# Justify a backward differential equation

Regards of 4.5.1, how we get 4.5.5?

• What book/paper is this from – Slade Nov 19 at 0:37
• It is a book by Jianfeng Zhang on BSDEs books.google.com/… – Alex C Nov 19 at 0:46
• Oh nice. It looks pretty thorough. I'll probably check it out. Thanks! – Slade Nov 19 at 1:04

Note that for the replicating portfolio to be self-financing it suffices that (1): $$\lambda_t=\frac{V_t-h_tS_t}{B_t}$$ where I have changed the notation by designating by $$B_t$$ the money market account: $$B_t=B_0e^{rt}$$ Hence, because the portfolio is self-financing, its dynamics are: \begin{align} dV_t&=\left(\frac{V_t-h_tS_t}{B_t}\right)dB_t+h_tdS_t \\[5pt] &=r(V_t-h_tS_t)dt+(\mu h_tS_tdt+\sigma h_tS_tdW_t) \end{align} Now, you can either be in a position to lend ($$V_t>h_tS_t$$) or borrow money. If the rates to lend $$r_1>0$$ and borrow $$r_2<0$$ are different then the equation above changes to: $$dV_t=r_1\max(V_t-h_tS_t,0)dt-r_2\min(V_t-h_tS_t,0)dt+h_tdS_t$$ You lend at $$r_1$$ if you have excess cash in your hedging account, namely the value of your hedge $$h_tS_t$$ is lower than the value of the derivative $$V_t$$, otherwise you borrow at $$r_2$$. Note that: \begin{align} V_t>h_tS_t \Leftrightarrow &\ r_1\max(V_t-h_tS_t,0)-r_2\min(V_t-h_tS_t,0) \\ &= r_1(V_t-h_tS_t)>0 \\[3pt] V_t Namely, the $$\max$$ and $$\min$$ functions allow to separate the excess/deficit of cash cases.
(1) Proving this is extremely cumbersome. Basically you need to prove that: $$B_td\lambda_t+d\lambda_tdB_t+S_tdh_t+dS_tdh_t=0$$ If I find the time (and energy) I'll try to post a derivation, otherwise you can check this answer to see how to do this.
• Thank you. but I am confusing that how the min/max appears in the equation. Is this mean that we want to maximize the value of $dV_t$? – jf1997 Nov 19 at 20:37
• @jf1997 see my updated answer, the $\max$ and $\min$ functions allow to separate the excess/deficit of cash cases. – Daneel Olivaw Nov 19 at 21:45
• @jf1997 I have slightly amended my answer as there was something wrong, strictly speaking the holding $\lambda_t$ of bank account has to be divided by $B_t$ (i.e. the value of the bank account) to ensure the replicating portfolio equals the derivative value $V_t$. – Daneel Olivaw Nov 21 at 16:08