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https://github.com/kinto-b/binomialpricing

Option pricing in the binomial model
https://github.com/kinto-b/binomialpricing

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Option pricing in the binomial model

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README 

        
Option pricing in the binomial model

================



![](https://img.shields.io/badge/lifecycle-experimental-orange.svg)



## Introduction



This package makes available a number of functions for pricing options

in the context of the binomial model, developed by Cox, Ross, and

Rubinstein (Cox, Ross, and Rubinstein 1979) and described in more detail

below. This model is attractive from a pedagogical angle because it

allows an instructor to exhibit the core ideas behind arbitrage pricing

in an intuitive environment. Moreover, it yields a pricing algorithm

that is simple to implement, and, in the limit, yields option values

which converge to those given by the more sophisticated Black-Scholes

model.



Included in this package are options for pricing common options. For

example, here is a vanilla European call:



``` r

# devtools::install_github("kinto-b/binomialpricing") # Uncomment to install

library(binomialpricing)



res <- crr_optionvalue(

  S0 = 62,

  u = 1.05943,

  d = 1/1.05943,

  r = 0.1/12, # 10% p.a compounded monthly

  expiry = 5,

  implicit_value = implicit_value(Sk=60, contract = "call"),

  american = FALSE

)

crr_plot(res)

```



![](README_files/figure-commonmark/unnamed-chunk-1-1.png)



You can use the `implicit_value()` function to pass through a range of

standard payoff functions, like a lookback payoff



``` r

res <- crr_optionvalue(

  S0 = 62,

  u = 1.05943,

  d = 1/1.05943,

  r = 0.1/12, # 10% p.a compounded monthly

  expiry = 4,

  implicit_value = implicit_value(Sk=60, contract = "call", exotic = "lookback_max"),

  american = FALSE

)

crr_plot(res)

```



![](README_files/figure-commonmark/unnamed-chunk-2-1.png)



Or a barrier payoff,



``` r

res <- crr_optionvalue(

  S0 = 62,

  u = 1.05943,

  d = 1/1.05943,

  r = 0.1/12, # 10% p.a compounded monthly

  expiry = 4,

  # ui denotes an up-and-in barrier

  implicit_value = implicit_value(Sk=60, contract = "call", barrier = "ui"), 

  american = FALSE

)

crr_plot(res)

```



![](README_files/figure-commonmark/unnamed-chunk-3-1.png)



But you can also roll your own payoff function to pass through. It just

needs to take in a matrix of prices, in which each column corresponds to

a time step and each row to a possible path through the lattice,



``` r

strangle_payoff <- function(prices) {

  pr <- prices[, ncol(prices)] # Latest price stored in last column

  

  vc <- pr - 65

  vp <- 55-pr

  v <- ifelse(vc X_0$ and they can settle up at $T$ like

I did.



It would seem, therefore, that the only rational price for $X$ should be

its manufacturing price. And indeed this is the case if the model

$\mathcal{M}$ is arbitrage-free.



**Definition** A market $\mathcal{M}$ is *arbitrage-free* if and only if

$\nexists \phi \in \Phi$ such that $V_0(\phi) = 0$ and $V_T(\phi) \ge 0$

a.s. and $\mathbb{P}(V_T(\phi) > 0) > 0$.



## The binomial model



In the binomial model, the price of the stock either goes up by a factor

$u$ or down by a factor $d$ at each time step,

i.e. $S_{k+1} = Z_{k+1}S_k$ where

$p = \mathbb{P}(Z_k = u) = 1-\mathbb{P}(Z_k = d)$, and bonds attract

simple interest so that $B_k = (1+r)^k = \hat r^k$. (Notice that in this

model the log-returns, $\rho_k = \ln S_k/S_{k-1}$ take the form of an

arithmetic random walk.)



### European options



#### One-step



To begin with, consider the special case in which $T=1$. Suppose that we

have an arbitrary *European* claim, $X$, paying $g(S_T)$ at expiration

so that $X_T = g(S_T)$. We know that the fair price for this claim will

be the manufacturing price, which is the cost of setting up a

replicating portfolio $\phi$. Recall that by definition

$V_T(\phi) = X_T$ a.s. In the context of our simple model this means

precisely that



$$

\begin{cases}

  \sigma_0 (u S_0) + \beta_0 \hat r = g(uS_0) \\

  \sigma_0 (d S_0) + \beta_0 \hat r = g(dS_0)

\end{cases}

$$



Solving this system of equations gives,



$$

\begin{cases}

  \sigma_0 = \frac{g(uS_0) - g(dS_0)}{(u-d)S_0} \\

  \beta_0 = \frac{ug(dS_0) - dg(uS_0)}{\hat r(u-d)}

\end{cases}

$$



If we define $\tilde p = \frac{\hat r - d}{u-d}$ and construct the

measure $\mathbb{\tilde P}$ s.t. $\mathbb{\tilde P}(Z_1 = u)= \tilde p$,

then substituting the solution above back into the value process we get

the following equation,



$$

\pi_0(X) = V_0(\phi) = \mathbb{\tilde E} (X_T / \hat r)  

$$



(Note that $\tilde p \in (0, 1)$ iff $d < \hat r < u$. We return to this

below.)



#### Multi-step



Now suppose that $T > 1$ and assume WLOG that our European claim has

payoff at maturity $g(S_1, ..., S_T)$. Let us consider what the price of

claim ought to be at time $T-1$. Given that at this point we know the

stock prices up to $T-1$, we are back in the setting of the one-step

model. The payoff is either $g(S_1, ..., uS_{T-1})$ or

$g(S_1, ..., dS_{T-1})$. Matching this to our portfolio value will give

us two equations in two unknowns, etc. etc. We know what the conclusion

will be, namely



$$

X_{T-1} = \mathbb{\tilde E} (X_T / \hat r | \mathcal{F}_{T-1})

$$



Moreover, at $T-2$ we may also consider ourselves in the setting of the

one-step model, but instead of looking for a portfolio $\phi$ where

$V_T(\phi) = X_T$ we look for a portfolio where

$V_{T-1}(\phi) = X_{T-1}$. Once again, we know the fair price:



$$

X_{T-2} = \mathbb{\tilde E} (X_{T-1} / \hat r | \mathcal{F}_{T-2})

  = \mathbb{\tilde E} (\mathbb{\tilde E} (X_T / \hat r | \mathcal{F}_{T-1}) / \hat r | \mathcal{F}_{T-2})

  = \mathbb{\tilde E} (X_T/ \hat r^{2} | \mathcal{F}_{T-2})

$$



Ultimately, we arrive at the result



$$

X_0 = \mathbb{\tilde E} (X_T/ \hat r^{T} | \mathcal{F}_{0}) = \mathbb{\tilde E} (X_T/ \hat r^{T})

$$



(We emphasise that this procedure allows us to replicate *any* arbitrary

European claim. We often express this by saying that any European claim

is *attainable* in the binomial model. Models with this property are

deemed *complete*.)



Or, more generally,



**Theorem 1** The value of a European option at time $T-m$ is given by

$X_{T-m} = \mathbb{\tilde E} (X_T / \hat r^{m} | \mathcal{F}_{T-m})$.



In this manner, we can compute the prices, $\pi_0(X), ..., \pi_{T}(X)$,

as well as the replicating process, $\phi_0, ..., \phi_{T}$, along the

entire lattice.



(Actually for European calls/puts we do not need to use the recursive

procedure since a closed for solution exists, the so-called

‘Cox-Ross-Rubinstein valuation formula’.)



In closing, we observe that the value of a European call is exactly

equal to the value of a European put.



**Theorem 2 (Put/call parity)** Given European call $C$ and put $P$ with

strike price $K$, at any time $T-k$,

$C_{T-k} - P_{T-k} = S_{T-k} - K \hat r^{-k}$.



*Proof* Notice that

$\mathbb{\tilde E}(S_T | F_{T-1}) = \tilde p uS_{T-1} + (1-\tilde p)dS_{T-1} = \hat r S_{T-1}$.

It is simple to extend this result to show

$\mathbb{\tilde E}(S_T | F_{T-k}) = \hat r^k S_{T-k}$. Consequently,



$$

C_{T-k} - P_{T-k} 

  = \hat r^{-k} \mathbb{\tilde E}(C_T - P_T | F_{T-k} ) 

  = \hat r^{-k} \mathbb{\tilde E}(S_T - K | F_{T-k} )

  = S_{T-k} - K \hat r^{-k}

$$



### American options



On the face of it, American options seem to be somewhat more complicated

than their European cousins. And, indeed, American puts *are* more

challenging to price than European puts. American calls, on the other

hand, are equivalent to European calls as the following theorem states,



**Theorem 3** Given European call $C$ and American call $C^a$ with

strike price $K$, at any time $T-k$, $C_k = C^a_k$.



*Proof* It is obvious that $C^a_k \ge C_k$. We need only consider the

possibility that $C^a_k > C_k$. Suppose this is the case.



Since $g_c(x) = (x - K)^+$ is convex, by Jensen’s inequality we have



Hence, it is clear that if interest rates are non-negative, i.e. if

$\hat r \ge 1$, it follows that $C_{T-k} \ge ( S_{T-k} - K)^+$.



Now suppose that at time $k$ I sell the American call, buy the European

call, and park the difference $\delta = C^a_k - C_k$ in bonds. If the

buyer of the American call holds to maturity, I carry off

$\delta \hat r^{T-k}$ profit since what I make off the European call

balances what I lose on the American one; if she exercises early, I rake

in even more since I can sell my European call for more than the

$(S_{k+t} - K)^+$ I owe her. In other words, if $C^a_k > C_k$ there is

an arbitrage opportunity.



Notice that for an American put a similar set of inequalities can be

formulated,



$$

P_{T-k} 

  = \mathbb{\tilde E}(\hat r^{-k} g_p(S_T) | F_{T-k}) 

  \ge \hat r^{-k} g_p(\mathbb{\tilde E}( S_T | F_{T-k}))

  = \hat r^{-k} (K-\hat r^{k} S_{T-k})^+

  = (\hat r^{-k} K - S_{T-k})^+

$$



However, when $\hat r \ge 1$, it does not follow from this that

$P_{T-k} \ge (K- S_{T-k})^+$. (Though when $\hat r < 1$ the argument

proceeds as above). The upshot is that the price of an American put can

diverge from the price of a European put.



Conceptually, an American option in discrete time is just like a

contract which ties together $T$ European options, with expiry times

$1, 2, ..., T$, under the restriction that only one of them may be

exercised. Consequently, it’s price at any time should be the same as

the highest priced European option. In other words,



**Theorem 4** Let

$\mathcal{T}_m = \{\tau: \tau \text{ is a ST and } m \le \tau \le T\}$

be the class of all stopping times bounded by $m$ and $T$. The value of

an American put at time $m$ is given by



$$

X_{m} = \max_{\tau \in \mathcal{T}_m} \mathbb{\tilde E} (g_p(S_\tau) / \hat r^{\tau - m} | \mathcal{F}_{m})

$$



Now, it is obvious that

$X_m \ge \mathbb{\tilde E}(X_{m+1}|\mathcal{F}_{m})$. For the

right-hand-side is exactly the price the option *would* have *if* we

were unable to exercise at time $m$, which cannot but be lower than the

price of an option which offers the additional freedom of exercising at

time $m$. More formally, we have the result



**Theorem 5** The value of an American put is a supermartingale



*Proof* Let $\tau_m^*$ be the ST that realises the maximum at time $m$.

Then



$$

\mathbb{\tilde E}(X_{m+1}|\mathcal{F}_{m})

  \overset{(*)}{=} \mathbb{\tilde E}(g_p(S_{\tau_{m+1}^*}) / \hat r^{\tau_{m+1}^* - m - 1} |\mathcal{F}_{m})

  \overset{(**)}{\le} \mathbb{\tilde E}(g_p(S_{\tau_{m}^*}) / \hat r^{\tau_{m}^* - m} |\mathcal{F}_{m})

  = X_m

$$



where (\*) is due to the tower property and (\*\*) is because $\tau_m^*$

is the ST which maximises the value of

$\mathbb{\tilde E} (g_p(S_\tau) / \hat r^{\tau - m} | \mathcal{F}_{m})$.



Hence by the optional sampling theorem we end up with a squeeze,



$$

X_0 

  \ge \mathbb{\tilde E}(X_{\tau^*}) 

  \ge \mathbb{\tilde E}(g_p(S_{\tau^*}) / \hat r^{\tau^*})

  = X_0

$$



Since $X_m \ge g_p(S_{m}) / \hat r^{m}$ for all $m$, the upshot is that

$X_{\tau^*} = g_p(S_{\tau^*}) / \hat r^{\tau^*}$ a.s. This result allows

us to establish,



**Theorem 6** Given an American put, the optimal exercise time is given

by $\tau^* = \inf \{k: X_k = g_p(S_k) / \hat r^{k}\}$



*Proof* We know that if $\tau'$ is a maximising ST, then

$X_{\tau'} = g_p(S_{\tau'}) / \hat r^{\tau'}$. Hence it follows that

$\tau^* \le \tau'$. Thus, by optional sampling theorem,



$$

g_p(S_{\tau^*}) / \hat r^{\tau^*} 

  = X_{\tau^*} 

  \ge \mathbb{\tilde E}(X_{\tau'}|\mathcal{F}_{\tau^*})  

  = \mathbb{\tilde E}( g_p(S_{\tau'}) / \hat r^{\tau'} |\mathcal{F}_{\tau^*})

$$



Taking expectations of both sides then we get



$$

\mathbb{\tilde E}( g_p(S_{\tau^*}) / \hat r^{\tau^*})

 \ge \mathbb{\tilde E}( g_p(S_{\tau'}) / \hat r^{\tau'})

$$



Hence $\tau^*$ must be the maximising ST.



## References









Cox, John C., Stephen A. Ross, and Mark Rubinstein. 1979. “Option

Pricing: A Simplified Approach.” *Journal of Financial Economics* 7 (3):

229–63. .