In chapter 5, we approximated the Black-Scholes model

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1 Chapter 7 The Black-Scholes Equation In chapter 5, we approximated the Black-Scholes model ds t /S t = µ dt + σ dx t 7.1) with a suitable Binomial model and were able to derive a pricing formula for option payoffs H = HS T ). The time 0 theoretical fair value is given by V BS 0 = e rt H S 0 e σ T y σ r e ) )T 1 e 1 y dy 7.) One can show that this quantity can also be obtained as the unique solution of the following partial differential equation: V BS 0 = V S = S 0, t = 0) where V S, t) is a solution of + σ V S + rs S S rv = 0 7.3) V S, t = T ) = HS) 7.4) Equation 7.3) is called the Black-Scholes equation. Instead of doing just a brute force calculation and checking that indeed 7.) is a solution of 7.3), which would give no further insight in the origin of 7.3), we will now derive 7.3,7.4) as the contiuous time limit of the recursion relations for the replicating portfolio values in the approximating Binomial model. ecall from Theorem.1 that the replicating strategy δ tk and the portfolio values can be inductively from k = N to k = 0) calculated through the following formulae: V tk where V tk = 1 d d ret down )V up 1 d d ret up )Vt down 7.5) V tn = H V up,down := V t Stk 1 + ret up,down ) ) 7.6) 43

2 44 Chapter 7 and, with time steps t k = k, the discount factor d k, = e r t k ) of Theorem.1 becomes d k, = e r 7.7) The delta s are obtained from δ tk = V Stk 1 + ret up ) ) V t Stk 1 + ret down ) ) S tk 1 + ret up ) S tk 1 + ret down ) 7.8) The Binomial model which approximates the Black-Scholes model 7.1) is given by The delta s of 7.8) simply become ret up = µ + σ 7.9) ret down = µ σ 7.10) δ t = S t, t) 7.11) in the contiuum limit 0. Now let us consider the continuum limit of 7.5). To get a feeling for the problem, let us first put the interest rates to zero, r = 0. In that case 7.5) reduces to V tk = ret downv up + ret up Vt down = µ + σ T )V up + µ + σ T )Vt down = V up + Vt down µ V up 7.1) Motivated by the Black-Scholes equation where a term sides of 7.1) the term V t S tk ), shows up, we subtract on both V tk S tk ) V t S tk ) = V up + Vt down We devide this by and obtain µ V up V t S tk ) = V up V t Stk 1 + µ) ) + Vt down µ V up σ + V t Stk 1 + µ) ) V t S tk ) 7.13) V tk S tk ) V t S tk ) = term 1 + term + term )

3 Chapter 7 45 with the following quantities: term 1 = V up V t S tk 1 + µ)) + Vt down = σ S t k 0 σ S t k V t S tk 1+µ+σ )) V t S tk 1+µ))+V t S tk 1+µ σ )) S tk σ ) V k ) 7.15) term = µ V t Stk 1 + µ + σ ) ) V t Stk 1 + µ σ ) ) V t Stk 1 + µ + σ ) ) V t Stk 1 + µ σ ) ) = µs tk S tk σ 0 µs tk k 7.16) term 3 = V Stk 1 + µ) ) V t S tk ) = S tk µ V Stk 1 + µ) ) V t S tk ) S tk µ 0 µs tk k 7.17) Thus, with the notation V = V S t, t) instead of V tk S tk ), we get or S t, t) V tk S tk ) V t S tk ) = lim 0 = σ S t S t, t) V S t, t) S t + σ S t µs t S t, t) + µs t S t, t) V S t, t) S t = ) which is the Black-Scholes equation for zero interest rates. To obtain the Black-Scholes equation with nonzero interest rates, we rewrite 7.5) as follows: V tk = 1 d d ret down )V up 1 d d ret up )V down = ret downv up + ret up Vt down 7.19) + 1 d d 1)ret down ) V up 1 d d 1)ret up ) V down

4 46 Chapter 7 The first term in 7.19) is the contribution for zero interest rates and has been considered following 7.1). The second term in 7.19), 1 d ) 1 + ret down)v up 1 + ret up )Vt down 7.0) is new. Thus, for non zero interest rates 7.14) changes to V tk S tk ) V t S tk ) with a fourth term given by term 4 = 1 d = 1 e r = 1 e r { 0 r S tk and 7.1) becomes = term 1 + term + term 3 + term 4 7.1) { V up t + ret down V up ret up V down } { Vt Stk 1 + µ + σ ) ) V t Stk 1 + µ σ ) ) + µ σ ) V up µ + σ ) Vt down } V t Stk 1 + µ + σ {S ) ) V t Stk 1 + µ σ ) ) tk S tk σ + µ V up } k } + 0 V V up + V down 7.) S t, t) = σ S t V S t, t) S t + r S t S t, t) r V 7.3) which is the Black-Scholes equation 7.3) with non zero interest rates. eduction of the Black-Scholes Equation to the Diffusion Equation Suppose that we would not know that the solution to the Black-Scholes equation is given by 7.), how would we proceed from 7.3,7.4) to obtain a solution? One possibility is to transform the Black-Scholes equation into the diffusion equation ux,t) = ux,t) x which can be solved, for example with Fourier transform, in a pretty straightforward way. This calculation goes as follows:

5 Chapter 7 47 The Black-Scholes equation for a european option with payoff HS T ) reads with the final condition + σ V S + rs S S rv = 0 7.4) V S, T ) = HS) 7.5) To turn 7.4) into a constant coefficient equation, we start by introducing the variables x and τ according to and write Because of = 7.4) becomes and S = e σx, τ = T t 7.6) V S, t) = V e σx, T τ) = e rτ vx, τ) 7.7) S = 1 σs rv v + σ S x, S = 1 σs 1 v σs x + 1 ) v + rs 1 σ S x σs 1 v x ) σ S x r x + σ σ v x rv = 0 ) v x = v 7.9) This looks almost like the diffusion equation. To eliminate the first derivative with respect to x we put k := r σ σ 7.30) and make the ansatz vx, τ) = e αx βτ ux, τ) 7.31) which gives With the choice 1 α u α u x + u ) + k αu + u ) = βu + u x x 1 u α ) + α + k) u x x + αk + β u = u α = k = r σ σ, β = k = 1 r ) σ r + σ 4 7.3) 7.33)

6 48 Chapter 7 we have to solve the diffusion equation with the initial condition 1 u x = u 7.34) ux, 0) = e kx He σx ) 7.35) The solution is ux, τ) = e ky He σy ) 1 τ e x y) τ dy 7.36) and we arrive at S = e σx, t = T τ) V S, t) = e 1 r σ + σ ) τ e r σ σ )y x) He σy 1 ) = 1 e 1 r σ + σ ) τ = 1 e 1 r σ + σ ) τ = 1 e rτ = 1 e rτ which coincides with 7.). τ e x y) τ HSe σ τy ) e y + r σ σ ) τy dy HSe σ τy )e 1 y r σ σ ) τy+ r σ σ ) τ) dy e 1 r σ σ ) τ HSe σ τy )e 1 y r σ σ ) τ) dy H S e σ τ y+r σ )τ) e y dy 7.37) dy