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This question is designed to make sure you know how to derive the Black-Scholes partial differential equation (pde)

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This question is designed to make sure you know how to derive the Black-Scholes partial differential equation (pde).
a) Why is Ito’s lemma needed?

b) Derive the Black-Scholes pde,

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a.) Ito's Lemma is a key component in the Ito Calculus, used to determine the derivative of a time-dependent function of a stochastic process. It performs the role of the chain rule in a stochastic setting, analogous to the chain rule in ordinary differential calculus. Ito's Lemma is a cornerstone of quantitative finance and it is intrinsic to the derivation of the Black-Scholes equation for contingent claims (options) pricing.

The Chain Rule

One of the most fundamental tools from ordinary calculus is the chain rule. It allows the calculation of the derivative of chained functional composition. Formally, if W(t) is a continuous function, and:

dW(t)=μ(W(t),t)dt

Then the chain rule states:

d(f(W(t)))=f′(W(t))μ(W(t),t)dt

When f has t as a direct dependent parameter also, we require additional terms and partial derivatives. In this instance, the chain rule is given by:

d(f(W(t),t))=(∂f∂w(W(t),t)μ(W(t),t)+∂f∂t(W(t),t))dt

In order to model an asset price distribution correctly in a log-normal fashion, a stochastic version of the chain rule will be used to solve a stochastic differential equation representing geometric Brownian motion.

The primary task is now to correctly extend the ordinary calculus version of the chain rule to be able to cope with random variables.

Ito's Lemma

Theorem (Ito's Lemma)

Let B(t) be a Brownian motion and W(t) be an Ito drift-diffusion process which satisfies the stochastic differential equation:

dW(t)=μ(W(t),t)dt+σ(W(t),t)dB(t)

If f(w,t)∈C2(R2,R) then f(W(t),t) is also an Ito drift-diffusion process, with its differential given by:

d(f(W(t),t))=∂f∂t(W(t),t)dt+f′(W(t),t)dW+12f″(W(t),t)dW(t)2

With dW(t)2 given by: dt2=0, dtdB(t)=0 and dB(t)2=dt.

b.)

The following derivation is given in Hull's Options, Futures, and Other Derivatives.That, in turn, is based on the classic argument in the original Black–Scholes paper.

Per the model assumptions above, the price of the underlying asset (typically a stock) follows a geometric Brownian motion. That is

${\frac {dS}{S}}=\mu \,dt+\sigma \,dW\,$

where W is a stochastic variable (Brownian motion). Note that W, and consequently its infinitesimal increment dW, represents the only source of uncertainty in the price history of the stock. Intuitively, W(t) is a process that "wiggles up and down" in such a random way that its expected change over any time interval is 0. (In addition, its variance over time T is equal to T; see Wiener process § Basic properties); a good discrete analogue for W is a simple random walk. Thus the above equation states that the infinitesimal rate of return on the stock has an expected value of μ dt and a variance of $\sigma ^{2}dt$ .

The payoff of an option $V(S,T)$ at maturity is known. To find its value at an earlier time we need to know how {\displaystyle V} $V$ evolves as a function of $S$ and $t$ . By Itô's lemma for two variables we have

$dV=\left(\mu S{\frac {\partial V}{\partial S}}+{\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}\right)dt+\sigma S{\frac {\partial V}{\partial S}}\,dW$

Now consider a certain portfolio, called the delta-hedge portfolio, consisting of being short one option and long ${\frac {\partial V}{\partial S}}$ shares at time $t$ . The value of these holdings is

$\Pi =-V+{\frac {\partial V}{\partial S}}S$

Over the time period $[t,t+\Delta t]$ , the total profit or loss from changes in the values of the holdings is (but see note below):

$\Delta \Pi =-\Delta V+{\frac {\partial V}{\partial S}}\,\Delta S$

Now discretize the equations for dS/S and dV by replacing differentials with deltas:

$\Delta S=\mu S\,\Delta t+\sigma S\,\Delta W\,$

$\Delta V=\left(\mu S{\frac {\partial V}{\partial S}}+{\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}\right)\Delta t+\sigma S{\frac {\partial V}{\partial S}}\,\Delta W$

and appropriately substitute them into the expression for $\Delta \Pi$ :

$\Delta \Pi =\left(-{\frac {\partial V}{\partial t}}-{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}\right)\Delta t$

Notice that the $\Delta W$ term has vanished. Thus uncertainty has been eliminated and the portfolio is effectively riskless. The rate of return on this portfolio must be equal to the rate of return on any other riskless instrument; otherwise, there would be opportunities for arbitrage. Now assuming the risk-free rate of return is $[t,t+\Delta t]$

$r\Pi \,\Delta t=\Delta \Pi$

If we now equate our two formulas for $\Delta \Pi$ we obtain:

$\left(-{\frac {\partial V}{\partial t}}-{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}\right)\Delta t=r\left(-V+S{\frac {\partial V}{\partial S}}\right)\Delta t$

Simplifying, we arrive at the celebrated Black–Scholes partial differential equation:

${\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0$

With the assumptions of the Black–Scholes model, this second-order partial differential equation holds for any type of option as long as its price function $V$ is twice differentiable with respect to $S$ and once with respect to $t$ . Different pricing formulae for various options will arise from the choice of payoff function at expiry and appropriate boundary conditions.

Technical note: A subtlety obscured by the discretization approach above is that the infinitesimal change in the portfolio value was due to only the infinitesimal changes in the values of the assets being held, not changes in the positions in the assets. In other words, the portfolio was assumed to be self-financing.

Alternative derivation

Here is an alternative derivation that can be utilized in situations where it is initially unclear what the hedging portfolio should be.

In the Black–Scholes model, assuming we have picked the risk-neutral probability measure, the underlying stock price S(t) is assumed to evolve as a geometric Brownian motion:

${\frac {dS(t)}{S(t)}}=r\ dt+\sigma dW(t)$

Since this stochastic differential equation (SDE) shows the stock price evolution is Markovian, any derivative on this underlying is a function of time t and the stock price at the current time, S(t). Then an application of Ito's lemma gives an SDE for the discounted derivative process $\exp(-rt)V(t,S(t))$ , which should be a martingale. In order for that to hold, the drift term must be zero, which implies the Black—Scholes PDE.

This derivation is basically an application of the Feynman-Kac formula and can be attempted whenever the underlying asset(s) evolve according to given SDE(s).

This question is designed to make sure you know how to derive the Black-Scholes partial differential equation (pde)

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