CS计算机代考程序代写 algorithm Financial Engineering

Financial Engineering
IC302
5. Basic Numerical Methods for Valuing Options (Binomial Trees)
David Oakes

Autumn Term 2020/1

Derivative Securities and Numerical Methods
2

Derivatives Securities and Models
• Derivative securities enrich financial markets by providing alternative trading venues for many sources of risk.
• This makes it easier for investors to take and hedge exposure to prices and interest rates, which expands investment opportunities and improves risk management.
• In order to make effective use of derivatives, we must understand how their prices are related to these risks.
• For this purpose, we often use models, especially when dealing with derivatives with contingent payments.
3

Models with Analytical Solutions
• Some models have an analytical solution: an explicit equation we can use to calculate the derivative’s value.
• The Black-Scholes-Merton formula (or Black-Scholes formula) for a European call option is a famous example.
• Often, however, the model that best describes a derivative does not have an analytical solution.
• When this occurs, we must use numerical methods to implement the model and calculate the derivative’s value.
4

Numerical Methods
• A numerical method is a procedure or algorithm for computing an approximate solution to a problem for which an analytical solution does not exist.
• We can think of numerical methods as recipes for finding the right answer when there is no explicit formula.
• Numerical methods are widely used in many applications in science and engineering, including the quantitative analysis of financial markets.
5

Numerical Methods
• Binomial trees (also trinomial trees)
• Monte Carlo simulation
• Finite difference methods
• Each of these methods uses risk-neutral valuation principles, and each has its advantages and disadvantages.
• We explain binomial methods in this lecture and Monte Carlo simulation in lecture 8.
6

No-Arbitrage Values for Derivatives
7

A Simple Derivatives Problem
• Consider a risky stock that is trading at a price of $100.
• Suppose that one year from now the price of the stock will either have moved up to $120 or down to $80.
• We are asked to value a one-year European call option on the stock with strike price $100.
• What price should we charge?

European Call Option Payoffs
• A call option gives the option holder the right but not the obligation to buy an underlying asset on the option exercise date at a specified strike price or exercise price.
• A European option can only be exercised on its expiry date In our example, the expiry date is one year from now.
• If the stock price moves up to $120 one year from now, the option holder will exercise the option, paying the strike price of $100 to acquire a share that is worth $120.
• The payoff to the option will be $20.
9

European Call Option Payoffs
• If, instead, the stock price moves down to $80, she will not exercise the option, and her payoff will be $0.
• This is illustrated in the diagram below:
10

Delta Hedging and No-Arbitrage
• We can value the option using a simple hedging argument.
• We are the seller of the option. We hedge our short position in the option by going long Δ shares of the underlying stock (Δ is the Greek capital letter delta).
• We choose Δ such that the value of our portfolio one year from now will be the same whether the stock price goes up to $120 or down to $80, so that the portfolio is riskless.
• In the absence of arbitrage opportunities, riskless portfolios must earn the risk-free interest rate.
11

Calculating Delta
• A portfolio that is short one call option and long Δ shares of the underlying stock will be worth:
120Δ − 20
one year from now if the stock price goes up to $120 and:
80Δ
if the stock price goes down to $80.
• We want to choose Δ such that these values are equal:
120Δ − 20 = 80Δ → Δ=0.5
12

Delta and the Riskless Portfolio
• We should go long 0.5 shares for each option that we sell.
• Substituting for Δ in the expressions for the portfolio value when the price goes up to $120 or down to $80 gives:
120Δ−20=120 0.5 −20=40 80Δ=80 0.5 =40
• A riskless portfolio must earn the risk-free rate. Suppose that the (continuously compounded) rate is 5% per year.
13

No-Arbitrage Option Price
• If there are no arbitrage opportunities, the value of the portfolio today must be:
𝑒!”#40 = 𝑒!$.$& ‘ 40 = 38.05
• The portfolio is short one option and long Δ = 0.5 shares of stock. The price of the stock today is $100. Denoting the value of the option as 𝑓, we must have that:
100Δ−𝑓=100 0.5 −𝑓=50−𝑓=38.05 → 𝑓=11.95
• The no-arbitrage price of the option is $11.95.
14

One-Period Binomial Model

One-Period Binomial Model
• We can generalize the argument of the last section to create a simple one-period model for valuing derivatives.
• Denote the current stock price by 𝑆0 and the current option value by 𝑓. The option expires after time 𝑇.
• At date 𝑇, the stock price will either increase to 𝑆0𝑢 or decrease to 𝑆0𝑑, where 𝑢 > 1 and 𝑑 < 1. • If the stock price moves up to 𝑆0𝑢 , the payoff to the option will be 𝑓1. If the stock price moves down to 𝑆0𝑑, the payoff to the option will be 𝑓2. One-Period Binomial Model • Because there are two possible outcomes for the stock price and the option at the end of the period, we call this kind of model a binomial model. 17 Delta and the Riskless Portfolio • We hedge our short position in the option by going long Δ shares of stock, and we choose delta so that the portfolio is riskless. Its value if the stock price goes up is: 𝑆$𝑢Δ − 𝑓( and its value if the stock price goes down is: 𝑆$𝑑Δ − 𝑓) • We can set the two expressions equal to find the value of Δ that makes the portfolio riskless: 𝑆$𝑢Δ − 𝑓( = 𝑆$𝑑Δ − 𝑓) → Δ= 𝑓(−𝑓) 𝑆$𝑢 − 𝑆$𝑑 18 Intrepreting Delta • Notice that Δ (delta) is the expected change in the option price for a small change in the stock price. • We can hedge a short position in a call option by going long Δ shares of stock. Similarly, we can hedge a short position in a put option by going short Δ shares of stock. • Delta is a key risk measure that plays an important role in pricing and hedging options. 19 No-Arbitrage Option Price • Denoting the (continuously compounded) risk-free interest rate by 𝑟, the present value of this portfolio is: 𝑒!"# 𝑆$𝑢Δ−𝑓( • The cost of setting up the portfolio today is: 𝑆$Δ − 𝑓 • If there are to be no arbitrage opportunities, these two must be equal: 𝑒!"# 𝑆$𝑢Δ−𝑓( =𝑆$Δ−𝑓 → 𝑓=𝑆$Δ1−𝑢𝑒!"# +𝑓(𝑒!"# 20 No-Arbitrage Option Price • Substituting for the value of delta that we found earlier, the no-arbitrage value of the option is: 𝑓=𝑆 𝑓(−𝑓) 1−𝑢𝑒!"# +𝑓𝑒!"#=𝑓( 1−𝑑𝑒!"# +𝑓) 𝑢𝑒!"#−1 $ 𝑆$𝑢−𝑆$𝑑 ( 𝑢−𝑑 • We could use this to calculate the option value directly, but it is more convenient to define a new variable 𝑞: 𝑞 = 𝑒"# − 𝑑 𝑢−𝑑 • Using this new variable, the option value is: 𝑓 = 𝑒!"# 𝑞𝑓( + 1 − 𝑞 𝑓) 21 Applying the Model • Consider again the example from the last section. • The initial stock price 𝑆0 = $100. Since the stock price one year from now may be either $120 or $80, we have 𝑢=1.2and𝑑=0.8. Therisk-freerateis𝑟=0.05and the time to expiry is 𝑇 = 1. The payoff to the option will be $20 if the stock price goes up and $0 if it goes down. • Using these values gives: 𝑞=𝑒"# −𝑑=𝑒$.$& ' −0.8=0.6282 𝑢 − 𝑑 1.2 − 0.8 22 No-Arbitrage Option Price • The no-arbitrage value of the option is: 𝑓=𝑒!"# 𝑞𝑓(+ 1−𝑞𝑓) =𝑒!$.$&' 0.628220 +0.37180 =11.95 just as we found in the previous section. 23 Risk-Neutral Valuation Derivatives and Hedging Costs • The prices of derivative securities are to a very considerable extent determined by hedging costs. • We see this in the one-period binomial model introduced in the previous section, where the no-arbitrage price of the option is determined entirely by a hedging argument. • If all market participants have the same hedging opportunities, they will agree on the price at which any derivative should trade, regardless of their preferences. Risk Neutrality • We can therefore calculate derivatives prices by imagining that investors have any arbitrary set of risk preferences. • For example, we could imagine that all investors are risk- neutral (i.e. that they are indifferent to risk). • If investors were risk-neutral, the expected return on every asset would be equal to the risk-free rate and all future cash flows would be discounted at this rate. • This greatly simplifies pricing because we don’t have to adjust for risk when discounting future cash flows. 26 Risk-Neutral Valuation • We can therefore calculate derivatives prices by assuming that all investors are risk-neutral and discounting expected payoffs under this assumption at the risk-free rate. • This important principle is called risk-neutral valuation. • It is central to how we value derivatives in practice, and we apply it repeatedly in this course. 27 Risk-Neutral Valuation • We are not saying that investors are actually risk-neutral. • If derivatives prices are determined by hedging costs, however, risk preferences don’t matter. • We therefore choose to calculate prices under risk preferences that make the calculation easier. • Risk-neutral valuation is a computational ‘trick’ we can use when derivatives prices are determined by hedging costs. 28 Risk-Neutral Probabilities • In a world of risk-neutral investors, the expected future cash flows from options and other assets would not be the same as in the real world, since the expected return on every asset would be equal to the risk-free rate. • In order to apply risk-neutral valuation, we need to adjust the probabilities associated with possible future payoffs to what they would be in a risk-neutral world. • We call these adjusted probabilities risk-neutral probabilities. 29 Risk-Neutral Probabilities • It turns out that this is exactly what we did in the one- period binomial model in the previous lesson. • The new variable 𝑞 that we introduced in order to simplify the valuation formula for the option is just the risk-neutral probability of an increase in the stock price at time 𝑇. • Toseethis,notethat,solongas𝑑<𝑒:; <𝑢,wecan interpret 𝑞 and 1 − 𝑞 as probabilities. 30 Risk-Neutral Probabilities • To prove that they are risk-neutral probabilities, we can use them to calculate the expected stock price at time 𝑇: 𝐸 𝑆# =𝑞𝑆$𝑢+ 1−𝑞 𝑆$𝑑=𝑞𝑆$ 𝑢−𝑑 +𝑆$𝑑 • Substituting in the definition of 𝑞 from earlier gives: 𝐸𝑆# =𝑞𝑆$ 𝑢−𝑑 +𝑆$𝑑=𝑒"#−𝑑𝑆$ 𝑢−𝑑 +𝑆$𝑑=𝑆$𝑒"# 𝑢−𝑑 • Under the probabilities 𝑞 and 1 − 𝑞, the expected return on the stock is the risk-free rate. • This is exactly as we would expect in a risk-neutral world. 31 Risk-Neutral Valuation Formula • The same is true of the option. As we have seen, its price is equal to its expected payoff under the risk-neutral probabilities, discounted at the risk-free rate: 𝑓 = 𝑒!"# 𝑞𝑓( + 1 − 𝑞 𝑓) • It follows that its expected return in the risk-neutral world is the risk-free interest rate. • We can use this risk-neutral valuation formula to calculate the price of the option directly. 32 Risk-Neutral Option Price • For the parameter values given earlier, we can calculate that the risk-neutral probabilities are: 𝑞=𝑒"# −𝑑=𝑒$.$& ' −0.8=0.6282 1−𝑞=0.3718 𝑢 − 𝑑 1.2 − 0.8 • Using these in the risk-neutral valuation formula gives: 𝑓=𝑒!"# 𝑞𝑓(+ 1−𝑞𝑓) =𝑒!$.$&' 0.628220 +0.37180 =11.95 • The risk-neutral option value is $11.95. 33 Multi-Period Binomial Model Extending the Binomial Model • We can extend the logic of the one-period binomial model to allow for multiple changes in the stock price over time. • As before, we denote the initial stock price by 𝑆0. • During each time step of length Δ𝑡 years, the price either increases by a factor 𝑢 or decreases by a factor 𝑑. • Now, however, these factors may apply repeatedly over multiple time steps. Extending the Binomial Model • For example, if after two time steps the price increased at each step, it will be 𝑆0𝑢=, but if it increased at the first time step and decreased at the second, it will be 𝑆0𝑢𝑑. • The result is a binomial ‘tree’ for the stock price, which branches out at each time step. • We denote the option price at each ‘node’ of the tree in a similar way, so that after two time steps it is 𝑓11 if the stock price increased at each time step, 𝑓12 if the stock price increased at the first step and decreased at the second time step, and so on. 36 Two-Step Binomial Tree • A binomial tree with two time steps is shown below: 37 Recombining Trees • Notice that, because 𝑢𝑑 = 𝑑𝑢, the tree is recombining: an upward movement followed by a downward movement has the same effect on the stock price as a downward movement followed by an upward movement. • For a recombining tree, the number of nodes at each time step increases linearly with the number of time steps. • This limits the amount of computation at each time step. • In a non-recombining tree the number of nodes increases geometrically and computation soon becomes infeasible. 38 Multi-Period Binomial Model • In this multi-period binomial model, we calculate the price of an option by starting at the end of the option’s life, working backwards in time through the tree. • We apply the risk-neutral valuation principle at each step. • Suppose, for example, that each time step Δ𝑡 is one year and that we wish to value a two-year call option. • The initial stock price is 𝑆0 = 100, the multiplicative factors for movements in the stock price are 𝑢 = 1.2 and 𝑑 = 0.8, and the risk-free interest rate 𝑟 is 5% per year. 39 Expiry Date Option Payoffs • If the strike price of the option is 𝐾 = $100, then the payoffs to the option for the three possible stock prices at the option expiry date will be as shown below: 40 Stepping Back Through the Tree • Now imagine that it is one year earlier, and we are at the upper of the two possible nodes of the tree at that point in time, where the stock price is $120. • One year later, the payoff to the option will be either $ 44 (if the stock price moves up) or $0 (if the stock price moves down). We can use the one-period risk-neutral valuation formula to find the current option price 𝑓1 : 𝑓( = 𝑒!"*+ 𝑞𝑓(( + 1−𝑞 𝑓() = 𝑒!$.$& ' 0.6282 44 +0.3718 0 = 26.29 41 Stepping Back Through the Tree • Notice that the risk-neutral probabilities are the same as in the one-period model, because the time step is one year, and the other parameter values are unchanged: 𝑞=𝑒"*+ −𝑑=𝑒$.$& ' −0.8=0.6282 1−𝑞=0.3718 𝑢 − 𝑑 1.2 − 0.8 • Similar calculations show that that value 𝑓2 of the option at the lower node of the tree at this same time step must be zero, since its payoff will be zero in each of the two possible outcomes for the stock price one year later. 42 Stepping Back Through the Tree • At this point in the process, the tree for the stock price and the option price is as shown below: 43 No-Arbitrage Option Price • Taking one more step back through the tree, we apply the risk-neutral valuation formula again to calculate the initial value 𝑓 of the option: 𝑓 = 𝑒!"*+ 𝑞𝑓( + 1−𝑞 𝑓) = 𝑒!$.$& ' 0.6282 26.29 +0.3718 0 = 15.71 • The no-arbitrage or risk-neutral price of the two-year call option is $15.71. • This process of pricing the option through risk-neutral valuation by stepping back in time through the tree is known as backwards induction. 44 Final Tree for Two-Period Model • The stock prices and option values for each node of the tree are shown below: 45 Implementing Binomial Models Implementing Binomial Trees • In practice, in order to make reliable estimates of derivatives prices we will need a tree with many possible terminal values for the underlying stock price. • This means dividing the time to the expiry date into many small steps, over each of which the price may change. • We also want the possible changes in the asset price over time to ‘look like’ those for the asset on which the derivative is a claim, in the sense that the volatility of the asset price implied by the tree converges to what we believe to be the actual volatility of the asset. Implementing Binomial Trees • We can do this by choosing suitable values for 𝑢 and 𝑑, since these control the volatility or dispersion of the stock price as we move forward in time through the tree. • Finally, in order to apply risk-neutral valuation, we must calculate the risk-neutral probabilities 𝑞 and 1 − 𝑞. • We do this in the same way as before. • As we know, under these probabilities the expected return on the stock is equal to the risk-free rate. 48 Cox-Ross-Rubinstein Model • There are many ways to construct a binomial model that meets these criteria. • One of the most popular was first introduced in a paper by Cox, Ross, and Rubinstein, and is often known as the CRR model. In this section, we explain how it works. 49 Matching Volatility with 𝑢 and 𝑑 • In the CRR model, we choose 𝑢 and 𝑑 as follows: 𝑢=𝑒, *- 𝑑=𝑒!, *- • This ensures that the variance of return on the stock over each time step of length Δ𝑡 is equal to 𝜎=Δ𝑡, where 𝜎 is the volatility of the stock (see Hull ch. 13). • This is consistent with how we typically model prices of risky assets when modelling derivatives and ensures that option prices calculated using the CRR model converge to those calculated using the Black-Scholes-Merton model in the limit, as the length of the time step Δ𝑡 goes to zero. 50 Risk-Neutral Probabilities • Once the tree has been constructed, we calculate the option payoffs at expiry and find the price through risk- neutral valuation and backward induction as before. • The risk-neutral probabilities 𝑞 and 1 − 𝑞 are calculated in the same way as in the previous section: 𝑞 = 𝑒"*+ − 𝑑 𝑢−𝑑 • Note, however, that the values of 𝑢 and 𝑑 in this formula are now calculated using the CRR formulae shown above, rather than chosen arbitrarily. 51 CRR Model Example • Suppose that we are asked to value a 1-year European call option on a risky stock for the following parameter values: Parameter Value 𝑆$ 100 𝐾 100 𝑟 0.06 𝑇 1 year 𝜎 0.20 (i.e. 20% per year) • 𝑆0 is the current stock price, 𝐾 is the exercise price, 𝑟 is the risk-free interest rate, and 𝑇 is the time to expiry. The only new variable is 𝜎, the volatility of the stock. • We will use three time steps in this simple example. 52 Calculating Key Variables • Denoting by 𝑛 the number of time steps, the length of each time step will be: Δ 𝑡 = 𝑇𝑛 = 13 The CRR values for 𝑢 and 𝑑 are: 𝑢=𝑒, *- =𝑒$.. '⁄0 =1.1224 𝑑=𝑒!, *- =𝑒!$.. '⁄0 =0.8909 The risk-neutral probabilities are: 𝑞=𝑒"*+ −𝑑=𝑒$.$1 '⁄0 −0.8909=0.5584 𝑢 − 𝑑 1.1224 − 0.8909 • • 1−𝑞=0.4416 53 Three-Step CRR Binomial Tree • The resulting three-step tree for the stock price is: 54 Terminal Option Payoffs • To price the option, we begin at the end of its life, calculating the payoffs for each terminal node of the tree. • This is illustrated in the tree on the next slide. 55 Terminal Option Payoffs 56 Backwards Induction • Suppose now that we are one time step before expiry, and the current stock price is $125.98. This puts us at the highest node of the tree at that point in time. • One time step later, the stock price will be either $141.40 (if the price moves up) or 112.24 (if the price moves down). We can use the risk-neutral valuation formula to calculate the option value 𝑓11 at the current node: 𝑓(( = 𝑒!"*+ 𝑞𝑓((( + 1 − 𝑞 𝑓(() = 𝑒!$.$1 '⁄0 0.5584 41.40 + 0.4416 12.24 = 27.96 57 Backwards Induction • Similar calculations show that the option values at the other nodes in the tree at this time step are: 𝑓() = 6.70 𝑓)) = 0.0 • The option values at this time step are shown in the tree on the next slide. 58 Backwards Induction 59 Completing the CRR Tree • Now we move one more time step backward through the tree and use the same risk-neutral valuation procedure to calculate the option values 𝑓1 and 𝑓2. • We continue in this way until we reach the initial date at the root of the tree. • The stock prices and option values at each node of the tree are shown on the next slide. 60 No-Arbitrage Option Price • The resulting no-arbitrage price of the option is $11.55. 61 CRR Implementation in Practice • In practice, we will need more than three time steps in order to ensure reasonable accuracy. • As few as 30 time steps may be sufficient, but we typically choose to use many more (e.g. 500 time steps). • For a European option, as the number of time steps is increased the price given by the CRR model converges to the Black-Scholes-Merton price. 62 American Options and Early Exercise American Options • An American option, unlike a European option, can be exercised at any time during its life. • Valuing American options is more difficult than valuing European options, because the investor must decide at each point in time whether it is better to exercise the American option immediately or to continue to hold it. • In this section, we show that the multi-period binomial model can be easily adapted to solve the ‘early exercise’ problem posed by American options. CRR Valuation of American Put • We will use the CRR model to value an American put option, using the same parameter values as previously: Parameter Value 𝑆$ 100 𝐾 100 𝑟 0.06 𝑇 1 year 𝜎 0.20 (i.e. 20% per year) • 𝑆0 is the current stock price, 𝐾 is the exercise price, 𝑟 is the risk-free interest rate, 𝑇 is the time to expiry, and 𝜎 is the volatility of the stock price. • We once again use a CRR model with three time steps. 65 Calculating Key Variables • Since the parameter values and the number of time steps are the same as in the earlier call option example, we will havethesamevaluesforΔ𝑡,𝑢,𝑑,𝑞,and1− 𝑞,sowedo not repeat the calculation of those values here. • The results are shown in the following table: Parameter Value Δ𝑡 1⁄3 𝑢 1.1224 𝑑 0.8909 𝑞 0.5584 1−𝑞 0.4416 66 Terminal Option Payoffs • The resulting stock price tree looks exactly like the tree in the call option example. • Once again, we calculate the option’s value by starting at the end of its life and working backwards in time through the tree, applying the risk-neutral valuation principle at each time step. • This time, however, the terminal payoffs will be those of a put option, as shown in the tree on the next slide. 67 Terminal Option Payoffs 68 Backwards Induction • Suppose now that it is one time step before expiry, and that the stock price is $79.38. This puts us at the lowest node in the tree at that point in time. • One time step later, the stock price will be either $89.09 (if the price moves up) or $70.72 (if the price moves down). Applying the risk-neutral valuation formula would give a value 𝑓22 for the option at this node of: 𝑓)) = 𝑒!"*+ 𝑞𝑓()) + 1 − 𝑞 𝑓))) = 𝑒!$.$1 '⁄0 0.5584 10.91 + 0.4416 29.28 = 18.65 69 Checking Early Exercise Condition • But this is an American option, so we must compare this with the amount that we would receive if we were to exercise the option immediately. • The strike price is $100, and the current stock price is $79.38, so the value from exercising the put immediately would be: 𝑚𝑎𝑥 𝐾−𝑆,0 = 100−79.38,0 =20.62 • This is more than the value we found using the risk- neutral valuation formula. 70 Checking Early Exercise Condition • It would be optimal for the option holder to exercise now, rather than waiting one more time step until the option expires. • We should therefore use the value from exercising the option, rather than the value from the risk-neutral valuation formula, at this node in the tree. • We use the same procedure to calculate the value of the option at each node of the tree. 71 Continuation and Exercise Values • First, we calculate the value using the risk-neutral valuation formula. Since this is the value of continuing to hold the option, we call it the continuation value. • We compare this with the value that could be obtained by exercising the option immediately, which we call the exercise value. • The option value is the higher of the exercise value and the continuation value: 𝑂𝑝𝑡𝑖𝑜𝑛 𝑉𝑎𝑙𝑢𝑒 = 𝑚𝑎𝑥 𝐸𝑥𝑒𝑟𝑐𝑖𝑠𝑒 𝑉𝑎𝑙𝑢𝑒, 𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑎𝑡𝑖𝑜𝑛 𝑉𝑎𝑙𝑢𝑒 72 American Option Valuation • At the current node of the tree, one time step before expiry and with the stock price equal to $79.38, this gives: 𝑓) ) = 𝑚 𝑎 𝑥 𝑚 𝑎 𝑥 𝐾 − 𝑆 $ 𝑑 . , 0 , 𝑒 ! " * + 𝑞 𝑓( ) ) + 1 − 𝑞 𝑓) ) ) = 𝑚𝑎𝑥 20.62, 18.65 = 20.62 • This is shown in the tree on the next slide. 73 American Option Valuation 74 Completing the Tree • We can continue to work backward through the tree, calculating the risk-neutral value and checking the early exercise condition at each node, until we arrive at the initial date at the root of the tree. • The stock prices and put option values at each node of the tree are shown on the next slide. 75 No-Arbitrage American Put Price • The no-arbitrage price of the American put is $6.10. 76 American Options in Practice • In practice, of course, we will need a tree with many more than three time steps to ensure reasonable accuracy. • Binomial trees are usually the most effective method for solving the early exercise problem for American options. • Monte Carlo simulation, which we discuss in lecture 8, is not well suited to valuing American-style options. • It can, however, be adapted to accommodate early exercise if needed. We show how this is done in lecture 8. 77 References Main reading: • Hull, ch. 13 and 21 78 79