{Si>K}µ(t){(St – K)+ + K}] dt + 10²C + (K, t)o² (K, t) K² dt 20K² = {E [(u(t) - ₂)e- Jo rotu (S₁ – K)+ in a similar way as we obtained the Fokker-Planck equation above, and noting that f(x)√k(x)dx = f(K) and D(0, T)P(St = K) = 2² c (K, t) from Section 3.1. In the presence of deterministic rates (including foreign rates and dividends), we do not have to worry about expectations of the prod- uct of rates and the underlying, and get² the much simpler + = {(µ(t) – r₁)C(K,t). + €¯forudhu 1 {S,>K\H(1)K] + 20162 (K, 1)0² (K, 1) Kª} dt rudu1{S₁>K}µ(t)K] K²}dt 20K² - So rudu1{S₁>K}µ(t) St 10²C 20K² o²(K,t) = 2- ac(K, t) Ət 10²C 20K² ac Ək (K, t)o² (K, t)K²dt ‚t)K²}, (K, t)u(t) K (K,t)o² (K,t)K². (u(t)- rt)C(K, t) - ac Ək (K, t)u(t)K ac (K, t) - (µ(t) - rt)C(K,t) + (K,t)µ(t)K 2² c K² (K, t) K² This is the familiar Dupire's equation for local volatility, which can be used to recover prices of options of all strikes and expiries observed in the market. We can also obtain an equation for local volatility from put prices, which should be obvious from put-call parity if not by a similar approach to the above.³ Alternatively, we could for- mulate local volatility as a function of implied volatilities and their derivatives since the price of a call or put option is a monotonic function of implied volatility. We shall not pursue the last possibil- ity here (as the algebra can be quite involved and little conceptual value is added from such an exercise).