3.1 Arbitrage Relations

In this section we consider the fundamental notion of no-arbitrage. An arbitrage opportunity arises if it is possible to make a riskless profit. In an ideal financial market, in which all investors dispose of the same pieces of information and in which all investors can react instantaneously, there should not be any arbitrage opportunity. Since otherwise each investor would try to realize the riskless profit instantaneously. The resulting transactions would change the prices of the involved financial instruments such that the arbitrage opportunity disappears.

Additionally to no-arbitrage we presume in the remaining chapter that the financial market fulfills further simplifying assumptions which are in this context of minor importance and solely serve to ease the argumentation. If these assumptions hold we speak of a perfect financial market.

ASSUMPTION (perfect financial market)
There are no arbitrage opportunities, no transaction costs, no taxes, and no restrictions on short selling. Lending rates equal borrowing rates and all securities are perfectly divisible.

The assumption of a perfect financial market is sufficient to determine the value of future and forward contracts as well as some important relations between the prices of some types of options. Above all no mathematical model for the price of the financial instrument is needed. However, in order to determine the value of options more than only economic assumptions are necessary. A detailed mathematical modelling becomes inevitable. Each mathematical approach though has to be in line with certain fundamental arbitrage relations being developed in this chapter. If the model implies values of future and forward contracts or option prices which do not fulfill these relations the model's assumptions must be wrong.

An important conclusion drawn from the assumption of a perfect financial market and thus from no-arbitrage will be used frequently in the proofs to come. It is the fact that two portfolios which have at a certain time $T$ the same value must have the same value at a prior time $t<T$ as well. Due to its importance we will further illustrate this reasoning. We proceed from two portfolios $A$ and $B$ consisting of arbitrary financial instruments. Their value in time $t$ will be denoted by $W_A(t)$ and $W_B(t)$ respectively. For any fixed point of time $T$, we assume that $W_A(T) = W_B(T)$ independently of the prior time $T$ values of each financial instrument contained in $A$ and $B.$ For any prior point of time $t<T$ we assume without loss of generality that $W_A(t) \leq W_B(t).$ In time $t$ an investor can construct without own financial resources a portfolio which is a combination of $A$ and $B$ by buying one unit of every instrument of $A,$ selling one unit of every instrument of $B$ (short selling) and by investing the difference $\Delta(t) = W_B(t) - W_A(t) \geq 0$ at a fixed rate $r.$ The combined portfolio has at time $t$ a value of

$$W_A(t) - W_B(t) + \Delta(t) = 0 ,$$

i.e. the investor has no initial costs. At time $T$ the part of the combined portfolio which is invested at rate $r$ has the compounded value
$\Delta(T) = \Delta(t) e^{r (T-t)},$ and hence the combined portfolio has a value of

$$W_A(T) - W_B(T) + \Delta(T) = \Delta(t) e^{r (T-t)} > 0 ,$$

if $\Delta(t) > 0.$ The investor made a riskless gain by investing in the combined portfolio which contradicts the no-arbitrage assumption. Therefore, it must hold $\Delta(t) = 0,$ i.e.  $W_A(t) = W_B(t).$

The previous reasoning can be used to determine the unknown value of a financial derivative. For this, a portfolio $A$ is constructed which contains instruments with known price along with one unit of the derivative under investigation. Portfolio $A$ will be compared to another portfolio $B$, called the duplicating portfolio, which contains exclusively instruments with known prices. Since the duplicating portfolio $B$ is constructed such that for certain it has the same value at a fixed point of time $T$ as portfolio $A$ the no-arbitrage assumption implies that both portfolios must have the same value at any prior point of time. The value of the financial derivative can thus be computed at any time $t \leq T.$ We illustrate this procedure in the following example of a forward contract.

Theorem 3.1  
We consider a long forward contract to buy an object which has a price of $S_t$ at time $t.$ Let $K$ be the delivery price, and let $T$ be the maturity date. $V(s,\tau)$ denotes the value of the long forward contract at time $t$ as a function of the current price $S_t=s$ and the time to maturity $\tau=T-t.$ We assume constant interest rates $r$ during the time to maturity.

1. If the underlying object does not pay any dividends and does not involve any costs during the time to maturity $\tau,$ then it holds

   $$V(S_t,\tau) = V_{K,T}(S_t,\tau) = S_t - Ke^{-r\tau}$$ (3.1)

   
   
   The forward price is equal to $F_t = S_t e^{r\tau}.$
2. If during the time to maturity the underlying pays at discrete time points dividends or involves any costs whose current time $t$ discounted total value is equal to $D_t,$ then it holds

   $$V(S_t,\tau) = V_{K,T}(S_t,\tau) = S_t - D_t - Ke^{-r\tau}$$ (3.2)

   
   
   The forward price is equal to $F_t = (S_t - D_t) e^{r\tau}.$
3. If the underlying involves continuous costs at rate $b,$ then it holds

   $$V(S_t,\tau) = V_{K,T}(S_t,\tau) = S_t e^{(b-r)\tau} - Ke^{-r\tau}$$ (3.3)

   
   
   The forward price is equal to $F_t = S_t e^{b\tau}.$

Proof:
For simplicity we assume the underlying object to be a stock paying either discrete dividend yields whose value discounted to time $t$ is $D_t$ or paying a continuous dividend yield at rate $b.$ In the latter case the stock involves continuous costs equal to $b=r-d.$ The investor having a long position in the stock gains dividends (as negative costs) at rate $d$ but simultaneously loses interests at rate $r$ since he invested his capital in the stock instead of in a bond with a fixed interest rate. In place of stocks, bonds, currencies or other simple instruments can be considered as well.

1. We consider at time $t$ the following two portfolios $A$ and $B$:

Portfolio $A$:

One long forward contract on a stock with delivery price $K$, maturing in time $T$.
One long zero bond with face value $K$, maturing in time $T$.

Portfolio $B$:

A long position in one unit of the stock.

At maturity $T$ portfolio $A$ contains a zero bond of value $K.$ Selling this zero bond for $K$ the obligation to buy the stock for $K$ can be fulfilled. Following these transactions portfolio $A$ consists as well as portfolio $B$ of one unit of the stock. Thus both portfolios have at time $T$ the same value and must therefore, due to the no-arbitrage assumption, have the same value at any time $t$ prior to $T:$

$$V(S_t,\tau) + Ke^{-r \tau} = S_t \; ,$$ (3.4)

since the value of the zero bond at time $t$ is given by discounting $K$ at rate $r, \, Ke^{-r \tau}.$ The forward price is by definition the solution of

$$0 = V_{F_t,T}(S_t,\tau) = S_t - F_te^{-r\tau} .$$

2. We consider at time $t$ the two portfolios $A$ and $B$ as given above and add one position to portfolio $B:$

Portfolio $B$:

A long position in one unit of the stock and one short position of size $D_t$ in a zero bond with interest rate $r$ (lending an amount of money of $D_t$).

At maturity $T$ the dividend yields of the stock in portfolio $B,$ which compounded to time $T$ amount to $D_te^{r\tau},$ are used to pay back the bond. Thus, both portfolios $A$ and $B$ consist again of one unit of the stock, and therefore they must have the same value at any time $t<T:$

$$V(S_t,\tau) + Ke^{-r \tau} = S_t - D_t \; .$$ (3.5)

The forward price results as in part 1 from the definition.

3. If the stock pays dividends continuously at a rate $d,$ then the reasoning is similar as in part 2. Once again, we consider at time $t$ two portfolios $A$ and $B.$ And again, $A$ is left unchanged, $B$ is now composed of the following position:

Portfolio $B$:

A long position in $e^{-d\tau}$ stocks.

Reinvesting the dividends yields continuously in the stock portfolio $B$ consists again of exactly one stock at time $T.$ Heuristically, this can be illustrated as follows: In the time interval $[t, t+\delta]$ the stock pays approximately, for a small $\delta,$ a dividend of $d \cdot \delta \cdot S_t.$ Thus, the current total amount of stocks in the portfolio, $e^{-d\tau} = e^{-d(T-t)},$ pays a total dividend yield of $d \cdot \delta \cdot S_t \cdot e^{-d(T-t)},$ which is reinvested in the stock. Assuming that the stock price does not change significantly in the interval $[t,t+\delta],$ i.e.  $S_{t+\delta} \approx S_t,$ portfolio $B$ contains in time $t+\delta$

$$(1 + d \cdot \delta ) \cdot e^{-d(T-t)} \approx e^{d\delta} \cdot e^{-d(T-t)} = e^{-d(T-t-\delta)}$$

stocks. The above reasoning can be done exactly by taking the limit $\delta \rightarrow 0,$ and it can be shown that portfolio $B$ contains at any time $s$ between $t$ and $T$ exactly $e^{-d(T-s)}$ stocks. That is, for $s=T$ portfolio $B$ is composed of exactly one stock. The same reasoning as in part 1 leads to the conclusion that portfolio $A$ and $B$ must have the same value at any time $t$. Thus, we have

$$V(S_t,\tau) + Ke^{-r\tau} = e^{-d\tau}S_t \; .$$ (3.6)

where we have to set $b=r-d.$ The forward price results as in part 1 from the definition. ${\Box}$

Example 3.1   We consider a long forward contract on a 5 year bond which is currently traded at a price of 900 EUR. The delivery price is 910 EUR, the time to maturity of the forward contract is one year. The coupon payments of the bond of 60 EUR occur after 6 and 12 months (the latter shortly before maturity of the forward contract). The continuously compounded annual interest rates for 6 and 12 months are 9% and 10% respectively. In this example we have

$$S_t = 900 \;,\; K = 910 \;,\; r = 0.10 \;,\; \tau = 1 \;,\; D_t = 60e^{-0.09\cdot\frac{1}{2}} + 60e^{-0.10} = 111.65$$ (3.7)

Thus, the value of the forward contract is given by

$$V(S_t,\tau) = 900 - 111.65 -910 e^{-0.10} = -35.05 .$$ (3.8)

The value of the respective short position in the forward contract is +35.05. The price $F_t$ of the forward contract is equal to $F_t = (S_t - D_t)e^{r\tau}= 871.26.$

Example 3.2   Consider a long forward contract to buy 1000 Dollar. If the investor buys the 1000 Dollar and invests this amount in a American bond, the American interest rate can be interpreted as a dividend yield $d$ which is continuously paid. Let $r$ be the home interest rate. The investment involves costs $b = r - d,$ which are the difference between the American and the home interest rate. Denoting the dollar exchange rate by $S_t$ the price of the forward contract is then given by

$$F_t = S_te^{b\tau} = S_te^{(r - d)\tau} .$$ (3.9)

While for $r>d$ a report $S_t < F_t$ results, for $r<d$ a backwardation $S_t > F_t$ results. If $r>d$ and the delivery price is chosen to equal the current exchange rate, i.e. $K=S_t,$ then the value of the forward contract is

$$V_{S_t,T}(S_t,\tau) = S_t (e^{-d\tau} - e^{-r\tau}) > 0 .$$

Buying the forward contract at a price of $S_t$ is thus more expensive than buying the dollars immediately for the same price since in the former case the investor can invest the money up to time $T$ in a domestic bond paying an interest rate which is higher than the American interest rate.

The following result states that forward and future contracts with the same delivery price and the same time to maturity are equal, if interest rates are constant during the contract period. We will use the fact that by definition forward and future contracts do not cost anything if the delivery price is chosen to be equal to the current price of the forward contract respectively the price of the future contract.

Theorem 3.2  
If interest rates are constant during contract period, then forward and future prices are equal.

Proof:
We proceed from the assumption that the future contract is agreed on at time 0, and that is has a time to maturity of $N$ days. We assume that profits and losses are settled (marked to market) on a daily basis at a daily interest rate of $\rho$. While the forward price at the end of day 0 is denoted by $F,$ the future price at the end of day $t, \, t = 0, 1, \cdots, N$ is denoted by $F_t.$ The goal is to show that $F=F_0.$ For that we construct two portfolios again:

Portfolio $A$:

A long position in $e^{N\rho}$ forward contracts with delivery price $F$ and maturity date $N.$
A long position in a zero bond with face value $Fe^{N\rho}$ maturing in $N$ days.

Portfolio $B$:

A long position in futures contracts with delivery price $F_t$ and maturity date $N$. The contracts are bought daily such that the portfolio contains at the end of the $t$-th day exactly $e^{(t+1)\rho}$ future contracts ( $t = 0, 1, \cdots, N$).
A long position in a zero bond with face value $F_{0}e^{N\rho}$ maturing in $N$ days.

Purchasing a forward or a future contract does not cost anything since their delivery prices are set to equal the current forward or future price. Due to the marking to market procedure the holder of portfolio $B$ receives from day $t-1$ to day $t$ for each future contract an amount of $F_t - F_{t-1}$ which can possibly be negative (i.e. he has to pay).

At maturity, i.e. at the end of day $N,$ the zero bond of portfolio $A$ is sold at the face value $Fe^{N\rho}$ to fulfill the terms of the forward contract and to buy $e^{N\rho}$ stocks at a the delivery price $F.$ Then $A$ contains exclusively these stocks and has a value of $S_Ne^{N\rho}.$ Following, we show that portfolio $B$ has the same value.

At the beginning of day $t$ portfolio $B$ contains $e^{t\rho}$ future contracts, and the holder receives due to the marking to market procedure the amount $(F_t - F_{t-1})e^{t\rho}$ which can possibly be negative. During the day he increases his long position in the future contracts at zero costs such that the portfolio contains $e^{(t+1)\rho}$ future contracts at the end of the day. The earnings at day $t$ compounded to the maturity date have a value of:

$$(F_t - F_{t-1})e^{t\rho}\cdot e^{(N-t)\rho}=(F_t - F_{t-1})e^{N\rho}.$$ (3.10)

At maturity the terms of the future contracts are fulfilled due to the marking to market procedure. All profits and losses compounded to day $N$ have a value of:

$$\sum_{t=1}^{N}(F_t - F_{t-1})e^{N\rho} = (F_N - F_0)e^{N\rho}.$$ (3.11)

Together with the zero bond portfolio $B$ has at day $N$ a value of

$$(F_N - F_0)e^{N\rho} + F_0e^{N\rho} = F_Ne^{N\rho} = S_Ne^{N\rho} ,$$

since at maturity the future price $F_N$ and the price $S_N$ of the underlying are obviously equal.

Hence, both portfolios have at day $N$ the same value and thus due to the no-arbitrage assumption their day 0 values must be equal as well. Since the forward contract with delivery price $F$ has a value of 0 at day 0 due to the definition of the forward price, the value of portfolio $A$ is equal to the value of the zero bond, i.e. F (face value $Fe^{N\rho}$ discounted to day 0). Correspondingly, the $e^\rho$ futures contained in portfolio $B$ have at the end of day 0 a value of 0 due to the definition of the future price. Again, the value of portfolio $B$ reduces to the value of the zero bond. The latter has a value of $F_0$ (face value $F_0e^{N\rho}$ discounted to day 0). Putting things together, we conclude that $F=F_0.$ ${\Box}$

Now, we want to proof some relationship between option prices using similar methods. The most elementary properties are summarized in the following remark without a proof. For that, we need the notion of the intrinsic value of an option.

Definition 3.1 (Intrinsic Value)  
The intrinsic value of a call option at time $t$ is given by $\max(S_t-K,0),$ the intrinsic value of a put option is given by $\max(K-S_t,0).$ If the intrinsic value of an option is positive we say that the option is in the money. If $S_t = K,$ then the option is at the money. If the intrinsic value is negative, then the option is said to be out of the money.

Remark 3.1  
Options satisfy the following elementary relations. $C(s,\tau) = C_{K,T}(s,\tau)$ and $P(s,\tau) = P_{K,T}(s,\tau)$ denote the time $t$ value of a call and a put with delivery price $K$ and maturity date $T$, if $\tau = T - t$ is the time to maturity and the price of the underlying is $s,$ i.e. $S_t = s.$

1. Option prices are non negative since an exercise only takes place if it is in the interest of the holder. An option gives the right to exercise. The holder is not obligated to do so.
2. American and European options have the same value at maturity $T$ since in $T$ they give the same rights to the holder. At maturity $T$ the value of the option is equal to the intrinsic value:
   $$C_{K,T}(S_T,0) = \max(S_T-K,0) \; , \quad P_{K,T}(S_T,0) = \max(K-S_T,0) .$$

3. An American option must be traded at least at its intrinsic value since otherwise a riskless profit can be realized by buying and immediately exercising the option. This relation does not hold in general for European options. The reason is that a European option can be exercised only indirectly by means of a future contract. The thereby involved discounting rate can possibly lead to the option being worth less than its intrinsic value.
4. The value of two American options which have different time to maturities, $T_1 \le T_2,$ is monotonous in time to maturity:
   $$C_{K,T_1}(s,T_1-t) \leq C_{K,T_2}(s,T_2-t) \; , \quad P_{K,T_1}(s,T_1-t) \leq P_{K,T_2}(s,T_2-t) .$$
   This follows, for calls, say, using 2., 3. from the inequality which holds at time $t = T_1$ with $s=S_{T_{1}}$

   $$C_{K,T_2}(s,T_2-T_1) \geq = \max(s-K,0) = C_{K,T_1}(s,0)$$ (3.12)

   
   
   Due to the no-arbitrage assumption the inequality must hold for any point in time $t \leq T_1.$ For European options this result does not hold in general.
5. An American option is at least as valuable as the identically specified European option since the American option gives more rights to the holder.
6. The value of a call is a monotonously decreasing function of the delivery price since the right to buy is the more valuable the lower the agreed upon delivery price. Accordingly, the value of a put is a monotonously increasing function of the delivery price.
   $$C_{K_1,T}(s,\tau) \geq C_{K_2,T}(s,\tau) \; , \quad P_{K_1,T}(s,\tau) \leq P_{K_2,T}(s,\tau)$$
   for $K_1 \leq K_2$. This holds for American as well as for European options.

The value of European call and put options on the same underlying with the same time to maturity and delivery price are closely linked to each other without using a complicated mathematical model.

Theorem 3.3 (Put-Call Parity for European Options)  
For the value of a European call and put option which have the same maturity date $T,$ the same delivery price $K,$ the same underlying the following holds (where $r$ denotes the continuous interest rate):

1. If the underlying pays a dividend yield with a time $t$ discounted total value of $D_t$ during the time to maturity $\tau = T - t$ then it holds

   $$C(S_t,\tau) = P(S_t,\tau) + S_t - D_t - K e^{-r\tau}$$ (3.13)

   
   
   SFEPutCall.xpl
2. If the underlying involves continuous costs of carry at rate $b$ during the time to maturity $\tau = T - t$ then it holds

   $$C(S_t,\tau) = P(S_t,\tau) + S_t e^{(b-r)\tau} - K e^{-r\tau}$$ (3.14)

   
   

Proof:
For simplicity, we again assume the underlying to be a stock. We consider a portfolio $A$ consisting of one call which will be duplicated by a suitable portfolio $B$ containing a put among others.

1. In the case of discrete dividend yields we consider at time $t$ the following portfolio $B:$

1. Buy the put.
2. Sell a zero bond with face value $K$ maturing $T.$
3. Buy one stock.
4. Sell a zero bond at the current price $D_t$.

The stock in portfolio $B$ pays dividends whose value discounted to time $t$ is $D_t.$ At time $T$ these dividend yields are used to pay back the zero bond of position d). Hence this position has a value of zero at time $T.$ Table 2.1 shows the value of portfolio $B$ at time $T$ where we distinguished the situations where the put is exercised ( $K \geq S_T$) and where it is not exercised.

Table 2.1: Value of portfolio $B$ at time $T$ (Theorem 2.3).

	Value at timet $T$
Position	$K<S_T$	$K \ge S_T$
a)	0	$K-S_T$
b)	$-K$	$-K$
c)	$S_T$	$S_T$
d)	0	0
Sum	$S_T-K$	0

At time $T$ portfolio $B$ has thus the same value $\max(S_T-K,0)$ as the call. To avoid arbitrage opportunities both portfolios $A$ and $B$ must have the same value at any time $t$ prior $T,$ that is it holds

$$C(S_t,\tau) = P(S_t,\tau) - K e^{-r\tau} + S_t - D_t$$ (3.15)

2. In the case of continuous dividends at rate $d$ and corresponding costs of carry $b=r-d$ we consider the same portfolio $B$ as in part 1. but this time without position d). Instead we buy $e^{-d\tau}$ stocks in position c) whose dividends are immediately reinvested in the same stock. If $d$ is negative, then the costs are financed by selling stocks. Thus, portfolio $B$ contains exactly one stock at time $T,$ and we conclude as in part 1. that the value of portfolio $B$ is at time $t$ equal to the value of the call. ${\Box}$

The proof of the put-call parity holds only for European options. For American options it may happen that the put or call are exercised prior maturity and that both portfolios are not hold until maturity.

The following result makes it possible to check whether prices of options on the same underlying are consistent. If the convexity formulated below is violated, then arbitrage opportunities arise as we will show in the example following the proof of the next theorem.

Theorem 3.4  
The price of a (American or European) Option is a convex function of the delivery price.

Proof:
It suffices to consider calls since the proof is analogous for puts. The put-call parity for European options is linear in the term which depends explicitly on $K.$ Hence, for European options it follows immediately that puts are convex in $K$ given that calls are convex in $K$.

For $0 \leq \lambda \leq 1$ and $K_1 < K_0$ we define $K_\lambda \stackrel{\mathrm{def}}{=}\lambda K_1 +(1-\lambda)K_0.$ We consider a portfolio $A$ which at time $t<T$ consists of one call with delivery price $K_{\lambda}$ and maturity date $T.$ At time $t$ we duplicate this portfolio by the following portfolio $B:$

1. A long position in $\lambda$ calls with delivery price $K_1$ maturing in $T$.
2. A long position in $(1 - \lambda)$ calls delivery price $K_0$ maturing in $T$.

By liquidating both portfolios at an arbitrary point of time $t', t \leq t' \leq T$ we can compute the difference in the values of portfolio $A$ and $B$ which is given in Table 2.2

Table 2.2: Difference in the values of portfolios $B$ and $A$ at time $t'$ (Theorem 2.4).

	Value at time $t'$
Position	$S_{t'} \le K_1$	$K_1 < S_{t'} \le K_\lambda$	$K_\lambda < S_{t'} \le K_0$	$K_0 < S_{t'}$
$B$ 1.	0	$\lambda (S_{t'} - K_1)$	$\lambda (S_{t'} - K_1)$	$\lambda (S_{t'} - K_1)$
$B$ 2.	0	0	0	$(1 - \lambda ) (S_{t'} - K_0)$
$- A$	0	0	$-(S_{t'} - K_\lambda)$	$-(S_{t'} - K_\lambda)$
Sum	0	$\lambda (S_{t'} - K_1)$	$(1 - \lambda ) (K_0 -S_{t'})$	0

Since $\lambda (S_{t'} - K_1) \ge 0$ und $(1 - \lambda )(K_0 -S_{t'}) ) \ge 0$ in the last row of Table 2.2 the difference in the values of portfolio $A$ and $B$ at time $t'$ and thus for any point of time $t<t'$ is greater than or equal to zero. Hence, denoting $\tau = T - t$ it holds

$$\lambda C_{K_1,T}(S_t,\tau) + (1 - \lambda )C_{K_0,T}(S_t,\tau) - C_{K_\lambda,T}(S_t,\tau) \ge 0$$ (3.16)

${\Box}$

Example 3.3  
We consider three European call options on the MD*TECH A.G. having all the same time to maturity and delivery prices $K_1 = 190,$ $K_\lambda = 200,$
$K_0 = 220$, i.e.  $\lambda = \frac{2}{3}.$ Table 2.3 shows the data of this example.

Table: Data of Example 2.3.

Delivery price	Option price
$K_1 = 190$	30.6 EUR
$K_\lambda = 200$	26.0 EUR
$K_0 = 220$	14.4 EUR

Due to the last theorem it must hold:

$$\textstyle\frac{2}{3} C_{K_1,T}(S_t,\tau) + \textstyle\frac{1}{3} C_{K_0,T}(S_t,\tau) \ge C_{K_\lambda,T}(S_t,\tau)$$ (3.17)

Since this condition is obviously violated an arbitrage opportunity exists, and with the following portfolio a riskless gain can be realized:

1. A long position in $\lambda = \frac{2}{3}$ calls with delivery price $K_1.$
2. A long position in $1- \lambda = \frac{1}{3}$ calls with delivery price $K_0.$
3. A short position in 1 call with delivery price $K_\lambda \stackrel{\mathrm{def}}{=}\frac{2}{3} K_1 +\frac{1}{3}K_0.$

By setting up this portfolio at the current time $t$ we realize an immediate profit of +0.80 EUR. The portfolio value at options' maturity $T$ is given by Table 2.4 from which we can extract that we realize further profits for stock prices $S_T$ between 190 and 220 of at most $$\frac{20}{3}$$ EUR.

Table 2.4: Portfolio value at time $T$ of Example 2.3.

	Value at time $T$
Position	$S_T \le 190$	$190 < S_T \le 200$	$200 < S_T \le 220$	$220 < S_T$
1.	0	$\frac{2}{3} (S_T - 190)$	$\frac{2}{3} (S_T - 190)$	$\frac{2}{3} (S_T - 190)$
2.	0	0	0	$\frac{1}{3} (S_T - 220)$
3.	0	0	$-(S_T - 200)$	$-(S_T - 200)$
Sum	0	$\frac{2}{3} (S_T - 190)$	$\frac{1}{3} ( 220 -S_T)$	0

We already said that option prices are monotonous functions of the delivery price. The following theorem for European options is more precise on this subject.

Theorem 3.5  
For two European calls (puts) with the same maturity date $T$ and delivery prices $K_1 \leq K_2$ it holds at time $t \leq T$:

$$0 \le C_{K_1,T}(S_t,\tau) - C_{K_2,T}(S_t,\tau) \le (K_2 - K_1)e^{-r\tau}$$ (3.18)

or

$$0 \le P_{K_2,T}(S_t,\tau) - P_{K_1,T}(S_t,\tau) \le (K_2 - K_1)e^{-r\tau}$$ (3.19)

with $\tau = T - t$ and $r$ denoting the time to maturity and the interest rate respectively. If call (put) option prices are differentiable as a function of the delivery price, then by taking the limit $K_2 - K_1 \rightarrow 0$ it follows

$$1 \le -e^{-r\tau} \le \frac{\partial C}{\partial K} \le 0 \quad 0 \le \frac{\partial P}{\partial K} \le e^{-r\tau} \le 1$$ (3.20)

Proof:
We proof the theorem for calls since for puts the reasoning is analogous. For this we consider a portfolio $A$ containing one call with delivery price $K_1$ which we compare to a duplicating portfolio $B.$ At time $t$ the latter portfolio consists of the following two positions:

1. A long position in one call with delivery price $K_2$.
2. A long position in one zero bond with face value $(K_2 - K_1)$ maturing in $T$.

The difference of the value of portfolios $B$ and $A$ at time $T$ is shown in Table 2.5.

Table: Difference in the values of portfolios $B$ and $A$ at time $T$ (Theorem 2.5).

	Value at time $T$
Position	$S_{T} \le K_1$	$K < S_{T} < K_2$	$K_2 \le S_{T}$
$B$ 1.	0	0	$S_{T} - K_2$
$B$ 2.	$K_2-K_1$	$K_2-K_1$	$K_2-K_1$
$- A$	0	$-(S_{T} - K_1)$	$-(S_{T} - K_1)$
Sum	$K_2-K_1$	$K_2-S_{T}$	0

At time $T$ portfolio $B$ is clearly as valuable as portfolio $A$ which given the no-arbitrage assumption must hold at time $t$ as well. We conclude:

$$C_{K_2,T}(S_t,\tau) + (K_2 - K_1)e^{-r\tau} \geq C_{K_1,T}(S_t,\tau) .$$

${\Box}$