Deriving the Black–Scholes PDE with a Delta-Hedged Option Portfolio
Summary
The document asks how a delta-hedged portfolio of two options leads to a Black–Scholes pricing equation, including a term for the underlying asset’s market price of risk. It writes the portfolio as one option minus a delta-scaled position in another, then uses Itô’s formula and the assumption that the hedged portfolio earns the risk-free rate to relate the options’ values and sensitivities.
The central question is why the resulting expression contains the asset’s expected return adjusted by its market price of risk, multiplied by the underlying price. It also notes that substituting the asset’s market price of risk yields the familiar risk-free-rate drift term in the pricing PDE. The document is a derivation question rather than a complete proof: it does not resolve the role of stochastic interest rates, nor spell out the assumptions required for the hedge and pricing argument.
Key ideas
- A delta hedge can remove the underlying asset’s instantaneous price risk from a portfolio of options.
- Applying Itô’s formula introduces time decay, curvature, and exposure to changes in the underlying price.
- The adjusted expected return term reflects compensation for the underlying asset’s market price of risk.
- Substituting the asset’s market price of risk gives the risk-free-rate drift term in the Black–Scholes PDE.
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# Black-Scholes with two options
# Black-Scholes with two options
I have got a Black-Scholes model with portoflio with two options which bond prices are $V_1$ and $V_2$ (with different maturities or strikes). The interest rate $r$ is stochastic and given by: $$ dr = u(r,t) dt + w(r,t)dW_t,$$ where $w$ and $u$ are some functions of $r$ and $t$.
The porfolio is given as follow: $$ \Pi = V_1 - \Delta V_2 $$
I need to prove that $$\boxed{\frac{\partial V}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (\mu -\lambda_S \sigma)S \frac{\partial V}{\partial S} - rV =0} \quad [1]$$
I found that problem in Paul Wilmott's book `"On quantitative finance"` on page `857`, but I dont understand everything.
We know $$dV = \frac{\partial V}{\partial t}dt + \frac12 \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}dt + \frac{\partial V}{\partial S}dS,\; \; \; [2]$$ and $$ d\Pi = dV_1 - \Delta d V_2, \; \; \; d\Pi = r\Pi dt. \; \; \; [3]$$
So after finding out form of $\Delta =\frac{\partial V_1/ \partial S}{\partial V_2/\partial S} $
and using $[2], \; [3]$ and $\Delta$ we can get
$$ \frac{1}{\partial V_1/ \partial S}\left[\frac{\partial V_1}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 V_1}{\partial S^2} - rV_1\right] = \frac{1}{\partial V_2/ \partial S}\left[\frac{\partial V_2}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 V_2}{\partial S^2} - rV_2\right]. \; \; \; [4]$$
And then the author in mentionet book writes it as
$$\frac{1}{\partial V_1/\partial S}\left[ \frac{\partial V_1}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 V_1}{\partial S^2} - rV_1\right] = (\mu - \lambda_S \sigma)S \; \; \; [5]$$
Which I don't get - why $(\mu - \lambda_S \sigma)$ it's multiplyed by $S$? How does $\lambda$ or $\mu$ looks like?
Later I'm finding out if $V=S$ we have $\lambda_S = \frac{\mu - r}{\sigma}$ and $ \lambda_S$ is market price of risk for asset, so if we put that in $[5]$ we get
$$ \frac{\partial V}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV =0. \; \; \; [6]$$
Is it a reason of why we wrote right side of $[4]$ that way? To get $[6]$ in that form?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.