### 金融代写|金融工程作业代写Financial Engineering代考| Black-Scholes Formula

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## 金融代写|金融工程作业代写Financial Engineering代考|Early Exercise of an American Call Option

When there are no dividends, it is never optimal to exercise an American call option before maturity and its value will be the same as a European call option. To see why, note that an American call option is a European call option with the additional feature that it can be exercised before maturity. Therefore, it must be as valuable as a European call option, i.e.,
$$C_{\text {American }}(t, s) \geq C_{\text {European }}(t, s) \text {, }$$
where $C_{\text {American }}(t, s)$ and $C_{E u r o p e a n}(t, s)$ are respectively American and European call options on the same underlying asset with the same strike price and maturity. Now, we can rewrite the put-call parity relationship (1.6) as
\begin{aligned} C_{\text {European }}(t, s) &=s-K e^{-r(T-t)}+\tilde{C}{\text {European }}(t, s) \ &=s-K+K\left{1-e^{-r(T-t)}\right}+\tilde{C}{\text {Éuropean }}(t, s)>s-K \end{aligned}
Putting these two observations together and remarking that $C(t, s)>0$, one obtains that
$$C_{\text {American }}(t, s) \geq C_{\text {European }}(t, s)>\max (s-K, 0)$$

Therefore, the value of an American call option prior to maturity is always higher than its immediate exercise value.

A similar reasoning shows that this is not true for put options. In this case, if the underlying price falls enough and the call option’s value is low enough, then it might become optimal to exercise a put option prior to maturity.

## 金融代写|金融工程作业代写Financial Engineering代考|Partial Differential Equation for Option Values

In Black and Scholes $[1973]$ and Merton [1974], it is shown that the value $C$ of a European call option with maturity $T$ satisfies the following partial differential equation:
$$\frac{\partial C}{\partial t}+r s \frac{\partial C}{\partial s}+\sigma^{2} \frac{s^{2}}{2} \frac{\partial^{2} C}{\partial s^{2}}=r C$$
with boundary condition $C(T, s)=\max (s-K, 0)$, for all $s \geq 0$ and for all $t \in(0, T)$. It can also be shown that the equation also holds for a general payoff $\Phi$.

A popular method for proving the validity of (1.7) is to construct a selffinancing portfolio $\Pi_{t}=\psi_{t 1} S_{t}+\psi_{t 0} e^{r t}$, so that at maturity $\Pi_{T}=\Phi{S(T)}$, where $\Phi{S(T)}$ is the payoff. In fact, $\psi_{t 1}=\left.\frac{\partial}{\partial s} C(t, s)\right|{s=S{t}}$. The justification of $(1.7)$ is given in Appendix 1.A.

In simple cases, numerical methods used for solving partial differential equations can be used to solve $(1.7)$; see, e.g., Wilmott [2006]. However, as we will see in later chapters, derivatives can depend on several risk factors or even depend on the path taken by the underlying asset. In such cases, numerical solution to partial differential equation can become cumbersome. Therefore, we now turn to a representation of the derivative’s price which is easier to handle.

## 金融代写|金融工程作业代写Financial Engineering代考|Option Value as an Expectation

Under the absence of arbitrage, there exists an equivalent probability measure under which the discounted value of an option is a martingale (see Appendix 1.B). Such a measure is called an equivalent martingale measure ${ }^{7}$ or risk-neutral measure, and one can show that it is unique for the BlackScholes model. In this case, the actual value of an option is simply the expected discounted value of the option at a later date, for example at maturity, under the equivalent martingale measure $Q$. The value of the option at time $t$ is thus given by
$$C(t, s)=e^{-r(T-t)} E[\Phi{\tilde{S}(T)} \mid \tilde{S}(t)=s]$$

where, under $Q, \tilde{W}$ is a Brownian motion and
$$\tilde{S}(u)=s e^{\left(r-\frac{a^{2}}{2}\right)(u-t)+\sigma{\tilde{W}(u)-\tilde{W}(t)}}, u \in[t, T]$$
Equivalently, one has
$$d \bar{S}(u)=r \bar{S}(u) d u+\sigma \tilde{S}(u) d \tilde{W}(u), \quad t \leq u \leq T$$
with $\tilde{S}(t)=s$.
Note that the law of $\tilde{S}$ is not the same as the law of $S, \mu$ being replaced by $r$. In practice, only the process $S$ is observed, not $S$.
Using the Feynman-Kac formula in Proposition 7.4.1, one can show that (1.8) is the solution of the partial differential equation (1.7).

Example 1.5.1 For example, for a European call option with strike price $K$, we have $\Phi(s)=\max (s-K, 0)$. One can then recover the Black-Scholes formula (1.4) using Proposition A.6.3 and the expectation formula (1.8).

Remark 1.5.2 Formula (1.8) can be extended to path-dependent options like Asian options, lookback options, etc. One has to estimate or evaluate the discounted payoff of the option under the dynamics $\tilde{S}$. Monte Carlo methods are then more than appropriate in this context since a Brownian motion is easy to simulate.

## 金融代写|金融工程作业代写Financial Engineering代考|Early Exercise of an American Call Option

C美国人 (吨,s)≥C欧洲的 (吨,s),

\begin{aligned} C_{\text {欧洲}}(t, s) &=sK e^{-r(Tt)}+\tilde{C}{\text {欧洲}}(t, s) \ & =s-K+K\left{1-e^{-r(Tt)}\right}+\tilde{C}{\text {Éuropean }}(t, s)>sK \end{aligned}\begin{aligned} C_{\text {欧洲}}(t, s) &=sK e^{-r(Tt)}+\tilde{C}{\text {欧洲}}(t, s) \ & =s-K+K\left{1-e^{-r(Tt)}\right}+\tilde{C}{\text {Éuropean }}(t, s)>sK \end{aligned}

C美国人 (吨,s)≥C欧洲的 (吨,s)>最大限度(s−ķ,0)

## 金融代写|金融工程作业代写Financial Engineering代考|Partial Differential Equation for Option Values

∂C∂吨+rs∂C∂s+σ2s22∂2C∂s2=rC

## 金融代写|金融工程作业代写Financial Engineering代考|Option Value as an Expectation

C(吨,s)=和−r(吨−吨)和[披小号~(吨)∣小号~(吨)=s]

d小号¯(在)=r小号¯(在)d在+σ小号~(在)d在~(在),吨≤在≤吨

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