Stochastic Calculus for Finance II : Continuous-Time Models

Steven E. Shreve

Chapter 5 Risk-Neutral Pricing - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider the discounted stock price $D(t) S(t)$ of (5.2.19). In this problem, we derive the formula $(5.2 .20)$ for $d(D(t) S(t))$ by two methods.
(i) Define $f(x)=S(0) e^x$ and set
$$
X(t)=\int_0^t \sigma(s) d W(s)+\int_0^t\left(\alpha(s)-R(s)-\frac{1}{2} \sigma^2(s)\right) d s
$$
so that $D(t) S(t)=f(X(t))$. Use the Itô-Doeblin formula to compute $d f(X(t))$.
(ii) According to Itô's product rule,
$$
d(D(t) S(t))=S(t) d D(t)+D(t) d S(t)+d D(t) d S(t) .
$$

Use (5.2.15) and (5.2.18) to work out the right-hand side of this equation.

Check back soon!

Problem 2

(State price density process). Show that the risk-neutral pricing formula $(5.2 .30)$ may be rewritten as
$$
D(t) Z(t) V(t)=\mathbb{E}[D(T) Z(T) V(T) \mid \mathcal{F}(t)]
$$

Here $Z(t)$ is the Radon-Nikodým derivative process (5.2.11) when the market price of risk process $\Theta(t)$ is given by (5.2.21) and the conditional expectation on the right-hand side of (5.9.1) is taken under the actual probability measure
252
5 Risk-Neutral Pricing
$\mathbb{P}$, not the risk-neutral measure $\tilde{\mathbb{P}}$. In particular, if for some $A \in \mathcal{F}(T)$ a derivative security pays off $\mathbb{I}_A$ (i.e., pays 1 if $A$ occurs and 0 if $A$ does not occur), then the value of this derivative security at time zero is $\mathbb{E}\left[D(T) Z(T) \mathbb{I}_A\right]$. The process $D(t) Z(t)$ appearing in (5.9.1) is called the state price density process.

Check back soon!

Problem 3

According to the Black-Scholes-Merton formula, the value at time zero of a European call on a stock whose initial price is $S(0)=x$ is given by
$$
c(0, x)=x N\left(d_{+}(T, x)\right)-K e^{-r T} N\left(d_{-}(T, x)\right),
$$
where
$$
\begin{aligned}
& d_{+}(T, x)=\frac{1}{\sigma \sqrt{T}}\left[\log \frac{x}{K}+\left(r+\frac{1}{2} \sigma^2\right) T\right], \\
& d_{-}(T, x)=d_{+}(T, x)-\sigma \sqrt{T} .
\end{aligned}
$$

The stock is modeled as a geometric Brownian motion with constant volatility $\sigma>0$, the interest rate is constant $r$, the call strike is $K$, and the call expiration time is $T$. This formula is obtained by computing the discounted expected payoff of the call under the risk-neutral measure,
$$
\begin{aligned}
c(0, x) & =\widetilde{\mathbb{E}}\left[e^{-r T}(S(T)-K)^{+}\right] \\
& =\widetilde{\mathbb{E}}\left[e^{-r T}\left(x \exp \left\{\sigma \widetilde{W}(T)+\left(r-\frac{1}{2} \sigma^2\right) T\right\}-K\right)^{+}\right],
\end{aligned}
$$

Check back soon!

Problem 4

(Black-Scholes-Merton formula for time-varying, nonrandom interest rate and volatility). Consider a stock whose price differential is
$$
d S(t)=r(t) S(t) d t+\sigma(t) d \widetilde{W}(t),
$$
where $r(t)$ and $\sigma(t)$ are nonrandom functions of $t$ and $\widetilde{W}$ is a Brownian motion under the risk-neutral measure $\tilde{\mathbb{P}}$. Let $T>0$ be given, and consider a European call, whose value at time zero is
$$
c(0, S(0))=\mathbf{E}\left[\exp \left\{-\int_0^T r(t) d t\right\}(S(T)-K)^{+}\right] .
$$
(i) Show that $S(T)$ is of the form $S(0) e^X$, where $X$ is a normal random variable, and determine the mean and variance of $X$.
(ii) Let
$$
\begin{aligned}
\operatorname{BSM}(T, x ; K, R, \Sigma)=x N & \left(\frac{1}{\Sigma \sqrt{T}}\left[\log \frac{x}{K}+\left(R+\Sigma^2 / 2\right) T\right]\right) \\
& -e^{-R T} K N\left(\frac{1}{\Sigma \sqrt{T}}\left[\log \frac{x}{K}+\left(R-\Sigma^2 / 2\right) T\right]\right)
\end{aligned}
$$
denote the value at time zero of a European call expiring at time $T$ when the underlying stock has constant volatility $\Sigma$ and the interest rate $R$ is constant. Show that
$$
c(0, S(0))=\operatorname{BSM}\left(S(0), T, \frac{1}{T} \int_0^T r(t) d t, \sqrt{\frac{1}{T} \int_0^T \sigma^2(t) d t}\right) .
$$

Check back soon!

Problem 5

Prove Corollary 5.3 .2 by the following steps.
(i) Compute the differential of $\frac{1}{Z(t)}$, where $Z(t)$ is given in Corollary 5.3.2.
(ii) Let $\widetilde{M}(t), 0 \leq t \leq T$, be a martingale under $\tilde{\mathbb{P}}$. Show that $M(t)=$ $Z(t) \widetilde{M}(t)$ is a martingale under $\mathbb{P}$.
(iii) According to Theorem 5.3.1, there is an adapted process $\Gamma(u), 0 \leq u \leq T$, such that
$$
M(t)=M(0)+\int_0^T \Gamma(u) d W(u), 0 \leq t \leq T .
$$

Write $\widetilde{M}(t)=M(t) \cdot \frac{1}{Z(t)}$ and take its differential using Itô's product rule.
(iv) Show that the differential of $\widetilde{M}(t)$ is the sum of an adapted process, which we call $\widetilde{\Gamma}(t)$, times $d \widetilde{W}(t)$, and zero times $d t$. Integrate to obtain (5.3.2).
254
5 Risk-Neutral Pricing

Check back soon!

03:30

Problem 6

. Use the two-dimensional Lévy Theorem, Theorem 4.6.5, to prove the two-dimensional Girsanov Theorem (i.e., Theorem 5.4.1 with $d=2$ ).

Jay Patel

Numerade Educator

Problem 7

(i) Suppose a multidimensional market model as described in Section 5.4.2 has an arbitrage. In other words, suppose there is a portfolio value process satisfying $X_1(0)=0$ and
$$
\mathbb{P}\left\{X_1(T) \geq 0\right\}=1, \quad \mathbb{P}\left\{X_1(T)>0\right\}>0,
$$
for some positive $T$. Show that if $X_2(0)$ is positive, then there exists a portfolio value process $X_2(t)$ starting at $X_2(0)$ and satisfying
$$
\mathbb{P}\left\{X_2(T) \geq \frac{X_2(0)}{D(T)}\right\}=1, \quad \mathbb{P}\left\{X_2(T)>\frac{X_2(0)}{D(T)}\right\}>0 .
$$
(ii) Show that if a multidimensional market model has a portfolio value process $X_2(t)$ such that $X_2(0)$ is positive and (5.4.24) holds, then the model has a portfolio value process $X_1(0)$ such that $X_1(0)=0$ and $(5.4 .23)$ holds.

Check back soon!

Problem 8

(Every strictly positive asset is a generalized geometric Brownian motion). Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space on which is defined a Brownian motion $W(t), 0 \leq t \leq T$. Let $\mathcal{F}(t), 0 \leq t \leq T$, be the filtration generated by this Brownian motion. Assume there is a unique risk-neutral measure $\widetilde{\mathbb{P}}$, and let $\widetilde{W}(t), 0 \leq t \leq T$, be the Brownian motion under $\widetilde{\mathbb{P}}$ obtained by an application of Girsanov's Theorem, Theorem 5.2.3.

Corollary 5.3.2 of the Martingale Representation Theorem asserts that every martingale $\widetilde{M}(t), 0 \leq t \leq T$, under $\widetilde{\mathbb{P}}$ can be written as a stochastic integral with respect to $\widetilde{W}(t), 0 \leq t \leq T$. In other words, there exists an adapted process $\widetilde{\Gamma}(t), 0 \leq t \leq T$, such that
$$
\widetilde{M}(t)=\widetilde{M}(0)+\int_0^t \widetilde{\Gamma}(u) d \widetilde{B}(u), \quad 0 \leq t \leq T .
$$

Now let $V(T)$ be an almost surely positive ("almost surely" means with probability one under both $\mathbb{P}$ and $\widetilde{\mathbb{P}}$ since these two measures are equivalent), $\mathcal{F}(T)$-measurable random variable. According to the risk-neutral pricing formula (5.2.31), the price at time $t$ of a security paying $V(T)$ at time $T$ is
$$
V(t)=\tilde{\mathbb{E}}\left[e^{-\int_t^T R(u) d u} V(T) \mid \mathcal{F}(t)\right], \quad 0 \leq t \leq T .
$$
(i) Show that there exists an adapted process $\tilde{\Gamma}(t), 0 \leq t \leq T$, such that
$$
d V(t)=R(t) V(t) d t+\frac{\widetilde{\Gamma}(t)}{D(t)} d \widetilde{W}(t), \quad 0 \leq t \leq T .
$$

Check back soon!

Problem 9

Exercise 5.9 (Implying the risk-neutral distribution). Let $S(t)$ be the price of an underlying asset, which is not necessarily a geometric Brownian motion (i.e., does not necessarily have constant volatility). With $S(0)=x$, the risk-neutral pricing formula for the price at time zero of a European call on this asset, paying $(S(T)-K)^{+}$at time $T$, is
$$
c(0, T, x, K)=\tilde{\mathbb{E}}\left[e^{-r T}(S(T)-K)^{+}\right] .
$$
(Normally we consider this as a function of the current time 0 and the current stock price $x$, but in this exercise we shall also treat the expiration time $T$ and the strike price $K$ as variables, and for that reason we include them as arguments of $c$.) We denote by $\tilde{p}(0, T, x, y)$ the risk-neutral density in the $y$ variable of the distribution of $S(T)$ when $S(0)=x$. Then we may rewrite the risk-neutral pricing formula as
$$
c(0, T, x, K)=e^{-r T} \int_K^{\infty}(y-K) \tilde{p}(0, T, x, y) d y .
$$

Suppose we know the market prices for calls of all strikes (i.e., we know $c(0, T, x, K)$ for all $K>0)^2$ We can then compute $c_K(0, T, x, K)$ and $c_{K K}(0, T, x, K)$, the first and second derivatives of the option price with respect to the strike. Differentiate (5.9.3) twice with respect to $K$ to obtain the equations
$$
\begin{aligned}
c_K(0, T, x, K) & =-e^{-r T} \int_K^{\infty} \tilde{p}(0, T, x, y) d y=-e^{-r T} \tilde{\mathbb{P}}\{S(T)>K\}, \\
c_K(0, T, x, K) & =e^{-r T} \tilde{p}(0, T, x, K) .
\end{aligned}
$$

The second of these equations provides a formula for the risk-neutral distribution of $S(T)$ in terms of call prices:
$$
\tilde{p}(0, T, x, K)=e^{r T} c_{K K}(0, T, x, K) \text { for all } K>0 .
$$
${ }^2$ In practice, we do not have this many prices. We have the prices of calls at some strikes, and we can infer the prices of calls at other strikes by knowing the prices of puts and using put-call parity. We must create prices for the calls of other strikes by interpolation of the prices we do have.

Check back soon!

Problem 10

Exercise 5.10 (Chooser option). Consider a model with a unique riskneutral measure $\widetilde{\mathbb{P}}$ and constant interest rate $r$. According to the risk-neutral pricing formula, for $0 \leq t \leq T$, the price at time $t$ of a European call expiring at time $T$ is
$$
C(t)=\tilde{\mathbb{E}}\left[e^{-r(T-t)}(S(T)-K)^{+} \mid \mathcal{F}(t)\right],
$$
where $S(T)$ is the underlying asset price at time $T$ and $K$ is the strike price of the call. Similarly, the price at time $t$ of a European put expiring at time $T$ is
$$
P(t)=\tilde{\mathbb{E}}\left[e^{-r(T-t)}(K-S(T))^{+} \mid \mathcal{F}(t)\right] .
$$

Finally, because $e^{-r t} S(t)$ is a martingale under $\tilde{\mathbb{P}}$, the price at time $t$ of a forward contract for delivery of one share of stock at time $T$ in exchange for a payment of $K$ at time $T$ is
$$
\begin{aligned}
F(t) & =\tilde{\mathbb{E}}\left[e^{-r(T-t)}(S(T)-K) \mid \mathcal{F}(t)\right] \\
& =e^{r t} \widetilde{\mathbb{E}}\left[e^{-r T} S(T) \mid \mathcal{F}(t)\right]-e^{-r(T-t)} K \\
& =S(t)-e^{-r(T-t)} K .
\end{aligned}
$$

Because
$$
(S(T)-K)^{+}-(K-S(T))^{+}=S(T)-K,
$$
we have the put-call parity relationship
$$
\begin{aligned}
C(t)-P(t) & =\tilde{\mathbb{E}}\left[e^{-r(T-t)}(S(T)-K)^{+}-e^{-r(T-t)}(K-S(T))^{+} \mid \mathcal{F}(t)\right] \\
& =\widetilde{\mathbb{E}}\left[e^{-r(T-t)}(S(T)-K) \mid \mathcal{F}(t)\right]=F(t) .
\end{aligned}
$$

Now consider a date $t_0$ between 0 and $T$, and consider a chooser option, which gives the right at time $t_0$ to choose to own either the call or the put.
(i) Show that at time $t_0$ the value of the chooser option is
$$
C\left(t_0\right)+\max \left\{0,-F\left(t_0\right)\right\}=C\left(t_0\right)+\left(e^{-r\left(T-t_0\right)} K-S\left(t_0\right)\right)^{+} .
$$
(ii) Show that the value of the chooser option at time 0 is the sum of the value of a call expiring at time $T$ with strike price $K$ and the value of a put expiring at time $t_0$ with strike price $e^{-r\left(T-t_0\right)} K$.

Check back soon!

Problem 11

(Hedging a cash flow). Let $W(t), 0 \leq t \leq T$, be a Brownian motion on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and let $\mathcal{F}(t), 0 \leq t \leq T$, be the filtration generated by this Brownian motion. Let the mean rate of return $\alpha(t)$, the interest rate $R(t)$, and the volatility $\sigma(t)$ be adapted processes, and assume that $\sigma(t)$ is never zero. Consider a stock price process whose differential is given by $(5.2 .15)$ :
5.9 Exercises
257
$$
d S(t)=\alpha(t) S(t) d t+\sigma(t) S(t) d W(t), \quad 0 \leq t \leq T .
$$

Suppose an agent must pay a cash flow at rate $C(t)$ at each time $t$, where $C(t), 0 \leq t \leq T$, is an adapted process. If the agent holds $\Delta(t)$ shares of stock at each time $t$, then the differential of her portfolio value will be
$$
d X(t)=\Delta(t) d S(t)+R(t)(X(t)-\Delta(t) S(t)) d t-C(t) d t .
$$

Show that there is a nonrandom value of $X(0)$ and a portfolio process $\Delta(t)$, $0 \leq t \leq T$, such that $X(T)=0$ almost surely. (Hint: Define the risk-neutral measure and apply Corollary 5.3.2 of the Martingale Representation Theorem to the process
$$
\widetilde{M}(t)=\widetilde{\mathbb{E}}\left[\int_0^T D(u) C(u) d u \mid \mathcal{F}(t)\right], \quad 0 \leq t \leq T,
$$
where $D(t)$ is the discount process (5.2.17).)

Check back soon!

Problem 12

(Correlation under change of measure). Consider the multidimensional market model of Subsection 5.4.2, and let $B_i(t)$ be defined by (5.4.7). Assume that the market price of risk equations (5.4.18) have a solution $\theta_1(t), \ldots, \Theta_d(t)$, and let $\widetilde{\mathbb{P}}$ be the corresponding risk-neutral measure under which
$$
\widetilde{W}_j(t)=W_j(t)+\int_0^t \Theta_j(u) d u, \quad j=1, \ldots, d,
$$
are independent Brownian motions.
(i) For $i=1, \ldots, d$, define $\gamma_i(t)=\sum_{j=1}^d \frac{\sigma_{0,}(t) \theta_j(t)}{\sigma_i(t)}$. Show that
$$
\widetilde{B}_i(t)=B_i(t)+\int_0^t \gamma_i(u) d u
$$
is a Brownian motion under $\widetilde{\mathbb{P}}$.
(ii) We saw in (5.4.8) that
$$
d S_i(t)=\alpha_i(t) S_i(t) d t+\sigma_i(t) S_i(t) d B_i(t), \quad i=1, \ldots, m .
$$

Show that
$$
d S_i(t)=R(t) S_i(t) d t+\sigma_i S_i(t) d \tilde{B}_i(t), \quad i=1, \ldots, m
$$

Check back soon!

Problem 13

In part (v) of Exercise 5.12, we saw that when we change measures and change Brownian motions, correlations can change if the instantaneous correlations are random. This exercise shows that a change of measure without a change of Brownian motions can change correlations if the market prices of risk are random
Let $W_1(t)$ and $W_2(t)$ be independent Brownian motions under a probability measure $\widetilde{\mathbb{P}}$. Take $\theta_1(t)=0$ and $\theta_2(t)=W_1(t)$ in the multidimensional Girsanov Theorem, Theorem 5.4.1. Then $\widetilde{W}_1(t)=W_1(t)$ and $\widetilde{W}_2(t)=W_2(t)+\int_0^t W_1(u) d u$.
(i) Because $\widetilde{W}_1(t)$ and $\widetilde{W}_2(t)$ are Brownian motions under $\widetilde{\mathbb{P}}$, the equation $\widetilde{\mathbb{E}} \widetilde{W}_1(t)=\widetilde{\mathbb{E}} W_2(t)=0$ must hold for all $t \in[0, T]$. Use this equation to conclude that
$$
\tilde{\mathbb{E}} W_1(t)=\tilde{\mathbb{E}} W_2(t)=0 \text { for all } t \in[0, T] .
$$
(ii) From Itô's product rule, we have
$$
d\left(W_1(t) W_2(t)\right)=W_1(t) d W_2(t)+W_2(t) d W_1(t) .
$$

Use this equation to show that
$$
\widetilde{\operatorname{Cov}}\left[W_1(T), W_2(T)\right]=\widetilde{\mathbb{E}}\left[W_1(T) W_2(T)\right]=-\frac{1}{2} T^2 .
$$

This is different from
$$
\operatorname{Cov}\left[W_1(T), W_2(T)\right]=\mathbb{E}\left[W_1(T) W_2(T)\right]=0 .
$$

Check back soon!

Problem 14

(Cost of carry). Consider a commodity whose unit price at time $t$ is $S(t)$. Ownership of a unit of this commodity requires payment at a rate $a$ per unit time (cost of carry) for storage. Note that this payment is per unit of commodity, not a fraction of the price of the commodity. Thus, the value of a portfolio that holds $\Delta(t)$ units of the commodity at time $t$ and also invests in a money market account with constant rate of interest $r$ has differential
$$
d X(t)=\Delta(t) d S(t)-a \Delta(t) d t+r(X(t)-\Delta(t) S(t)) d t .
$$

As with the dividend-paying stock in Section 5.5, we must choose the riskneutral measure so that the discounted portfolio value $e^{-r t} X(t)$ is a martingale. We shall assume a constant volatility, so in place of (5.5.6) we have
$$
d S(t)=r S(t) d t+\sigma S(t) d \widetilde{W}(t)+a d t,
$$
where $\widetilde{W}(t)$ is a Brownian motion under the risk-neutral measure $\widetilde{\mathbb{P}}$.
(i) Show that when $d S(t)$ is given by (5.9.7), then under $\widetilde{\mathbb{P}}$ the discounted portfolio value process $e^{-r t} X(t)$, where $X(t)$ is given by (5.9.6), is a martingale.

Check back soon!