Let \(V(S,t)\) be a smooth function and is defined over \(0 \leq S_t< \infty\) and \(t \leq T\). Using Taylor expansion, and recalling the values of \(u\) and \(l\), we can write $$ V^+=V(S_t u, t+\Delta t)=V(S_t+ S_t(u-1), t+\Delta t)=V(S_t+ S_t(\mu \Delta t + \sigma \sqrt{\Delta t} H), t+\Delta t) $$ $$ =V(S_t,t)+\frac{\partial V}{\partial S} S_t(\mu \Delta t + \sigma \sqrt{\Delta t} H)+ \frac{\partial V}{\partial t} \Delta t + \frac{1}{2}\frac{\partial^2 V}{\partial S^2} S_t^2 \sigma^2 H^2 \Delta t+ \mathcal{O}{(\Delta t)^{3/2}} $$ and $$ V^-=V(S_t l, t+\Delta t)=V(S_t+ S_t(l-1), t+\Delta t)=V(S_t+ S_t(\mu \Delta t - \sigma \sqrt{\Delta t} H), t+\Delta t)$$ $$ =V(S_t,t)+\frac{\partial V}{\partial S} S_t(\mu \Delta t - \sigma \sqrt{\Delta t} H)+ \frac{\partial V}{\partial t} \Delta t + \frac{1}{2}\frac{\partial^2 V}{\partial S^2} S_t^2 \sigma^2 H^2 \Delta t+ \mathcal{O}{(\Delta t)^{3/2}} $$ Subtracting \(V^-\) from \(V^+\) gives $$ V^+-V^-=2\frac{\partial V}{\partial S} S_t\sigma \sqrt{\Delta t} G+ \frac{1}{2}\frac{\partial^2 V}{\partial S^2} S_t^2 \sigma^2 (H^2-H^{-2}) \Delta t + \mathcal{O}{(\Delta t)^{3/2}} \label{Diff_V} $$ Substituting \( V+ \), \(V^-\) and (\(V^+-V^-\)) into \(V_t\), gives $$ V_t (1+r\Delta t)=(1+r\Delta t)\frac{V^+-V^-}{u-l} +\frac{u V^-- l V^+ }{u-l} +\mathcal{O}{(\Delta t)^{2}}. $$ This gives $$ - r V_t + \frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2} S_t^2 \sigma^2 \frac{\partial^2 V}{\partial S^2}= 0, $$ which is called the Black-Scholes partial differential equation. The Black-Scholes PDE can be transformed into the heat equation, which can be solved analytically.
Another derivation of the Black-Scholes partial differential equation
Assume that the security price follows a Geometric Brownian motion with a constant drift \(\mu\) and a constant variance of \(\sigma^2\). Use Ito's lemma on \(V(t,S)\) $$ \mathrm{d} V(t,S)= \mathrm{d} S \frac{\partial V}{\partial S} + \mathrm{d} t \frac{\partial V}{\partial t} + (\mathrm{d} S)^2 ~\frac{1}{2} \frac{\partial^2 V}{\partial S^2} $$ Substitute \(\mathrm{d} S\) into \(\mathrm{d} V(t,S)\) and use the Heuristic rules to get $$ \mathrm{d} V(t,S)= (\mu~ S ~\mathrm{d} t+ \sigma~ \mathrm{d} \mathbf{B}_t) \frac{\partial V}{\partial S} + \mathrm{d} t \frac{\partial V}{\partial t} + (\sigma^2~ S^2 ~\mathrm{d} t) \frac{1}{2} \frac{\partial^2 V}{\partial S^2} $$ Consider the portfolio of $$ \Pi= V + \Theta~ S $$ $$ \mathrm{d} \Pi= \mathrm{d} V + \Theta~ \mathrm{d} S= \mathrm{d} V + \Theta (\mu~ S ~\mathrm{d} t+ \sigma ~\mathrm{d} \mathbf{B}_t) $$ \(\Pi\) growth with risk-free interest rate such that to \(\mathrm{d} \Pi=r \Pi \mathrm{d} t\) $$ \mathrm{d}\Pi=(r-1)~ (V + \Theta~ S)= \mathrm{d}V + \Theta ~(\mu ~S ~\mathrm{d}t+ \sigma ~\mathrm{d}\mathbf{B}_t).$$ This gives $$ (r~\mathrm{d}t)~(V~+~\Theta~S)=(\mu~S~\mathrm{d}t+~\sigma~\mathrm{d}\mathbf{B}_t)~\frac{\partial~V}{\partial~S}~+~\mathrm{d}t~\frac{\partial~V}{\partial~t}~+~(\sigma^2~S^2~\mathrm{d}t)~\frac{1}{2}~\frac{\partial^2~V}{\partial~S^2}+~\Theta~(\mu~S~\mathrm{d}t+~\sigma~\mathrm{d}\mathbf{B}_t). $$ Simplify to get $$ -r~\mathrm{d}t~V~-~r~\mathrm{d}t~\Theta~S~+~(\mu~S~\mathrm{d}t+~\sigma~\mathrm{d}\mathbf{B}_t)~\frac{\partial~V}{\partial~S}~+~\mathrm{d}t~\frac{\partial~V}{\partial~t}~+~(\sigma^2~S^2~\mathrm{d}t)~\frac{1}{2}~\frac{\partial^2~V}{\partial~S^2}+~\Theta~(\mu~S~\mathrm{d}t+~\sigma~\mathrm{d}\mathbf{B}_t)=0. $$ Re-arrange the above equation to get $$ -r~\mathrm{d}t~V~-~r~\mathrm{d}t~\Theta~S~+~\mu~S~\mathrm{d}t~\frac{\partial~V}{\partial~S}~+~\mathrm{d}t~\frac{\partial~V}{\partial~t}~+~(\sigma^2~S^2~\mathrm{d}t)~\frac{1}{2}\frac{\partial^2~V}{\partial~S^2}+~\Theta~\mu~S~\mathrm{d}t=-\sigma~\mathrm{d}\mathbf{B}_t~\frac{\partial~V}{\partial~S}-\Theta~\sigma~\mathrm{d}\mathbf{B}_t . $$ We then Hedge and choose \(\Theta=-\frac{\partial V}{\partial S}\) (short selling) to eliminate the stochastic term such that $$ %-r ~\mathrm{d} t~ V + r ~\mathrm{d} t ~S \frac{\partial V}{\partial S} + \mu~ S~ \mathrm{d} t~ \frac{\partial V}{\partial S} + \mathrm{d} t~ \frac{\partial V}{\partial t} + (\sigma^2 ~S^2~ \mathrm{d} t) \frac{1}{2} \frac{\partial^2 V}{\partial S^2}-\frac{\partial V}{\partial S} ~\mu ~S ~\mathrm{d} t~=0 -r ~\mathrm{d} t ~V + r~ \mathrm{d} t~ S \frac{\partial V}{\partial S} + \mathrm{d} t~ \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 ~S^2~ \mathrm{d} t ~\frac{\partial^2 V}{\partial S^2}=0 $$ Dividing the above equation by \(\mathrm{d} t\) gives the Black-Scholes PDE.
References
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The pricing of options and corporate liabilities
, Journal of Political Economy 81(3), 637-659.
- Cox, J. C., Ross, S. A., and Rubinstein, M. (1979).
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Theory of Rational Option Pricing
, The Bell Journal of Economics and Management Science , Vol. 4, No. 1 (Spring, 1973), pp. 141-183
- Tang, Q.
Mathematical Models in Industry and Finance
, Lecture Notes University of Sussex (2008).
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The Mathematics of Financial Derivatives: A Student Introduction
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