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A multiplicative factor converting a value into units of a chosen initial account. With the continuous-time bank account $B$, $D_t=1/B_t$; discounted traded asset prices are martingales under the corresponding equivalent martingale measure.

The terminal density is strictly positive and has expectation one under the usual deterministic initial bond-price convention. Its density process isFor $0\le s\le t\le T$, the Bayes formula for conditional expectation under a change of measure givesThe same calculation at $s=0$ gives the finite expectation $1/P(0,T)$, so this is a true martingale, not just a formal conditional identity. Thus the continuous-time bank account measured in units of the maturity-$T$ bond is a ${\mathbb{Q}}_T$-martingale. This is the forward measure change of numéraire. If the initial bond price were random rather than given, integrability of its reciprocal would need to be included for this true-martingale assertion.

To avoid confusing the continuous-time bank account with the coefficient of $r_t$, denote the latter by $b(\tau )$ and the other coefficient by $a(\tau )$, where $\tau =T-t$. Apply the Itô formula to $D_t[a(\tau )+b(\tau )r_t]$. Since the expression is affine in $r$, its second rate derivative is zero. Its drift isThe quadratic terms cancel. The remaining expression isIt vanishes when $a^{\prime }=0$ and $b^{\prime }=-a-b$. For a unit bond payoff choose terminal conditions $a(0)=1$, $b(0)=0$, givingThe resulting local martingale is $D_t[1-(1-e^{-\tau })r_t]$. The allowed bound $0\le r_t\le 1$ puts it between zero and one, so the bounded local martingale criterion makes it a true martingale. At maturity it equals $D_T$. Comparing with part (a) therefore givesIn particular $e^{-(T-t)}\le P(t,T)\le 1$, and $P(T,T)=1$. This is linear bond pricing in a bounded short-rate diffusion; choosing zero coefficients would produce a local martingale but would not price the required terminal payoff.

Choose the market price of risk ${\lambda }_t=(\mu -r{S}_t)/\sigma$. The Girsanov theorem, with the hypotheses allowed in the question, gives an equivalent martingale measure $\mathbb{Q}$ under whichThus the stock under the risk-neutral measure is a linear Gaussian diffusion. In particular, for $\tau =T-t$ its conditional mean and standard deviation arePut $\varphi (x)=(2\pi {)}^{-1/2}{e}^{-x^2/2}$ and let $\Phi$ be the standard normal distribution function. For $N\sim N(0,1)$,since ${\int }_{-d}^{\infty }x\varphi (x)dx=\varphi (d)$. Discounting this expectation gives the call price in an arithmetic stock model with interest. A particularly convenient expression isAt maturity define $C(T,s)=(s-K{)}^{+}$. This value is nonnegative because it is a discounted expectation of a nonnegative payoff.

For $t<T$, hold ${\pi }_t=C_s(t,S_t)$ shares and hold ${\beta }_t=[C(t,S_t)-{\pi }_t{S}_t]/B_t$ units of the continuous-time bank account. The pricing function solvesThe Itô formula under the physical measure therefore givesThis proves self-financing and terminal replication, with wealth always $C(t,S_t)\ge 0$. The coefficients are locally smooth before maturity, and the strategy extends to maturity through its continuous wealth limit and the square-integrable discounted payoff representation.

To see minimality, any other nonnegative self-financing portfolio replicating the payoff has discounted wealth a nonnegative local martingale under $\mathbb{Q}$, hence a supermartingale. Its initial capital $x$ must satisfy $x\ge {\mathbb{E}}^{\mathbb{Q}}[e^{-rT}(S_T-K{)}^{+}]=C(0,S_0)$. The strategy constructed above attains equality. ThusThe additive physical diffusion may take negative stock values; the formula and nonnegative replicating wealth remain valid. Replacing it by a multiplicative Black–Scholes diffusion would give the wrong price and hedge.

For fixed maturity $T$, define $b_t^T={\int }_t^T\sigma (t,u)du$ and $I_t^T={\int }_t^{T}f(t,u)du$. The stochastic Fubini theorem and the moving lower endpoint giveThe factor $1/2$ is the integral over one of the two triangles in the square $[t,T{]}^2$. Applying the Itô formula to $P(t,T)=e^{-I_t^T}$, its quadratic-variation correction cancels that factor:Set $D_t=e^{-{\int }_0^t{r}_{s}ds}$, the reciprocal of the continuous-time bank account. The Itô product rule then givesIf $|\sigma |\le M$, then $|b_t^T|\le MT$ on this finite horizon. The Novikov condition holds, so this stochastic exponential is a true martingale, not merely a local martingale. The discounted price is thereforeThe authoritative PDF discounts to $t$ in this part. The TeX transcription's upper endpoint $T$ would include future short rates and is incorrect here.

The instantaneous continuously compounded interest rate. The continuous-time bank account grows at this rate; it equals the instantaneous forward rate at current maturity.