Math Problems Vault

Problem 13.10.1

(a)(b)

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(a)

Let $Z$ have the $N\left(\gamma , \tau^{2}\right)$ distribution. Show that

\mathbb {E}\left[\left(a e^{Z}-K\right)^{+}\right] = a e^{\gamma +\frac{1}{2} \tau ^{2}} \Phi \left(\frac{\log (a / K)+\gamma }{\tau }+\tau \right)-K \Phi \left(\frac{\log (a / K)+\gamma }{\tau }\right)

where $\Phi$ is the $N(0,1)$ distribution function.

(b)

Let $\mathbb {Q}$ be a probability measure under which $\sigma W$ is a Wiener process with drift $r-\mu$ and instantaneous variance $\sigma^{2}$ . Show for $0 \leq t \leq T$ that

\mathbb {E}_{\mathbb {Q}}\left[ \left(S_{T}-K\right)^{+} \mid \mathcal{F}_{t} \right] = S_{t} e^{r(T-t)} \Phi \left(d_{1}\left(t, S_{t}\right)\right)-K \Phi \left(d_{2}\left(t, S_{t}\right)\right)

where

d_{1}(t, x) = \frac{\log (x / K)+\left(r+\frac{1}{2} \sigma ^{2}\right)(T-t)}{\sigma \sqrt{T-t}}, \quad d_{2}(t, x) = d_{1}(t, x)-\sigma \sqrt{T-t}

Problem 13.10.2

(a)(b)(c)

Consider a portfolio which, at time $t$ , holds $\xi (t, S)$ units of stock and $\psi (t, S)$ units of bond, and assume these quantities depend only on the values of $S_{u}$ for $0 \leq u \leq t$ . Find the function $\psi$ such that the portfolio is self-financing in the three cases:

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(a)

$\xi (t, S) = 1$ for all $t, S$ ,

(b)

$\xi (t, S) = S_{t}$ ,

(c)

$\xi (t, S) = \int_{0}^{t} S_{v} d v$ .

Problem 13.10.3

(a)(b)(c)(d)

Suppose the stock price $S_{t}$ is itself a Wiener process and the interest rate $r$ equals $0$ , so that a unit of bond has unit value for all time. In the notation of Exercise (13.10.2), which of the following define self-financing portfolios?

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(a)

$\xi (t, S) = \psi (t, S) = 1$ for all $t, S$ ,

(b)

$\xi (t, S) = 2 S_{t}, \psi (t, S) = -S_{t}^{2}-t$ ,

(c)

$\xi (t, S) = -t, \psi (t, S) = \int_{0}^{t} S_{s} d s$ ,

(d)

$\xi (t, S) = \int_{0}^{t} S_{s} d s, \psi (t, S) = -\int_{0}^{t} S_{s}^{2} d s$ .

Problem 13.10.4

An 'American call option' differs from a European call option in that it may be exercised by the buyer at any time up to the expiry date. Show that the value of the American call option is the same as that of the corresponding European call option, and that there is no advantage to the holder of such an option to exercise it strictly before its expiry date.

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Problem 13.10.5

Show that the Black-Scholes value at time 0 of the European call option is an increasing function of the initial stock price, the exercise date, the interest rate, and the volatility, and is a decreasing function of the strike price.

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