# 金融代写|期权理论代写Mathematical Introduction to Options代考|MATH485

#### Doug I. Jones

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## 金融代写|期权理论代写Mathematical Introduction to Options代考|GREEKS

(i) A deeper understanding of the material of the last couple of chapters is obtained by considering the various partial derivatives of the stock price. Let us return to the Taylor expansion of Section 3.4:
$$\delta f_{S_t t}=\frac{\partial f_{S_t t}}{\partial S_t} \delta S_t+\frac{\partial f_{S_S t}}{\partial t} \delta t+\frac{1}{2}\left{\frac{\partial^2 f_{S_t t}}{\partial S_t^2} \delta S_t^2+\frac{\partial^2 f_{S_t t}}{\partial S_t \partial t} \delta S_t \delta t+\frac{\partial^2 f_{S_t t}}{\partial t^2} \delta t^2\right}+\cdots$$
Using the substitution $\delta S_t^2 \rightarrow \sigma^2 S_t^2 \delta t$ which is explained in Section 3.3, and retaining only terms of first order in $\delta t$ gives
$$\delta f_{S_t t}=\frac{\partial f_{S_t t}}{\partial S_t} \delta S_t+\left{\frac{\partial f_{S_t t}}{\partial t}+\frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 f_{S_t t}}{\partial S_t^2}\right} \delta t$$
In Section 4.2(vi) we defined the delta of a derivative by $\Delta_{S_t t}=\partial f_{S_t t} / \partial S_t$; two other partial derivatives are now defined as follows.
Theta: $\theta_{S_t t}={ }{\partial t}^{\partial f{s t}}$ This is the rate at which the value of an option changes over time.

Gamma: $\Gamma_{S_t t}=\frac{\partial \Delta_{S_t t}}{\partial S_t}=\frac{\partial^2 f_{s_s t}}{\partial S_t^2}$ The second derivative is a measure of the rate of change of the slope of the curve of $f_{S_t t}$ against $S_t$, i.e. it measures the sharpness of the curvature of the curve.

Lightening up on the notation a little, the Taylor expansion can now be written in Greek letters:
$$\delta f_t=\Delta_t \delta S_t+\left{\theta_t+\frac{1}{2} \sigma^2 S_t^2 \Gamma_t\right} \delta t$$
(ii) This last equation is illustrated in Figure 4.2, and may be given the following physical interpretation: in the time period $\delta t$, the stock price moves by $\delta S_t$; over the same interval, the derivative price moves by $\delta f_t=f_{S_t+\delta S_t t+\delta t}-f_{S_t t}$. This is represented by a move from the point A to the point $\mathrm{A}^{\prime}$ in the graph, with the value of $\delta f_t$ represented by the distance $\mathrm{A}^{\prime} \mathrm{D}$. This is made up of three distinct parts, which are easiest to understand if we think in terms of replication of an option rather than hedging.

## 金融代写|期权理论代写Mathematical Introduction to Options代考|DERIVATION OF MODEL FROM EXPECTED VALUES

(i) Risk neutrality tells us that the value at time $t=0$ of a call option maturing at time $T=0$ is given by
$$C_0=\mathrm{e}^{-r T} \mathrm{E}\left[C_T\right]{\text {risk neutral }}=\mathrm{e}^{-r T} \mathrm{E}\left[\max \left[\left(S_T-X\right), 0\right]\right]{\text {risk neutral }}$$
For notional simplicity, the risk-neutral suffix will be dropped but it must always be remembered that we are dealing with pseudo-probabilities and pseudo-expectations.

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